# 1300 math formulas

```1300 Math Formulas
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fp_k= =VVQVNMTTQN=
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`&ccedil;&eacute;&oacute;&ecirc;&aacute;&Ouml;&Uuml;&iacute;=&laquo;=OMMQ=^Kp&icirc;&aacute;&ecirc;&aacute;&aring;K=^&auml;&auml;=o&aacute;&Ouml;&Uuml;&iacute;&euml;=o&Eacute;&euml;&Eacute;&ecirc;&icirc;&Eacute;&Ccedil;K=
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q&Uuml;&aacute;&euml;=&eacute;~&Ouml;&Eacute;=&aacute;&euml;=&aacute;&aring;&iacute;&Eacute;&aring;&iacute;&aacute;&ccedil;&aring;~&auml;&auml;&oacute;=&auml;&Eacute;&Ntilde;&iacute;=&Auml;&auml;~&aring;&acirc;K=
i
Preface
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q&Uuml;&aacute;&euml;= &Uuml;~&aring;&Ccedil;&Auml;&ccedil;&ccedil;&acirc;= &aacute;&euml;= ~= &Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&iacute;&Eacute;= &Ccedil;&Eacute;&euml;&acirc;&iacute;&ccedil;&eacute;= &ecirc;&Eacute;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;= &Ntilde;&ccedil;&ecirc;= &euml;&iacute;&igrave;&Ccedil;&Eacute;&aring;&iacute;&euml;= ~&aring;&Ccedil;= &Eacute;&aring;&Ouml;&aacute;&aring;&Eacute;&Eacute;&ecirc;&euml;K= f&iacute;= &Uuml;~&euml;= &Eacute;&icirc;&Eacute;&ecirc;&oacute;&iacute;&Uuml;&aacute;&aring;&Ouml;= &Ntilde;&ecirc;&ccedil;&atilde;= &Uuml;&aacute;&Ouml;&Uuml;= &euml;&Aring;&Uuml;&ccedil;&ccedil;&auml;=
&atilde;~&iacute;&Uuml;=&iacute;&ccedil;=&atilde;~&iacute;&Uuml;=&Ntilde;&ccedil;&ecirc;=~&Ccedil;&icirc;~&aring;&Aring;&Eacute;&Ccedil;=&igrave;&aring;&Ccedil;&Eacute;&ecirc;&Ouml;&ecirc;~&Ccedil;&igrave;~&iacute;&Eacute;&euml;=&aacute;&aring;=&Eacute;&aring;&Ouml;&aacute;&aring;&Eacute;&Eacute;&ecirc;&aacute;&aring;&Ouml;I=
&Eacute;&Aring;&ccedil;&aring;&ccedil;&atilde;&aacute;&Aring;&euml;I=&eacute;&Uuml;&oacute;&euml;&aacute;&Aring;~&auml;=&euml;&Aring;&aacute;&Eacute;&aring;&Aring;&Eacute;&euml;I=~&aring;&Ccedil;=&atilde;~&iacute;&Uuml;&Eacute;&atilde;~&iacute;&aacute;&Aring;&euml;K=q&Uuml;&Eacute;=&Eacute;&Auml;&ccedil;&ccedil;&acirc;=
&Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&euml;= &Uuml;&igrave;&aring;&Ccedil;&ecirc;&Eacute;&Ccedil;&euml;= &ccedil;&Ntilde;= &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;I= &iacute;~&Auml;&auml;&Eacute;&euml;I= ~&aring;&Ccedil;= &Ntilde;&aacute;&Ouml;&igrave;&ecirc;&Eacute;&euml;= &Ntilde;&ecirc;&ccedil;&atilde;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;= p&Eacute;&iacute;&euml;I= ^&auml;&Ouml;&Eacute;&Auml;&ecirc;~I= d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I= q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I= j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;=
~&aring;&Ccedil;= a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;I= s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;I= ^&aring;~&auml;&oacute;&iacute;&aacute;&Aring;= d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&oacute;I= `~&auml;&Aring;&igrave;&auml;&igrave;&euml;I=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;I=p&Eacute;&ecirc;&aacute;&Eacute;&euml;I=~&aring;&Ccedil;=m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=q&Uuml;&Eacute;&ccedil;&ecirc;&oacute;K==
q&Uuml;&Eacute;= &euml;&iacute;&ecirc;&igrave;&Aring;&iacute;&igrave;&ecirc;&Eacute;&Ccedil;= &iacute;~&Auml;&auml;&Eacute;= &ccedil;&Ntilde;= &Aring;&ccedil;&aring;&iacute;&Eacute;&aring;&iacute;&euml;I= &auml;&aacute;&aring;&acirc;&euml;I= ~&aring;&Ccedil;= &auml;~&oacute;&ccedil;&igrave;&iacute;= &atilde;~&acirc;&Eacute;=
&Ntilde;&aacute;&aring;&Ccedil;&aacute;&aring;&Ouml;= &iacute;&Uuml;&Eacute;= &ecirc;&Eacute;&auml;&Eacute;&icirc;~&aring;&iacute;= &aacute;&aring;&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring;= &egrave;&igrave;&aacute;&Aring;&acirc;= ~&aring;&Ccedil;= &eacute;~&aacute;&aring;&auml;&Eacute;&euml;&euml;I= &euml;&ccedil;= &aacute;&iacute;=
&Aring;~&aring;=&Auml;&Eacute;=&igrave;&euml;&Eacute;&Ccedil;=~&euml;=~&aring;=&Eacute;&icirc;&Eacute;&ecirc;&oacute;&Ccedil;~&oacute;=&ccedil;&aring;&auml;&aacute;&aring;&Eacute;=&ecirc;&Eacute;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;=&Ouml;&igrave;&aacute;&Ccedil;&Eacute;K===
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ii
Contents
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1 krj_bo=pbqp=
NKN= p&Eacute;&iacute;=f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml;==1=
NKO= p&Eacute;&iacute;&euml;=&ccedil;&Ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;==5=
NKP= _~&euml;&aacute;&Aring;=f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml;==7=
NKQ= `&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;==8=
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2 ^idb_o^=
OKN= c~&Aring;&iacute;&ccedil;&ecirc;&aacute;&aring;&Ouml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==12=
OKO= m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==13=
OKP= m&ccedil;&iuml;&Eacute;&ecirc;&euml;==14=
OKQ= o&ccedil;&ccedil;&iacute;&euml;==15=
OKR= i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&euml;==16=
OKS= b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==18=
OKT= f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&aacute;&Eacute;&euml;==19=
OKU= `&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;=f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==22=
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3 dbljbqov=
PKN= o&aacute;&Ouml;&Uuml;&iacute;=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;==24=
PKO= f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml;=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;==27=
PKP= b&egrave;&igrave;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;==28=
PKQ= p&Aring;~&auml;&Eacute;&aring;&Eacute;=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;==29=
PKR= p&egrave;&igrave;~&ecirc;&Eacute;==33=
PKS= o&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;==34=
PKT= m~&ecirc;~&auml;&auml;&Eacute;&auml;&ccedil;&Ouml;&ecirc;~&atilde;==35=
PKU= o&Uuml;&ccedil;&atilde;&Auml;&igrave;&euml;==36=
PKV= q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;==37=
PKNM= f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml;=q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;==38=
PKNN= f&euml;&ccedil;&euml;&Aring;&Eacute;&auml;&Eacute;&euml;=q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;=&iuml;&aacute;&iacute;&Uuml;=f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=`&aacute;&ecirc;&Aring;&auml;&Eacute;==40=
PKNO= q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;=&iuml;&aacute;&iacute;&Uuml;=f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=`&aacute;&ecirc;&Aring;&auml;&Eacute;==41=
iii
PKNP= h&aacute;&iacute;&Eacute;==42=
PKNQ= `&oacute;&Aring;&auml;&aacute;&Aring;=n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;==43=
PKNR= q~&aring;&Ouml;&Eacute;&aring;&iacute;&aacute;~&auml;=n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;==45=
PKNS= d&Eacute;&aring;&Eacute;&ecirc;~&auml;=n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;==46=
PKNT= o&Eacute;&Ouml;&igrave;&auml;~&ecirc;=e&Eacute;&ntilde;~&Ouml;&ccedil;&aring;==47=
PKNU= o&Eacute;&Ouml;&igrave;&auml;~&ecirc;=m&ccedil;&auml;&oacute;&Ouml;&ccedil;&aring;==48=
PKNV= `&aacute;&ecirc;&Aring;&auml;&Eacute;==50=
PKOM= p&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=~=`&aacute;&ecirc;&Aring;&auml;&Eacute;==53=
PKON= p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=`&aacute;&ecirc;&Aring;&auml;&Eacute;==54=
PKOO= `&igrave;&Auml;&Eacute;==55=
PKOP= o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;==56=
PKOQ= m&ecirc;&aacute;&euml;&atilde;==57=
PKOR= o&Eacute;&Ouml;&igrave;&auml;~&ecirc;=q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;==58=
PKOS= o&Eacute;&Ouml;&igrave;&auml;~&ecirc;=m&oacute;&ecirc;~&atilde;&aacute;&Ccedil;==59=
PKOT= c&ecirc;&igrave;&euml;&iacute;&igrave;&atilde;=&ccedil;&Ntilde;=~=o&Eacute;&Ouml;&igrave;&auml;~&ecirc;=m&oacute;&ecirc;~&atilde;&aacute;&Ccedil;==61=
PKOU= o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=o&aacute;&Ouml;&Uuml;&iacute;=t&Eacute;&Ccedil;&Ouml;&Eacute;==62=
PKOV= m&auml;~&iacute;&ccedil;&aring;&aacute;&Aring;=p&ccedil;&auml;&aacute;&Ccedil;&euml;==63=
PKPM= o&aacute;&Ouml;&Uuml;&iacute;=`&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;==66=
PKPN= o&aacute;&Ouml;&Uuml;&iacute;=`&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=&iuml;&aacute;&iacute;&Uuml;=~&aring;=l&Auml;&auml;&aacute;&egrave;&igrave;&Eacute;=m&auml;~&aring;&Eacute;=c~&Aring;&Eacute;==68=
PKPO= o&aacute;&Ouml;&Uuml;&iacute;=`&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc;=`&ccedil;&aring;&Eacute;==69=
PKPP= c&ecirc;&igrave;&euml;&iacute;&igrave;&atilde;=&ccedil;&Ntilde;=~=o&aacute;&Ouml;&Uuml;&iacute;=`&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc;=`&ccedil;&aring;&Eacute;==70=
PKPQ= p&eacute;&Uuml;&Eacute;&ecirc;&Eacute;==72=
PKPR= p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=`~&eacute;==72=
PKPS= p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=p&Eacute;&Aring;&iacute;&ccedil;&ecirc;==73=
PKPT= p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;==74=
PKPU= p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=t&Eacute;&Ccedil;&Ouml;&Eacute;==75=
PKPV= b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;==76=
PKQM= `&aacute;&ecirc;&Aring;&igrave;&auml;~&ecirc;=q&ccedil;&ecirc;&igrave;&euml;==78=
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4 qofdlkljbqov=
QKN= o~&Ccedil;&aacute;~&aring;=~&aring;&Ccedil;=a&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;=j&Eacute;~&euml;&igrave;&ecirc;&Eacute;&euml;=&ccedil;&Ntilde;=^&aring;&Ouml;&auml;&Eacute;&euml;==80=
QKO= a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;&euml;=~&aring;&Ccedil;=d&ecirc;~&eacute;&Uuml;&euml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==81=
QKP= p&aacute;&Ouml;&aring;&euml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==86=
QKQ= q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=`&ccedil;&atilde;&atilde;&ccedil;&aring;=^&aring;&Ouml;&auml;&Eacute;&euml;==87=
QKR= j&ccedil;&euml;&iacute;=f&atilde;&eacute;&ccedil;&ecirc;&iacute;~&aring;&iacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==88=
iv
QKS= o&Eacute;&Ccedil;&igrave;&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==89=
QKT= m&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;&aacute;&Aring;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==90=
QKU= o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==90=
QKV= ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;=~&aring;&Ccedil;=p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==91=
QKNM= a&ccedil;&igrave;&Auml;&auml;&Eacute;=^&aring;&Ouml;&auml;&Eacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==92=
QKNN= j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&Eacute;=^&aring;&Ouml;&auml;&Eacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==93=
QKNO= e~&auml;&Ntilde;=^&aring;&Ouml;&auml;&Eacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==94=
QKNP= e~&auml;&Ntilde;=^&aring;&Ouml;&auml;&Eacute;=q~&aring;&Ouml;&Eacute;&aring;&iacute;=f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&aacute;&Eacute;&euml;==94=
QKNQ= q&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;&aacute;&aring;&Ouml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=b&ntilde;&eacute;&ecirc;&Eacute;&euml;&euml;&aacute;&ccedil;&aring;&euml;=&iacute;&ccedil;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;==95=
QKNR= q&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;&aacute;&aring;&Ouml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=b&ntilde;&eacute;&ecirc;&Eacute;&euml;&euml;&aacute;&ccedil;&aring;&euml;=&iacute;&ccedil;=p&igrave;&atilde;==97===
QKNS= m&ccedil;&iuml;&Eacute;&ecirc;&euml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==98=
QKNT= d&ecirc;~&eacute;&Uuml;&euml;=&ccedil;&Ntilde;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==99=
QKNU= m&ecirc;&aacute;&aring;&Aring;&aacute;&eacute;~&auml;=s~&auml;&igrave;&Eacute;&euml;=&ccedil;&Ntilde;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==102=
QKNV= o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==103=
QKOM= q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==106=
QKON= o&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;&euml;=&iacute;&ccedil;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==106=
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5 j^qof`bp=^ka=abqbojfk^kqp=
RKN= a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;==107=
RKO= m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;==109=
RKP= j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;==110=
RKQ= l&eacute;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;=&iuml;&aacute;&iacute;&Uuml;=j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;==111=
RKR= p&oacute;&euml;&iacute;&Eacute;&atilde;&euml;=&ccedil;&Ntilde;=i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==114=
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6 sb`qlop=
SKN= s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;==118=
SKO= s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;==120=
SKP= s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring;==122=
SKQ= p&Aring;~&auml;&aacute;&aring;&Ouml;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;==122=
SKR= p&Aring;~&auml;~&ecirc;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;==123=
SKS= s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;==125=
SKT= q&ecirc;&aacute;&eacute;&auml;&Eacute;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=127=
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7 ^k^ivqf`=dbljbqov=
TKN= l&aring;&Eacute;=-a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=p&oacute;&euml;&iacute;&Eacute;&atilde;==130=
v
TKO= q&iuml;&ccedil;=-a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=p&oacute;&euml;&iacute;&Eacute;&atilde;==131=
TKP= p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=&aacute;&aring;=m&auml;~&aring;&Eacute;==139=
TKQ= `&aacute;&ecirc;&Aring;&auml;&Eacute;==149=
TKR= b&auml;&auml;&aacute;&eacute;&euml;&Eacute;==152=
TKS= e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~==154=
TKT= m~&ecirc;~&Auml;&ccedil;&auml;~==158=
TKU= q&Uuml;&ecirc;&Eacute;&Eacute;=-a&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=p&oacute;&euml;&iacute;&Eacute;&atilde;==161=
TKV= m&auml;~&aring;&Eacute;==165=
TKNM= p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=&aacute;&aring;=p&eacute;~&Aring;&Eacute;==175=
TKNN= n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml;==180=
TKNO= p&eacute;&Uuml;&Eacute;&ecirc;&Eacute;==189=
=
=
8 afccbobkqf^i=`^i`rirp=
UKN= c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=~&aring;&Ccedil;=q&Uuml;&Eacute;&aacute;&ecirc;=d&ecirc;~&eacute;&Uuml;&euml;==191=
UKO= i&aacute;&atilde;&aacute;&iacute;&euml;=&ccedil;&Ntilde;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==208=
UKP= a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=~&aring;&Ccedil;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;==209=
UKQ= q~&Auml;&auml;&Eacute;=&ccedil;&Ntilde;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;==211=
UKR= e&aacute;&Ouml;&Uuml;&Eacute;&ecirc;=l&ecirc;&Ccedil;&Eacute;&ecirc;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;==215=
UKS= ^&eacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;==217=
UKT= a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;==221=
UKU= j&igrave;&auml;&iacute;&aacute;&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==222=
UKV= a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=l&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;&euml;==225=
=
=
9 fkqbdo^i=`^i`rirp=
VKN= f&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==227=
VKO= f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=o~&iacute;&aacute;&ccedil;&aring;~&auml;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==228=
VKP= f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=f&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==231=
VKQ= f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=q&ecirc;&aacute;&Ouml;&ccedil;&aring;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==237=
VKR= f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==241=
VKS= f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml;=~&aring;&Ccedil;=i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==242=
VKT= o&Eacute;&Ccedil;&igrave;&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==243=
VKU= a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==247=
VKV= f&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==253=
VKNM= a&ccedil;&igrave;&Auml;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==257=
VKNN= q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==269=
vi
VKNO= i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==275=
VKNP= p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==285=
=
=
10 afccbobkqf^i=bnr^qflkp=
NMKN= c&aacute;&ecirc;&euml;&iacute;=l&ecirc;&Ccedil;&Eacute;&ecirc;=l&ecirc;&Ccedil;&aacute;&aring;~&ecirc;&oacute;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==295=
NMKO= p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=l&ecirc;&Ccedil;&Eacute;&ecirc;=l&ecirc;&Ccedil;&aacute;&aring;~&ecirc;&oacute;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==298=
NMKP= p&ccedil;&atilde;&Eacute;=m~&ecirc;&iacute;&aacute;~&auml;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==302=
=
=
11 pbofbp=
NNKN= ^&ecirc;&aacute;&iacute;&Uuml;&atilde;&Eacute;&iacute;&aacute;&Aring;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==304=
NNKO= d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==305=
NNKP= p&ccedil;&atilde;&Eacute;=c&aacute;&aring;&aacute;&iacute;&Eacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==305=
NNKQ= f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==307=
NNKR= m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==307=
NNKS= `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=q&Eacute;&euml;&iacute;&euml;==308=
NNKT= ^&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==310=
NNKU= m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==311=
NNKV= a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring;=~&aring;&Ccedil;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==312=
NNKNM= q~&oacute;&auml;&ccedil;&ecirc;=~&aring;&Ccedil;=j~&Aring;&auml;~&igrave;&ecirc;&aacute;&aring;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==313=
NNKNN= m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=b&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring;&euml;=&Ntilde;&ccedil;&ecirc;=p&ccedil;&atilde;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==314=
NNKNO= _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==316=
NNKNP= c&ccedil;&igrave;&ecirc;&aacute;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;==316=
=
=
12 mol_^_fifqv=
NOKN= m&Eacute;&ecirc;&atilde;&igrave;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;=~&aring;&Ccedil;=`&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml;==318=
NOKO= m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;==319=
=
=
=
=
=
vii
=
q&Uuml;&aacute;&euml;=&eacute;~&Ouml;&Eacute;=&aacute;&euml;=&aacute;&aring;&iacute;&Eacute;&aring;&iacute;&aacute;&ccedil;&aring;~&auml;&auml;&oacute;=&auml;&Eacute;&Ntilde;&iacute;=&Auml;&auml;~&aring;&acirc;K=
=
viii
Chapter 1
Number Sets
=
=
=
=
1.1 Set Identities
=
p&Eacute;&iacute;&euml;W=^I=_I=`=
r&aring;&aacute;&icirc;&Eacute;&ecirc;&euml;~&auml;=&euml;&Eacute;&iacute;W=f=
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=W= ^′ =
m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;=&euml;&igrave;&Auml;&euml;&Eacute;&iacute;W= ^ ⊂ _ ==
b&atilde;&eacute;&iacute;&oacute;=&euml;&Eacute;&iacute;W= ∅ =
r&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&euml;&Eacute;&iacute;&euml;W= ^ ∪ _ =
f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&euml;&Eacute;&iacute;&euml;W= ^ ∩ _ =
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=&euml;&Eacute;&iacute;&euml;W= ^ y _ =
=
=
1.
=
2.
=
3.
4.
5.
^ ⊂ f=
^ ⊂ ^=
^ = _ =&aacute;&Ntilde;= ^ ⊂ _ =~&aring;&Ccedil;= _ ⊂ ^ .=
=
b&atilde;&eacute;&iacute;&oacute;=p&Eacute;&iacute;=
∅⊂^=
=
r&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=p&Eacute;&iacute;&euml;==
` = ^ ∪ _ = {&ntilde; &ouml; &ntilde; ∈ ^ &ccedil;&ecirc; &ntilde; ∈ _}=
=
1
CHAPTER 1. NUMBER SETS
=
=====
=
Figure 1.
6.
=
7.
=
8.
=
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=
^∪_ = _∪^=
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=
^ ∪ (_ ∪ ` ) = (^ ∪ _ ) ∪ ` =
f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=p&Eacute;&iacute;&euml;=
` = ^ ∪ _ = {&ntilde; &ouml; &ntilde; ∈ ^ ~&aring;&Ccedil; &ntilde; ∈ _} =
=
=
=====
=
Figure 2.
9.
=
10.
=
=
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=
^∩_ = _∩^=
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=
^ ∩ (_ ∩ ` ) = (^ ∩ _ ) ∩ ` =
=
2
CHAPTER 1. NUMBER SETS
11.
=
12.
=
13.
=
14.
a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=
^ ∪ (_ ∩ ` ) = (^ ∪ _ ) ∩ (^ ∪ ` ) I=
^ ∩ (_ ∪ ` ) = (^ ∩ _ ) ∪ (^ ∩ ` ) K=
f&Ccedil;&Eacute;&atilde;&eacute;&ccedil;&iacute;&Eacute;&aring;&Aring;&oacute;=
^ ∩ ^ = ^ I==
^∪^ = ^=
a&ccedil;&atilde;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;=
^ ∩ ∅ = ∅ I=
^∪f= f=
f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;=
^ ∪ ∅ = ^ I==
^∩f= ^
=
15.
16.
17.
18.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=
^′ = {&ntilde; ∈ f &ouml; &ntilde; ∉ ^}
=
`&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=~&aring;&Ccedil;=r&aring;&aacute;&ccedil;&aring;
^ ∪ ^′ = f I==
^ ∩ ^′ = ∅ =
=
a&Eacute;=j&ccedil;&ecirc;&Ouml;~&aring;∞&euml;=i~&iuml;&euml;
(^ ∪ _ )′ = ^′ ∩ _′ I==
(^ ∩ _ )′ = ^′ ∪ _′ =
=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=p&Eacute;&iacute;&euml;
` = _ y ^ = {&ntilde; &ouml; &ntilde; ∈ _ ~&aring;&Ccedil; &ntilde; ∉ ^} =
=
3
CHAPTER 1. NUMBER SETS
=
=====
=
Figure 3.
=
19.
_ y ^ = _ y (^ ∩ _ )
=
20.
_ y ^ = _ ∩ ^′
21.
^y^=∅
22.
^ y _ = ^ =&aacute;&Ntilde;= ^ ∩ _ = ∅ .
=
=
=
=====
=
Figure 4.
=
23.
(^ y _) ∩ ` = (^ ∩ `) y (_ ∩ `)
24.
^′ = f y ^
25.
`~&ecirc;&iacute;&Eacute;&euml;&aacute;~&aring;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;
` = ^ &times; _ = {(&ntilde; I &oacute; ) &ouml; &ntilde; ∈ ^ ~&aring;&Ccedil; &oacute; ∈ _}
=
=
4
=
CHAPTER 1. NUMBER SETS
1.2 Sets of Numbers
=
26.
27.
=
28.
=
29.
=
30.
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=k=
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W= kM =
f&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W=w=
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W= w + =
k&Eacute;&Ouml;~&iacute;&aacute;&icirc;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;W= w − =
o~&iacute;&aacute;&ccedil;&aring;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=n=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=o==
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=`==
=
=
k~&iacute;&igrave;&ecirc;~&auml;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
`&ccedil;&igrave;&aring;&iacute;&aacute;&aring;&Ouml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W k = {NI OI PI K} K=
t&Uuml;&ccedil;&auml;&Eacute;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
`&ccedil;&igrave;&aring;&iacute;&aacute;&aring;&Ouml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=~&aring;&Ccedil;=&ograve;&Eacute;&ecirc;&ccedil;W= k M = {MI NI OI PI K} K=
f&aring;&iacute;&Eacute;&Ouml;&Eacute;&ecirc;&euml;
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;&aacute;&ecirc;=&ccedil;&eacute;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute;&euml;=~&aring;&Ccedil;=&ograve;&Eacute;&ecirc;&ccedil;W=
w + = k = {NI OI PI K}I=
w − = {KI − PI − OI − N} I=
w = w − ∪ {M} ∪ w + = {KI − PI − OI − NI MI NI OI PI K} K=
o~&iacute;&aacute;&ccedil;&aring;~&auml;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
o&Eacute;&eacute;&Eacute;~&iacute;&aacute;&aring;&Ouml;=&ccedil;&ecirc;=&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&iacute;&aacute;&aring;&Ouml;=&Ccedil;&Eacute;&Aring;&aacute;&atilde;~&auml;&euml;W==
~


n = &ntilde; &ouml; &ntilde; = ~&aring;&Ccedil; ~ ∈ w ~&aring;&Ccedil; &Auml; ∈ w ~&aring;&Ccedil; &Auml; ≠ M K=
&Auml;


f&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
k&ccedil;&aring;&ecirc;&Eacute;&eacute;&Eacute;~&iacute;&aacute;&aring;&Ouml;=~&aring;&Ccedil;=&aring;&ccedil;&aring;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&iacute;&aacute;&aring;&Ouml;=&Ccedil;&Eacute;&Aring;&aacute;&atilde;~&auml;&euml;K
=
5
CHAPTER 1. NUMBER SETS
31.
o&Eacute;~&auml;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;==
r&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=~&aring;&Ccedil;=&aacute;&ecirc;&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=oK=
=
32.
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;
` = {&ntilde; + &aacute;&oacute; &ouml; &ntilde; ∈ o ~&aring;&Ccedil; &oacute; ∈ o}I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&aacute;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=&igrave;&aring;&aacute;&iacute;K
=
33.
k⊂ w⊂n⊂ o ⊂ `=
=
===
=
=
Figure 5.
=
=
=
=
=
=
6
CHAPTER 1. NUMBER SETS
1.3 Basic Identities
=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;=
=
=
34.
^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&icirc;&Eacute;=f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;=
~+M=~ =
=
35.
^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&icirc;&Eacute;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=
~ + (− ~ ) = M =
=
36.
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;=
~ +&Auml;= &Auml;+~ =
37.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;=
(~ + &Auml;) + &Aring; = ~ + (&Auml; + &Aring; ) =
=
=
38.
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=p&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&aacute;&ccedil;&aring;=
~ − &Auml; = ~ + (− &Auml;) =
=
39.
=
40.
41.
42.
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&icirc;&Eacute;=f&Ccedil;&Eacute;&aring;&iacute;&aacute;&iacute;&oacute;=
~ ⋅N = ~ =
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&icirc;&Eacute;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=
N
~ ⋅ = N I= ~ ≠ M
~
=
j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=q&aacute;&atilde;&Eacute;&euml;=M
~ ⋅M = M
=
`&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=
~ ⋅&Auml; = &Auml;⋅~
=
=
7
CHAPTER 1. NUMBER SETS
43.
^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=
(~ ⋅ &Auml;)⋅ &Aring; = ~ ⋅ (&Auml; ⋅ &Aring; )
=
a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute;=i~&iuml;=
~ (&Auml; + &Aring; ) = ~&Auml; + ~&Aring; =
44.
=
45.
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=a&aacute;&icirc;&aacute;&euml;&aacute;&ccedil;&aring;=
~
N
= ~⋅ =
&Auml;
&Auml;
=
=
=
1.4 Complex Numbers
=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=&igrave;&aring;&aacute;&iacute;W=&aacute;=
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&ograve;=
o&Eacute;~&auml;=&eacute;~&ecirc;&iacute;W=~I=&Aring;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=&eacute;~&ecirc;&iacute;W=&Auml;&aacute;I=&Ccedil;&aacute;=
j&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml;=&ccedil;&Ntilde;=~=&Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&ecirc;I= &ecirc;N I= &ecirc;O =
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=&Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W= ϕ I= ϕN I= ϕO =
=
=
46.
=
47.
=
48.
&aacute;N = &aacute; =
&aacute; O = −N =
&aacute; P = −&aacute; =
&aacute;Q = N=
&aacute;R = &aacute; =
&aacute; S = −N =
&aacute; T = −&aacute; =
&aacute;U = N =
&aacute; Q &aring; +N = &aacute; =
&aacute; Q &aring;+ O = −N =
&aacute; Q &aring; + P = −&aacute; =
&aacute; Q&aring; = N =
&ograve; = ~ + &Auml;&aacute; =
`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=m&auml;~&aring;&Eacute;=
=
8
CHAPTER 1. NUMBER SETS
=
=====
=
Figure 6.
=
49.
=
50.
=
51.
=
(~ + &Auml;&aacute; ) − (&Aring; + &Ccedil;&aacute; ) = (~ − &Aring; ) + (&Auml; − &Ccedil;)&aacute; =
(~ + &Auml;&aacute; )(&Aring; + &Ccedil;&aacute; ) = (~&Aring; − &Auml;&Ccedil; ) + (~&Ccedil; + &Auml;&Aring; )&aacute; =
~ + &Auml;&aacute; ~&Aring; + &Auml;&Ccedil; &Auml;&Aring; − ~&Ccedil;
=
+
⋅&aacute; =
&Aring; + &Ccedil;&aacute; &Aring; O + &Ccedil; O &Aring; O + &Ccedil; O
52.
=
53.
(~ + &Auml;&aacute; ) + (&Aring; + &Ccedil;&aacute; ) = (~ + &Aring; ) + (&Auml; + &Ccedil; )&aacute; =
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute;=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=
|||||||
~ + &Auml;&aacute; = ~ − &Auml;&aacute; =
=
54.
~ = &ecirc; &Aring;&ccedil;&euml; ϕ I= &Auml; = &ecirc; &euml;&aacute;&aring; ϕ ==
=
9
CHAPTER 1. NUMBER SETS
=
=
Figure 7.
55.
=
56.
=
m&ccedil;&auml;~&ecirc;=m&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=
~ + &Auml;&aacute; = &ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) =
j&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml;=~&aring;&Ccedil;=^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
f&Ntilde;= ~ + &Auml;&aacute; =&aacute;&euml;=~=&Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;I=&iacute;&Uuml;&Eacute;&aring;=
&ecirc; = ~ O + &Auml;O =E&atilde;&ccedil;&Ccedil;&igrave;&auml;&igrave;&euml;FI==
&Auml;
ϕ = ~&ecirc;&Aring;&iacute;~&aring; =E~&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;FK=
~
=
57.
=
58.
m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;=
&ograve; N ⋅ &ograve; O = &ecirc;N (&Aring;&ccedil;&euml; ϕN + &aacute; &euml;&aacute;&aring; ϕN ) ⋅ &ecirc;O (&Aring;&ccedil;&euml; ϕO + &aacute; &euml;&aacute;&aring; ϕO ) =
= &ecirc;N&ecirc;O [&Aring;&ccedil;&euml;(ϕN + ϕO ) + &aacute; &euml;&aacute;&aring;(ϕN + ϕO )] =
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;=
|||||||||||||||||||||
&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) = &ecirc;[&Aring;&ccedil;&euml;(− ϕ) + &aacute; &euml;&aacute;&aring;(− ϕ)] =
=
59.
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&ccedil;&Ntilde;=~=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;=
N
N
= [&Aring;&ccedil;&euml;(− ϕ) + &aacute; &euml;&aacute;&aring;(− ϕ)] =
&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) &ecirc;
10
CHAPTER 1. NUMBER SETS
60.
=
61.
=
62.
=
63.
=
64.
n&igrave;&ccedil;&iacute;&aacute;&Eacute;&aring;&iacute;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=o&Eacute;&eacute;&ecirc;&Eacute;&euml;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;=
&ograve; N &ecirc;N (&Aring;&ccedil;&euml; ϕN + &aacute; &euml;&aacute;&aring; ϕN ) &ecirc;N
= [&Aring;&ccedil;&euml;(ϕN − ϕO ) + &aacute; &euml;&aacute;&aring;(ϕN − ϕO )] =
=
&ograve; O &ecirc;O (&Aring;&ccedil;&euml; ϕO + &aacute; &euml;&aacute;&aring; ϕO ) &ecirc;O
m&ccedil;&iuml;&Eacute;&ecirc;=&ccedil;&Ntilde;=~=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
&aring;
&ograve; &aring; = [&ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ)] = &ecirc; &aring; [&Aring;&ccedil;&euml;(&aring;ϕ) + &aacute; &euml;&aacute;&aring;(&aring;ϕ)] =
c&ccedil;&ecirc;&atilde;&igrave;&auml;~=&plusmn;a&Eacute;=j&ccedil;&aacute;&icirc;&ecirc;&Eacute;≤=
(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ)&aring; = &Aring;&ccedil;&euml;(&aring;ϕ) + &aacute; &euml;&aacute;&aring;(&aring;ϕ) =
k&iacute;&Uuml;=o&ccedil;&ccedil;&iacute;=&ccedil;&Ntilde;=~=`&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
ϕ + Oπ&acirc;
ϕ + Oπ&acirc; 

&aring;
&ograve; = &aring; &ecirc;(&Aring;&ccedil;&euml; ϕ + &aacute; &euml;&aacute;&aring; ϕ) = &aring; &ecirc;  &Aring;&ccedil;&euml;
+ &aacute; &euml;&aacute;&aring;
 I==
&aring;
&aring; 

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&acirc; = MI NI OI KI &aring; − N K==
b&igrave;&auml;&Eacute;&ecirc;∞&euml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~=
&Eacute; &aacute;&ntilde; = &Aring;&ccedil;&euml; &ntilde; + &aacute; &euml;&aacute;&aring; &ntilde; =
=
=
11
Chapter 2
Algebra
=
=
=
=
2.1 Factoring Formulas
=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;==
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
=
=
65.
=
66.
=
67.
=
68.
=
69.
=
70.
=
71.
=
72.
~ O − &Auml;O = (~ + &Auml;)(~ − &Auml;) =
~ P − &Auml;P = (~ − &Auml;)(~ O + ~&Auml; + &Auml;O ) =
~ P + &Auml;P = (~ + &Auml;)(~ O − ~&Auml; + &Auml;O ) =
~ Q − &Auml;Q = (~ O − &Auml;O )(~ O + &Auml;O ) = (~ − &Auml;)(~ + &Auml;)(~ O + &Auml;O ) =
~ R − &Auml;R = (~ − &Auml;)(~ Q + ~ P &Auml; + ~ O &Auml;O + ~&Auml;P + &Auml;Q ) =
~ R + &Auml;R = (~ + &Auml;)(~ Q − ~ P &Auml; + ~ O &Auml;O − ~&Auml;P + &Auml;Q ) =
f&Ntilde;=&aring;=&aacute;&euml;=&ccedil;&Ccedil;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;=
~ &aring; + &Auml;&aring; = (~ + &Auml;)(~ &aring;−N − ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O − K − ~&Auml;&aring; −O + &Auml;&aring; −N ) K==
f&Ntilde;=&aring;=&aacute;&euml;=&Eacute;&icirc;&Eacute;&aring;I=&iacute;&Uuml;&Eacute;&aring;==
~ &aring; − &Auml;&aring; = (~ − &Auml;)(~ &aring; −N + ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O + K + ~&Auml;&aring;−O + &Auml;&aring; −N ) I==
12
CHAPTER 2. ALGEBRA
~ &aring; + &Auml;&aring; = (~ + &Auml;)(~ &aring;−N − ~ &aring; −O &Auml; + ~ &aring; −P &Auml;O − K + ~&Auml;&aring;−O − &Auml;&aring; −N ) K=
=
=
=
2.2 Product Formulas
73.
=
74.
=
75.
=
76.
=
77.
=
78.
=
79.
=
80.
=
81.
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;==
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&aring;I=&acirc;=
=
=
(~ − &Auml;)O = ~ O − O~&Auml; + &Auml;O =
(~ + &Auml;)O = ~ O + O~&Auml; + &Auml;O =
(~ − &Auml;)P = ~ P − P~ O &Auml; + P~&Auml;O − &Auml;P =
(~ + &Auml;)P = ~ P + P~ O&Auml; + P~&Auml;O + &Auml;P =
(~ − &Auml;)Q = ~ Q − Q~ P &Auml; + S~ O &Auml;O − Q~&Auml;P + &Auml;Q =
(~ + &Auml;)Q = ~ Q + Q~ P &Auml; + S~ O&Auml;O + Q~&Auml;P + &Auml;Q =
_&aacute;&aring;&ccedil;&atilde;&aacute;~&auml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~=
(~ + &Auml;)&aring; = &aring;` M~ &aring; + &aring;`N~ &aring;−N&Auml; + &aring;` O~ &aring;−O&Auml;O + K + &aring;` &aring;−N~&Auml;&aring;−N + &aring;` &aring; &Auml;&aring; I
&aring;&gt;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &aring; ` &acirc; =
=~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Auml;&aacute;&aring;&ccedil;&atilde;&aacute;~&auml;=&Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;K=
&acirc;&gt; (&aring; − &acirc; )&gt;
(~ + &Auml; + &Aring; )O = ~ O + &Auml;O + &Aring; O + O~&Auml; + O~&Aring; + O&Auml;&Aring; =
(~ + &Auml; + &Aring; + K + &igrave; + &icirc; )O = ~ O + &Auml;O + &Aring; O + K + &igrave; O + &icirc; O + =
+ O(~&Auml; + ~&Aring; + K + ~&igrave; + ~&icirc; + &Auml;&Aring; + K + &Auml;&igrave; + &Auml;&icirc; + K + &igrave;&icirc; ) =
13
CHAPTER 2. ALGEBRA
2.3 Powers
=
_~&euml;&Eacute;&euml;=E&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW=~I=&Auml;==
m&ccedil;&iuml;&Eacute;&ecirc;&euml;=E&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW=&aring;I=&atilde;=
=
=
~ &atilde; ~ &aring; = ~ &atilde;+&aring; =
82.
=
83.
~&atilde;
= ~ &atilde; −&aring; =
&aring;
~
=
84.
=
(~&Auml;)&atilde; = ~ &atilde; &Auml;&atilde; =
85.
~&atilde;
~
  = &atilde; =
&Auml;
 &Auml;
&atilde;
=
86.
=
87.
=
88.
=
(~ )
&atilde; &aring;
= ~ &atilde;&aring; =
~ M = N I= ~ ≠ M =
~N = N =
~ −&atilde; =
89.
N
=
~&atilde;
=
&atilde;
&aring;
~ = &aring; ~&atilde; =
90.
=
=
=
=
=
14
CHAPTER 2. ALGEBRA
2.4 Roots
=
91.
=
_~&euml;&Eacute;&euml;W=~I=&Auml;==
m&ccedil;&iuml;&Eacute;&ecirc;&euml;=E&ecirc;~&iacute;&aacute;&ccedil;&aring;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;FW=&aring;I=&atilde;=
~ I &Auml; ≥ M =&Ntilde;&ccedil;&ecirc;=&Eacute;&icirc;&Eacute;&aring;=&ecirc;&ccedil;&ccedil;&iacute;&euml;=E &aring; = O&acirc; I= &acirc; ∈ k F=
=
=
&aring;
~&Auml; = &aring; ~ &aring; &Auml; =
92.
=
&aring;
~ &atilde; &Auml; = &aring;&atilde; ~ &atilde; &Auml;&aring; =
93.
&aring;
~ &aring;~
=
I= &Auml; ≠ M =
&Auml; &aring;&Auml;
=
94.
=
95.
=
96.
=
~ &aring;&atilde; ~ &atilde; &aring;&atilde; ~ &atilde;
I= &Auml; ≠ M K=
=
=
&atilde;
&Auml;&aring;
&Auml; &aring;&atilde; &Auml;&aring;
&aring;
(~ )
&aring;
&atilde;
( ~)
&aring;
&aring;
&eacute;
= &aring; ~ &atilde;&eacute; =
=~=
&aring;&eacute;
97.
=
&aring;
~&atilde; =
98.
=
&aring;
~ =~ =
99.
=
&atilde; &aring;
100.
=
&atilde;
&aring;
&atilde;
~ = &atilde;&aring; ~ =
( ~)
&aring;
~ &atilde;&eacute; =
&atilde;
= &aring; ~&atilde; =
15
CHAPTER 2. ALGEBRA
N &aring; ~ &aring; −N
=
I= ~ ≠ M K=
&aring;
~
~
101.
=
~&plusmn; &Auml; =
102.
~ + ~O − &Auml;
~ − ~O − &Auml;
&plusmn;
=
O
O
=
N
~m &Auml;
=
=
~−&Auml;
~&plusmn; &Auml;
103.
=
=
=
2.5 Logarithms
=
104.
105.
106.
107.
108.
109.
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&ntilde;I=&oacute;I=~I=&Aring;I=&acirc;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;==
=
=
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;=
&oacute; = &auml;&ccedil;&Ouml; ~ &ntilde; =&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;= &ntilde; = ~ &oacute; I= ~ &gt; M I= ~ ≠ N K=
=
&auml;&ccedil;&Ouml; ~ N = M =
=
&auml;&ccedil;&Ouml; ~ ~ = N =
=
− ∞ &aacute;&Ntilde; ~ &gt; N
&auml;&ccedil;&Ouml; ~ M = 
=
+ ∞ &aacute;&Ntilde; ~ &lt; N
=
&auml;&ccedil;&Ouml; ~ (&ntilde;&oacute; ) = &auml;&ccedil;&Ouml; ~ &ntilde; + &auml;&ccedil;&Ouml; ~ &oacute; =
=
&ntilde;
&auml;&ccedil;&Ouml; ~ = &auml;&ccedil;&Ouml; ~ &ntilde; − &auml;&ccedil;&Ouml; ~ &oacute; =
&oacute;
16
CHAPTER 2. ALGEBRA
110. &auml;&ccedil;&Ouml; ~ (&ntilde; &aring; ) = &aring; &auml;&ccedil;&Ouml; ~ &ntilde; =
=
N
111. &auml;&ccedil;&Ouml; ~ &aring; &ntilde; = &auml;&ccedil;&Ouml; ~ &ntilde; =
&aring;
=
&auml;&ccedil;&Ouml; &Aring; &ntilde;
112. &auml;&ccedil;&Ouml; ~ &ntilde; =
= &auml;&ccedil;&Ouml; &Aring; &ntilde; ⋅ &auml;&ccedil;&Ouml; ~ &Aring; I= &Aring; &gt; M I= &Aring; ≠ N K=
&auml;&ccedil;&Ouml; &Aring; ~
=
N
113. &auml;&ccedil;&Ouml; ~ &Aring; =
=
&auml;&ccedil;&Ouml; &Aring; ~
=
114. &ntilde; = ~ &auml;&ccedil;&Ouml; ~ &ntilde; =
=
115. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;=&iacute;&ccedil;=_~&euml;&Eacute;=NM=
&auml;&ccedil;&Ouml; NM &ntilde; = &auml;&ccedil;&Ouml; &ntilde; =
=
116. k~&iacute;&igrave;&ecirc;~&auml;=i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;=
&auml;&ccedil;&Ouml; &Eacute; &ntilde; = &auml;&aring; &ntilde; I==
&acirc;
 N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &Eacute; = &auml;&aacute;&atilde; N +  = OKTNUOUNUOUK =
&acirc; →∞
 &acirc;
=
N
117. &auml;&ccedil;&Ouml; &ntilde; =
&auml;&aring; &ntilde; = MKQPQOVQ &auml;&aring; &ntilde; =
&auml;&aring; NM
=
N
118. &auml;&aring; &ntilde; =
&auml;&ccedil;&Ouml; &ntilde; = OKPMORUR &auml;&ccedil;&Ouml; &ntilde; =
&auml;&ccedil;&Ouml; &Eacute;
=
=
=
=
=
17
CHAPTER 2. ALGEBRA
2.6 Equations
=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;I=&eacute;I=&egrave;I=&igrave;I=&icirc;=
p&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;W= &ntilde; N I= &ntilde; O I= &oacute; N I= &oacute; O I= &oacute; P =
=
=
119. i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&aacute;&aring;=l&aring;&Eacute;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;=
&Auml;
~&ntilde; + &Auml; = M I= &ntilde; = − K==
~
=
120. n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
− &Auml; &plusmn; &Auml;O − Q~&Aring;
~&ntilde; + &Auml;&ntilde; + &Aring; = M I= &ntilde; NI O =
K=
O~
=
121. a&aacute;&euml;&Aring;&ecirc;&aacute;&atilde;&aacute;&aring;~&aring;&iacute;=
a = &Auml;O − Q~&Aring; =
=
122. s&aacute;&Eacute;&iacute;&Eacute;∞&euml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;=
f&Ntilde;= &ntilde; O + &eacute;&ntilde; + &egrave; = M I=&iacute;&Uuml;&Eacute;&aring;==
&ntilde; N + &ntilde; O = −&eacute;
K=

&ntilde;
&ntilde;
=
&egrave;
N
O

=
&Auml;
123. ~&ntilde; O + &Auml;&ntilde; = M I= &ntilde; N = M I= &ntilde; O = − K=
~
=
&Aring;
124. ~&ntilde; O + &Aring; = M I= &ntilde; NI O = &plusmn; − K=
~
=
125. `&igrave;&Auml;&aacute;&Aring;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K=`~&ecirc;&Ccedil;~&aring;&ccedil;∞&euml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~K==
&oacute; P + &eacute;&oacute; + &egrave; = M I==
O
18
CHAPTER 2. ALGEBRA
&oacute; N = &igrave; + &icirc; I= &oacute; OI P = −
N
(&igrave; + &icirc; ) &plusmn; P (&igrave; + &icirc; ) &aacute; I==
O
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
O
&igrave;=P −
O
O
O
&egrave;
&egrave;
&egrave;  &eacute;
 &egrave;  &eacute;
+   +   I= &icirc; = P − −   +   K==
O
O
 O P
 O P
=
=
2.7 Inequalities
s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;=
~ I &Auml;I &Aring;I &Ccedil;
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W= 
I=&atilde;I=&aring;=
~N I ~ O I ~ P I KI ~ &aring;
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;W=aI= a&ntilde; I= a&oacute; I= a&ograve; ==
=
=
126. f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&aacute;&Eacute;&euml;I=f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=k&ccedil;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;=~&aring;&Ccedil;=d&ecirc;~&eacute;&Uuml;&euml;==
=
f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;=
f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=k&ccedil;&iacute;~&iacute;&aacute;&ccedil;&aring;=
d&ecirc;~&eacute;&Uuml;=
[~I &Auml;]=
~ ≤ &ntilde; ≤ &Auml;=
~ &lt; &ntilde; ≤ &Auml;=
(~I &Auml;] =
=
~ ≤ &ntilde; &lt; &Auml;=
[~I &Auml;) =
=
~ &lt; &ntilde; &lt; &Auml;=
(~I &Auml;) =
=
− ∞ &lt; &ntilde; ≤ &Auml; I=
&ntilde;≤&Auml;=
− ∞ &lt; &ntilde; &lt; &Auml; I=
&ntilde;&lt;&Auml;=
~ ≤ &ntilde; &lt; ∞ I=
&ntilde;≥~=
~ &lt; &ntilde; &lt; ∞ I=
&ntilde; &gt;~=
(− ∞I &Auml;] =
=
=
(− ∞I &Auml;) =
=
[~I ∞ ) =
=
(~I ∞ ) =
=
19
CHAPTER 2. ALGEBRA
127.
=
128.
=
129.
=
130.
=
131.
=
132.
=
133.
=
f&Ntilde;= ~ &gt; &Auml; I=&iacute;&Uuml;&Eacute;&aring;= &Auml; &lt; ~ K=
f&Ntilde;= ~ &gt; &Auml; I=&iacute;&Uuml;&Eacute;&aring;= ~ − &Auml; &gt; M =&ccedil;&ecirc;= &Auml; − ~ &lt; M K=
f&Ntilde;= ~ &gt; &Auml; I=&iacute;&Uuml;&Eacute;&aring;= ~ + &Aring; &gt; &Auml; + &Aring; K=
f&Ntilde;= ~ &gt; &Auml; I=&iacute;&Uuml;&Eacute;&aring;= ~ − &Aring; &gt; &Auml; − &Aring; K=
f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &Aring; &gt; &Ccedil; I=&iacute;&Uuml;&Eacute;&aring;= ~ + &Aring; &gt; &Auml; + &Ccedil; K=
f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &Aring; &gt; &Ccedil; I=&iacute;&Uuml;&Eacute;&aring;= ~ − &Ccedil; &gt; &Auml; − &Aring; K=
f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &atilde; &gt; M I=&iacute;&Uuml;&Eacute;&aring;= &atilde;~ &gt; &atilde;&Auml; K=
134. f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &atilde; &gt; M I=&iacute;&Uuml;&Eacute;&aring;=
~ &Auml;
&gt; K=
&atilde; &atilde;
=
135. f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &atilde; &lt; M I=&iacute;&Uuml;&Eacute;&aring;= &atilde;~ &lt; &atilde;&Auml; K=
=
~ &Auml;
136. f&Ntilde;= ~ &gt; &Auml; =~&aring;&Ccedil;= &atilde; &lt; M I=&iacute;&Uuml;&Eacute;&aring;= &lt; K=
&atilde; &atilde;
=
137. f&Ntilde;= M &lt; ~ &lt; &Auml; =~&aring;&Ccedil;= &aring; &gt; M I=&iacute;&Uuml;&Eacute;&aring;= ~ &aring; &lt; &Auml;&aring; K=
=
138. f&Ntilde;= M &lt; ~ &lt; &Auml; =~&aring;&Ccedil;= &aring; &lt; M I=&iacute;&Uuml;&Eacute;&aring;= ~ &aring; &gt; &Auml;&aring; K=
=
139. f&Ntilde;= M &lt; ~ &lt; &Auml; I=&iacute;&Uuml;&Eacute;&aring;= &aring; ~ &lt; &aring; &Auml; K=
=
~+&Auml;
I==
140.
~&Auml; ≤
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M =I= &Auml; &gt; M X=~&aring;=&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;=&aacute;&euml;=&icirc;~&auml;&aacute;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;= ~ = &Auml; K==
=
N
141. ~ + ≥ O I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M X=~&aring;=&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;=&iacute;~&acirc;&Eacute;&euml;=&eacute;&auml;~&Aring;&Eacute;=&ccedil;&aring;&auml;&oacute;=~&iacute;= ~ = N K=
~
20
CHAPTER 2. ALGEBRA
142.
&aring;
~N~ O K~ &aring; ≤
~N + ~ O + K + ~ &aring;
I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~N I ~ O I KI ~ &aring; &gt; M K=
&aring;
=
&Auml;
143. f&Ntilde;= ~&ntilde; + &Auml; &gt; M =~&aring;&Ccedil;= ~ &gt; M I=&iacute;&Uuml;&Eacute;&aring;= &ntilde; &gt; − K=
~
=
&Auml;
144. f&Ntilde;= ~&ntilde; + &Auml; &gt; M =~&aring;&Ccedil;= ~ &lt; M I=&iacute;&Uuml;&Eacute;&aring;= &ntilde; &lt; − K==
~
=
145. ~&ntilde; O + &Auml;&ntilde; + &Aring; &gt; M =
=
=
~ &gt; M=
=
=
=
=
a&gt;M=
=
=
=
a=M=
=
=
=
a&lt;M=
=
&ntilde; &lt; &ntilde; N I= &ntilde; &gt; &ntilde; O =
=
&ntilde; N &lt; &ntilde; I= &ntilde; &gt; &ntilde; N =
=
=
−∞&lt; &ntilde; &lt;∞=
=
21
~ &lt;M=
=
=
&ntilde;N &lt; &ntilde; &lt; &ntilde; O =
=
&ntilde; ∈∅ =
=
=
&ntilde; ∈∅ =
=
=
=
CHAPTER 2. ALGEBRA
~+&Auml; ≤ ~ + &Auml; =
146.
=
147.
=
148.
=
149.
=
150.
=
f&Ntilde;= &ntilde; &lt; ~ I=&iacute;&Uuml;&Eacute;&aring;= − ~ &lt; &ntilde; &lt; ~ I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M K=
f&Ntilde;= &ntilde; &gt; ~ I=&iacute;&Uuml;&Eacute;&aring;= &ntilde; &lt; −~ =~&aring;&Ccedil;= &ntilde; &gt; ~ I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M K=
f&Ntilde;= &ntilde; O &lt; ~ I=&iacute;&Uuml;&Eacute;&aring;= &ntilde; &lt; ~ I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M K=
f&Ntilde;= &ntilde; O &gt; ~ I=&iacute;&Uuml;&Eacute;&aring;= &ntilde; &gt; ~ I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ &gt; M K=
151. f&Ntilde;=
=
&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; ) &gt; M
&Ntilde; (&ntilde; )
&gt; M I=&iacute;&Uuml;&Eacute;&aring;= 
K=
&Ouml; (&ntilde; )
&Ouml; (&ntilde; ) ≠ M
&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; ) &lt; M
&Ntilde; (&ntilde; )
&lt; M I=&iacute;&Uuml;&Eacute;&aring;= 
K=
&Ouml; (&ntilde; )
&Ouml; (&ntilde; ) ≠ M
152.
=
=
=
2.8 Compound Interest Formulas
=
c&igrave;&iacute;&igrave;&ecirc;&Eacute;=&icirc;~&auml;&igrave;&Eacute;W=^=
f&aring;&aacute;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&eacute;&ccedil;&euml;&aacute;&iacute;W=`=
^&aring;&aring;&igrave;~&auml;=&ecirc;~&iacute;&Eacute;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;W=&ecirc;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&oacute;&Eacute;~&ecirc;&euml;=&aacute;&aring;&icirc;&Eacute;&euml;&iacute;&Eacute;&Ccedil;W=&iacute;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&aacute;&atilde;&Eacute;&euml;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&eacute;&Eacute;&ecirc;=&oacute;&Eacute;~&ecirc;W=&aring;=
=
=
153. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=`&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;=f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~=
&aring;&iacute;
 &ecirc;
^ = ` N +  =
 &aring;
=
22
CHAPTER 2. ALGEBRA
154. p&aacute;&atilde;&eacute;&auml;&aacute;&Ntilde;&aacute;&Eacute;&Ccedil;=`&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;=f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~=
f&Ntilde;=&aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=&aacute;&euml;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&ccedil;&aring;&Aring;&Eacute;=&eacute;&Eacute;&ecirc;=&oacute;&Eacute;~&ecirc;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&Eacute;&icirc;&aacute;&ccedil;&igrave;&euml;=
&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=&euml;&aacute;&atilde;&eacute;&auml;&aacute;&Ntilde;&aacute;&Eacute;&euml;=&iacute;&ccedil;W=
&iacute;
^ = `(N + &ecirc; ) K=
=
155. `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=`&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;=f&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=
f&Ntilde;=&aacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&euml;&iacute;=&aacute;&euml;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;~&auml;&auml;&oacute;=E &aring; → ∞ FI=&iacute;&Uuml;&Eacute;&aring;==
^ = `&Eacute; &ecirc;&iacute; K=
=
=
23
Chapter 3
Geometry
=
=
=
=
3.1 Right Triangle
=
i&Eacute;&Ouml;&euml;=&ccedil;&Ntilde;=~=&ecirc;&aacute;&Ouml;&Uuml;&iacute;=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W=~I=&Auml;=
e&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute;W=&Aring;=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
j&Eacute;&Ccedil;&aacute;~&aring;&euml;W= &atilde; ~ I= &atilde; &Auml; I= &atilde; &Aring; =
^&aring;&Ouml;&auml;&Eacute;&euml;W= α I β =
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 8.
=
156. α + β = VM&deg; =
=
24
CHAPTER 3. GEOMETRY
157. &euml;&aacute;&aring; α =
~
= &Aring;&ccedil;&euml; β =
&Aring;
=
158. &Aring;&ccedil;&euml; α =
&Auml;
= &euml;&aacute;&aring; β =
&Aring;
=
159. &iacute;~&aring; α =
~
= &Aring;&ccedil;&iacute; β =
&Auml;
=
&Auml;
160. &Aring;&ccedil;&iacute; α = = &iacute;~&aring; β =
~
=
&Aring;
161. &euml;&Eacute;&Aring; α = = &Aring;&ccedil;&euml; &Eacute;&Aring; β =
&Auml;
=
162. &Aring;&ccedil;&euml; &Eacute;&Aring; α =
&Aring;
= &euml;&Eacute;&Aring; β =
~
=
163. m&oacute;&iacute;&Uuml;~&Ouml;&ccedil;&ecirc;&Eacute;~&aring;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
~ O + &Auml;O = &Aring; O =
=
164. ~ = &Ntilde;&Aring; I= &Auml; = &Ouml;&Aring; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &Ntilde;= ~&aring;&Ccedil;= &Aring;= ~&ecirc;&Eacute;= &eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &auml;&Eacute;&Ouml;&euml;= ~= ~&aring;&Ccedil;= &Auml;I= &ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;I=&ccedil;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Uuml;&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute;=&Aring;K=
=
O
=
O
=
=====
Figure 9.
=
25
CHAPTER 3. GEOMETRY
165. &Uuml; O = &Ntilde;&Ouml; I===
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Uuml;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&Ntilde;&ecirc;&ccedil;&atilde;=&iacute;&Uuml;&Eacute;=&ecirc;&aacute;&Ouml;&Uuml;&iacute;=~&aring;&Ouml;&auml;&Eacute;K==
=
O
O
~
&Auml;
166. &atilde; O~ = &Auml;O − I= &atilde; O&Auml; = ~ O − I===
Q
Q
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &atilde; ~ =~&aring;&Ccedil;= &atilde; &Auml; =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&atilde;&Eacute;&Ccedil;&aacute;~&aring;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&Eacute;&Ouml;&euml;=~=~&aring;&Ccedil;=&Auml;K==
=
=
=
Figure 10.
=
&Aring;
167. &atilde; &Aring; = I==
O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &atilde; &Aring; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&atilde;&Eacute;&Ccedil;&aacute;~&aring;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Uuml;&oacute;&eacute;&ccedil;&iacute;&Eacute;&aring;&igrave;&euml;&Eacute;=&Aring;K=
=
&Aring;
168. o = = &atilde; &Aring; =
O
=
~ +&Auml;−&Aring;
~&Auml;
=
=
169. &ecirc; =
O
~ +&Auml;+&Aring;
=
170. ~&Auml; = &Aring;&Uuml; =
=
=
26
CHAPTER 3. GEOMETRY
171. p =
~&Auml; &Aring;&Uuml;
=
=
O
O
=
=
=
3.2 Isosceles Triangle
=
_~&euml;&Eacute;W=~=
i&Eacute;&Ouml;&euml;W=&Auml;=
_~&euml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;W= β =
s&Eacute;&ecirc;&iacute;&Eacute;&ntilde;=~&aring;&Ouml;&auml;&Eacute;W= α =
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Auml;~&euml;&Eacute;W=&Uuml;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 11.
=
172. β = VM&deg; −
α
=
O
=
173. &Uuml; O = &Auml;O −
O
~
=
Q
27
CHAPTER 3. GEOMETRY
174. i = ~ + O&Auml; =
=
175. p =
O
~&Uuml; &Auml;
= &euml;&aacute;&aring; α =
O
O
=
=
=
3.3 Equilateral Triangle
=
p&aacute;&Ccedil;&Eacute;=&ccedil;&Ntilde;=~=&Eacute;&egrave;&igrave;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W=~=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 12.
=
176. &Uuml; =
~ P
=
O
=
28
CHAPTER 3. GEOMETRY
O
~ P
=
177. o = &Uuml; =
P
P
=
N
~ P o
= =
178. &ecirc; = &Uuml; =
P
S
O
=
179. i = P~ =
=
180. p =
O
~&Uuml; ~ P
=
=
O
Q
=
=
=
3.4 Scalene Triangle
E^=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=&iuml;&aacute;&iacute;&Uuml;=&aring;&ccedil;=&iacute;&iuml;&ccedil;=&euml;&aacute;&Ccedil;&Eacute;&euml;=&Eacute;&egrave;&igrave;~&auml;F=
=
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W=~I=&Auml;I=&Aring;=
~ +&Auml;+&Aring;
==
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W= &eacute; =
O
^&aring;&Ouml;&auml;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;W= αI βI γ =
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&euml;&aacute;&Ccedil;&Eacute;&euml;=~I=&Auml;I=&Aring;W= &Uuml; ~ I &Uuml; &Auml; I &Uuml; &Aring; =
j&Eacute;&Ccedil;&aacute;~&aring;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&euml;&aacute;&Ccedil;&Eacute;&euml;=~I=&Auml;I=&Aring;W= &atilde; ~ I &atilde; &Auml; I &atilde; &Aring; =
_&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;&euml;= αI βI γ W= &iacute; ~ I &iacute; &Auml; I &iacute; &Aring; =
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
^&ecirc;&Eacute;~W=p=
=
=
29
CHAPTER 3. GEOMETRY
=
=====
=
Figure 13.
=
181. α + β + γ = NUM&deg; =
182. ~ + &Auml; &gt; &Aring; I==
&Auml; + &Aring; &gt; ~ I==
~ + &Aring; &gt; &Auml; K=
=
183. ~ − &Auml; &lt; &Aring; I==
&Auml; − &Aring; &lt; ~ I==
~ − &Aring; &lt; &Auml; K=
=
=
184. j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;=
~
&egrave; = I= &egrave; &ouml;&ouml; ~ K=
O
=
=
=
=====
Figure 14.
=
30
CHAPTER 3. GEOMETRY
185. i~&iuml;=&ccedil;&Ntilde;=`&ccedil;&euml;&aacute;&aring;&Eacute;&euml;=
~ O = &Auml;O + &Aring; O − O&Auml;&Aring; &Aring;&ccedil;&euml; α I=
&Auml;O = ~ O + &Aring; O − O~&Aring; &Aring;&ccedil;&euml; β I=
&Aring; O = ~ O + &Auml;O − O~&Auml; &Aring;&ccedil;&euml; γ K=
=
186. i~&iuml;=&ccedil;&Ntilde;=p&aacute;&aring;&Eacute;&euml;=
~
&Auml;
&Aring;
=
=
= Oo I==
&euml;&aacute;&aring; α &euml;&aacute;&aring; β &euml;&aacute;&aring; γ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=o=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;K==
=
~
&Auml;
&Aring;
&Auml;&Aring;
~&Aring;
~&Auml; ~&Auml;&Aring;
=
=
=
=
=
=
187. o =
=
O &euml;&aacute;&aring; α O &euml;&aacute;&aring; β O &euml;&aacute;&aring; γ O&Uuml; ~ O&Uuml; &Auml; O&Uuml; &Aring; Qp
=
(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I==
188. &ecirc; O =
&eacute;
N N
N
N
= +
+ K=
&ecirc; &Uuml;~ &Uuml;&Auml; &Uuml;&Aring;
=
(&eacute; − &Auml;)(&eacute; − &Aring; ) I=
α
189. &euml;&aacute;&aring; =
O
&Auml;&Aring;
&Aring;&ccedil;&euml;
α
&eacute;(&eacute; − ~ )
I=
=
O
&Auml;&Aring;
&iacute;~&aring;
α
=
O
(&eacute; − &Auml;)(&eacute; − &Aring; ) K=
&eacute;(&eacute; − ~ )
=
O
190. &Uuml; ~ =
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I=
~
O
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) I=
&Uuml;&Auml; =
&Auml;
O
&Uuml;&Aring; =
&eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) K=
&Aring;
31
CHAPTER 3. GEOMETRY
191. &Uuml; ~ = &Auml; &euml;&aacute;&aring; γ = &Aring; &euml;&aacute;&aring; β I=
&Uuml; &Auml; = ~ &euml;&aacute;&aring; γ = &Aring; &euml;&aacute;&aring; α I=
&Uuml; &Aring; = ~ &euml;&aacute;&aring; β = &Auml; &euml;&aacute;&aring; α K=
=
&Auml; +&Aring; ~
− I==
O
Q
O
O
~ + &Aring; &Auml;O
&atilde; O&Auml; =
− I==
O
Q
O
O
~ + &Auml; &Aring;O
O
&atilde;&Aring; =
− K=
O
Q
192. &atilde; O~ =
O
O
O
=
=
=
=====
Figure 15.
=
O
O
O
193. ^j = &atilde; ~ I= _j = &atilde; &Auml; I= `j = &atilde; &Aring; =Ec&aacute;&Ouml;KNRFK=
P
P
P
=
Q&Auml;&Aring;&eacute;(&eacute; − ~ )
194. &iacute; O~ =
I==
(&Auml; + &Aring; )O
Q~&Aring;&eacute;(&eacute; − &Auml;)
&iacute; O&Auml; =
I==
(~ + &Aring; )O
Q~&Auml;&eacute;(&eacute; − &Aring; )
&iacute; O&Aring; =
K=
(~ + &Auml;)O
=
32
CHAPTER 3. GEOMETRY
~&Uuml; ~ &Auml;&Uuml; &Auml; &Aring;&Uuml; &Aring;
=
=
I==
O
O
O
~&Auml; &euml;&aacute;&aring; γ ~&Aring; &euml;&aacute;&aring; β &Auml;&Aring; &euml;&aacute;&aring; α
I==
p=
=
=
O
O
O
p = &eacute;(&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; ) =Ee&Eacute;&ecirc;&ccedil;&aring;∞&euml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~FI=
p = &eacute;&ecirc; I==
~&Auml;&Aring;
p=
I=
Qo
p = Oo O &euml;&aacute;&aring; α &euml;&aacute;&aring; β &euml;&aacute;&aring; γ I=
α
β
γ
p = &eacute;O &iacute;~&aring; &iacute;~&aring; &iacute;~&aring; K=
O
O
O
195. p =
=
=
=
3.5 Square
p&aacute;&Ccedil;&Eacute;=&ccedil;&Ntilde;=~=&euml;&egrave;&igrave;~&ecirc;&Eacute;W=~=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
Figure 16.
33
CHAPTER 3. GEOMETRY
196. &Ccedil; = ~ O ==
=
197. o =
&Ccedil; ~ O
=
=
O
O
=
~
198. &ecirc; = =
O
199. i = Q~ =
=
=
200. p = ~ =
=
=
=
O
3.6 Rectangle
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;W=~I=&Auml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 17.
=
201. &Ccedil; = ~ O + &Auml;O ==
34
CHAPTER 3. GEOMETRY
202. o =
&Ccedil;
=
O
=
203. i = O(~ + &Auml;) =
=
204. p = ~&Auml; =
=
=
=
3.7 Parallelogram
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&ccedil;&Ouml;&ecirc;~&atilde;W=~I=&Auml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
`&ccedil;&aring;&euml;&Eacute;&Aring;&igrave;&iacute;&aacute;&icirc;&Eacute;=~&aring;&Ouml;&auml;&Eacute;&euml;W= αI β =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= ϕ =
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;==
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=====
=
Figure 18.
=
205. α + β = NUM&deg; =
206. &Ccedil; + &Ccedil; = O(~ + &Auml; ) =
O
N
O
O
O
=
O
=
35
CHAPTER 3. GEOMETRY
207. &Uuml; = &Auml; &euml;&aacute;&aring; α = &Auml; &euml;&aacute;&aring; β =
208. i = O(~ + &Auml;) =
209. p = ~&Uuml; = ~&Auml; &euml;&aacute;&aring; α I==
N
p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ K=
O
=
=
=
=
=
3.8 Rhombus
=
p&aacute;&Ccedil;&Eacute;=&ccedil;&Ntilde;=~=&ecirc;&Uuml;&ccedil;&atilde;&Auml;&igrave;&euml;W=~=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
`&ccedil;&aring;&euml;&Eacute;&Aring;&igrave;&iacute;&aacute;&icirc;&Eacute;=~&aring;&Ouml;&auml;&Eacute;&euml;W= αI β =
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=e=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
=====
Figure 19.
=
36
CHAPTER 3. GEOMETRY
210. α + β = NUM&deg; =
=
211. &Ccedil; + &Ccedil; = Q~ =
O
N
O
O
O
=
212. &Uuml; = ~ &euml;&aacute;&aring; α =
&Ccedil;N&Ccedil; O
=
O~
=
&Uuml; &Ccedil;&Ccedil;
~ &euml;&aacute;&aring; α
213. &ecirc; = = N O =
=
O
Q~
O
=
214. i = Q~ =
=
215. p = ~&Uuml; = ~ &euml;&aacute;&aring; α I==
N
p = &Ccedil;N&Ccedil; O K=
O
=
=
=
O
3.9 Trapezoid
=
_~&euml;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W=~I=&Auml;=
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W=&egrave;=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
^&ecirc;&Eacute;~W=p=
=
=
37
CHAPTER 3. GEOMETRY
=
=
Figure 20.
=
216. &egrave; =
217. p =
~+&Auml;
=
O
~+&Auml;
⋅ &Uuml; = &egrave;&Uuml; =
O
=
=
=
=
3.10 Isosceles Trapezoid
=
_~&euml;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W=~I=&Auml;=
i&Eacute;&Ouml;W=&Aring;=
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W=&egrave;=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
^&ecirc;&Eacute;~W=p=
=
=
38
CHAPTER 3. GEOMETRY
=
=
Figure 21.
=
218. &egrave; =
~+&Auml;
=
O
=
219. &Ccedil; = ~&Auml; + &Aring; =
=
N
O
220. &Uuml; = &Aring; O − (&Auml; − ~ ) =
Q
O
=
&Aring; ~&Auml; + &Aring; O
=
(O&Aring; − ~ + &Auml;)(O&Aring; + ~ − &Auml;)
=
~+&Auml;
222. p =
⋅ &Uuml; = &egrave;&Uuml; =
O
=
=
=
=
=
=
221. o =
39
CHAPTER 3. GEOMETRY
3.11 Isosceles Trapezoid with
Inscribed Circle
=
_~&euml;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W=~I=&Auml;=
i&Eacute;&Ouml;W=&Aring;=
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W=&egrave;=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 22.
=
223. ~ + &Auml; = O&Aring; =
=
~+&Auml;
224. &egrave; =
=&Aring;=
O
=
225. &Ccedil; = &Uuml; + &Aring; =
O
O
O
=
40
CHAPTER 3. GEOMETRY
226. &ecirc; =
&Uuml;
~&Auml;
=
=
O
O
=
&Auml;
&Aring;&Ccedil; &Aring;&Ccedil; &Aring;
&Aring;
&Aring;
~+&Auml; ~
N+
&Uuml;O + &Aring; O =
=
=
=
+S+ =
O&Uuml; Q&ecirc; O
~&Auml; O&Uuml;
U
&Auml;
~
=
228. i = O(~ + &Auml;) = Q&Aring; =
=
(~ + &Auml;) ~&Auml; = &egrave;&Uuml; = &Aring;&Uuml; = i&ecirc; ==
~+&Auml;
⋅&Uuml; =
229. p =
O
O
O
=
=
=
227. o =
O
3.12 Trapezoid with Inscribed Circle
=
_~&euml;&Eacute;&euml;=&ccedil;&Ntilde;=~=&iacute;&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;W=~I=&Auml;=
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&aacute;&Ccedil;&Eacute;&euml;W=&Aring;I=&Ccedil;=
j&aacute;&Ccedil;&auml;&aacute;&aring;&Eacute;W=&egrave;=
^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;W=&Uuml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= ϕ =
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
41
CHAPTER 3. GEOMETRY
=
=
Figure 23.
=
230. ~ + &Auml; = &Aring; + &Ccedil; =
~+&Auml; &Aring;+&Ccedil;
=
=
231. &egrave; =
O
O
232. i = O(~ + &Auml;) = O(&Aring; + &Ccedil; ) =
=
=
=
~+&Auml;
&Aring;+&Ccedil;
⋅&Uuml; =
⋅ &Uuml; = &egrave;&Uuml; I==
O
O
N
p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ K=
O
233. p =
=
=
=
3.13 Kite
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&acirc;&aacute;&iacute;&Eacute;W=~I=&Auml;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
^&aring;&Ouml;&auml;&Eacute;&euml;W= αI βI γ =
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
42
CHAPTER 3. GEOMETRY
=
=
Figure 24.
=
234. α + β + Oγ = PSM&deg; =
235. i = O(~ + &Auml;) =
=
=
236. p =
&Ccedil;N&Ccedil; O
=
O
=
=
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W=~I=&Auml;I=&Aring;I=&Ccedil;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= ϕ =
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml;=~&aring;&Ouml;&auml;&Eacute;&euml;W= αI βI γ I δ =
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&eacute;==
^&ecirc;&Eacute;~W=p=
43
CHAPTER 3. GEOMETRY
=
=
Figure 25.
=
237. α + γ = β + δ = NUM&deg; =
=
238. m&iacute;&ccedil;&auml;&Eacute;&atilde;&oacute;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
~&Aring; + &Auml;&Ccedil; = &Ccedil;N&Ccedil; O =
239. i = ~ + &Auml; + &Aring; + &Ccedil; =
=
=
N (~&Aring; + &Auml;&Ccedil; )(~&Ccedil; + &Auml;&Aring; )(~&Auml; + &Aring;&Ccedil; )
I==
240. o =
Q (&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; )(&eacute; − &Ccedil; )
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &eacute; = K=
O
=
N
241. p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ I==
O
p = (&eacute; − ~ )(&eacute; − &Auml;)(&eacute; − &Aring; )(&eacute; − &Ccedil; ) I==
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &eacute; = K=
O
=
=
=
44
CHAPTER 3. GEOMETRY
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W=~I=&Auml;I=&Aring;I=&Ccedil;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= ϕ =
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&eacute;==
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 26.
=
242. ~ + &Aring; = &Auml; + &Ccedil; =
=
243. i = ~ + &Auml; + &Aring; + &Ccedil; = O(~ + &Aring; ) = O(&Auml; + &Ccedil; ) =
=
&Ccedil;NO&Ccedil; OO − (~ − &Auml;) (~ + &Auml; − &eacute; )
I==
O&eacute;
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &eacute; = K==
O
=
O
O
244. &ecirc; =
45
CHAPTER 3. GEOMETRY
N
245. p = &eacute;&ecirc; = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ =
O
=
=
=
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=~=&egrave;&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;W=~I=&Auml;I=&Aring;I=&Ccedil;=
a&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= &Ccedil;N I &Ccedil; O =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;&euml;W= ϕ =
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml;=~&aring;&Ouml;&auml;&Eacute;&euml;W= αI βI γ I δ =
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
=======
Figure 27.
=
246. α + β + γ + δ = PSM&deg; =
247. i = ~ + &Auml; + &Aring; + &Ccedil; =
=
=
46
CHAPTER 3. GEOMETRY
N
248. p = &Ccedil;N&Ccedil; O &euml;&aacute;&aring; ϕ =
O
=
=
=
3.17 Regular Hexagon
=
p&aacute;&Ccedil;&Eacute;W=~=
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml;=~&aring;&Ouml;&auml;&Eacute;W= α =
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&eacute;==
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 28.
=
249. α = NOM&deg; =
=
250. &ecirc; = &atilde; =
~ P
=
O
47
CHAPTER 3. GEOMETRY
251. o = ~ =
=
252. i = S~ =
=
O
~ P P
I==
O
i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &eacute; = K=
O
=
=
=
253. p = &eacute;&ecirc; =
3.18 Regular Polygon
=
p&aacute;&Ccedil;&Eacute;W=~=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&euml;&aacute;&Ccedil;&Eacute;&euml;W=&aring;=
f&aring;&iacute;&Eacute;&ecirc;&aring;~&auml;=~&aring;&Ouml;&auml;&Eacute;W= α =
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&eacute;==
^&ecirc;&Eacute;~W=p=
=
=
48
CHAPTER 3. GEOMETRY
=
=
Figure 29.
=
254. α =
255. α =
&aring;−O
⋅ NUM&deg; =
O
=
&aring;−O
⋅ NUM&deg; =
O
=
256. o =
~
π
O &euml;&aacute;&aring;
&aring;
=
=
257. &ecirc; = &atilde; =
~
O &iacute;~&aring;
π
&aring;
= oO −
~O
=
Q
=
258. i = &aring;~ =
=
259. p =
&aring;o
Oπ
&euml;&aacute;&aring; I==
O
&aring;
O
p = &eacute;&ecirc; = &eacute; o O −
~O
I==
Q
49
CHAPTER 3. GEOMETRY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &eacute; =
i
K==
O
=
=
=
3.19 Circle
=
o~&Ccedil;&aacute;&igrave;&euml;W=o=
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&Ccedil;=
`&Uuml;&ccedil;&ecirc;&Ccedil;W=~=
p&Eacute;&Aring;~&aring;&iacute;=&euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;&euml;W=&Eacute;I=&Ntilde;=
q~&aring;&Ouml;&Eacute;&aring;&iacute;=&euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;W=&Ouml;=
`&Eacute;&aring;&iacute;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;W= α =
f&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=~&aring;&Ouml;&auml;&Eacute;W= β =
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
α
260. ~ = Oo &euml;&aacute;&aring; =
O
=
=
=
Figure 30.
=
50
CHAPTER 3. GEOMETRY
261. ~N~ O = &Auml;N&Auml;O =
=
=
=
Figure 31.
=
262. &Eacute;&Eacute;N = &Ntilde;&Ntilde;N =
=
=
=
=====
Figure 32.
=
263. &Ouml; O = &Ntilde;&Ntilde;N =
=
51
CHAPTER 3. GEOMETRY
=
=====
=
Figure 33.
=
264. β =
α
=
O
=
=
=
Figure 34.
=
265. i = Oπo = π&Ccedil; =
=
266. p = πo O =
io
π&Ccedil;
=
==
Q
O
O
=
52
CHAPTER 3. GEOMETRY
3.20 Sector of a Circle
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=~=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
^&ecirc;&Aring;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W=&euml;=
`&Eacute;&aring;&iacute;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=E&aacute;&aring;=&ecirc;~&Ccedil;&aacute;~&aring;&euml;FW=&ntilde;=
`&Eacute;&aring;&iacute;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=E&aacute;&aring;=&Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;FW= α =
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 35.
=
267. &euml; = o&ntilde; =
268. &euml; =
=
πoα
=
NUM&deg;
=
269. i = &euml; + Oo =
=
270. p =
o&euml; o &ntilde; πo α
=
=
==
O
O
PSM&deg;
O
O
=
=
53
CHAPTER 3. GEOMETRY
3.21 Segment of a Circle
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=~=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
^&ecirc;&Aring;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W=&euml;=
`&Uuml;&ccedil;&ecirc;&Ccedil;W=~=
`&Eacute;&aring;&iacute;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=E&aacute;&aring;=&ecirc;~&Ccedil;&aacute;~&aring;&euml;FW=&ntilde;=
`&Eacute;&aring;&iacute;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=E&aacute;&aring;=&Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;FW= α =
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;W=&Uuml;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
=
=
Figure 36.
=
271. ~ = O O&Uuml;o − &Uuml; O =
=
N
272. &Uuml; = o −
Qo O − ~ O I= &Uuml; &lt; o =
O
=
273. i = &euml; + ~ =
=
54
CHAPTER 3. GEOMETRY
O
O
N
[&euml;o − ~(o − &Uuml; )] = o  απ − &euml;&aacute;&aring; α  = o (&ntilde; − &euml;&aacute;&aring; &ntilde; ) I==
O
O  NUM&deg;
 O
O
p ≈ &Uuml;~ K=
P
274. p =
=
=
=
3.22 Cube
=
b&Ccedil;&Ouml;&Eacute;W=~==
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=&ecirc;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=&ecirc;=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
===
Figure 37.
=
275. &Ccedil; = ~ P =
=
~
276. &ecirc; = =
O
=
55
CHAPTER 3. GEOMETRY
277. o =
~ P
=
O
=
278. p = S~ =
O
=
279. s = ~ ==
=
=
=
P
3.23 Rectangular Parallelepiped
=
b&Ccedil;&Ouml;&Eacute;&euml;W=~I=&Auml;I=&Aring;==
a&aacute;~&Ouml;&ccedil;&aring;~&auml;W=&Ccedil;=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
=====
Figure 38.
=
280. &Ccedil; = ~ O + &Auml;O + &Aring; O =
281. p = O(~&Auml; + ~&Aring; + &Auml;&Aring; ) =
282. s = ~&Auml;&Aring; ==
=
=
56
CHAPTER 3. GEOMETRY
3.24 Prism
=
i~&iacute;&Eacute;&ecirc;~&auml;=&Eacute;&Ccedil;&Ouml;&Eacute;W=&auml;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
i~&iacute;&Eacute;&ecirc;~&auml;=~&ecirc;&Eacute;~W= p i =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
=====
Figure 39.
=
283. p = p i + Op_ K==
=
284. i~&iacute;&Eacute;&ecirc;~&auml;=^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=o&aacute;&Ouml;&Uuml;&iacute;=m&ecirc;&aacute;&euml;&atilde;=
p i = (~ N + ~ O + ~ P + K + ~ &aring; )&auml; =
=
285. i~&iacute;&Eacute;&ecirc;~&auml;=^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~&aring;=l&Auml;&auml;&aacute;&egrave;&igrave;&Eacute;=m&ecirc;&aacute;&euml;&atilde;=
p i = &eacute;&auml; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&eacute;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&ecirc;&ccedil;&euml;&euml;=&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;K=
=
57
CHAPTER 3. GEOMETRY
286. s = p_ &Uuml; =
=
287. `~&icirc;~&auml;&aacute;&Eacute;&ecirc;&aacute;D&euml;=m&ecirc;&aacute;&aring;&Aring;&aacute;&eacute;&auml;&Eacute;==
d&aacute;&icirc;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&euml;&ccedil;&auml;&aacute;&Ccedil;&euml;=&aacute;&aring;&Aring;&auml;&igrave;&Ccedil;&Eacute;&Ccedil;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&eacute;&auml;~&aring;&Eacute;&euml;K=f&Ntilde;=&Eacute;&icirc;&Eacute;&ecirc;&oacute;=
&eacute;&auml;~&aring;&Eacute;=&Aring;&ecirc;&ccedil;&euml;&euml;=&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&eacute;&auml;~&aring;&Eacute;&euml;=&Uuml;~&euml;=&iacute;&Uuml;&Eacute;=&euml;~&atilde;&Eacute;=
~&ecirc;&Eacute;~=&aacute;&aring;=&Auml;&ccedil;&iacute;&Uuml;=&euml;&ccedil;&auml;&aacute;&Ccedil;&euml;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&ccedil;&auml;&aacute;&Ccedil;&euml;=~&ecirc;&Eacute;=&Eacute;&egrave;&igrave;~&auml;K=
=
=
=
3.25 Regular Tetrahedron
=
q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=&euml;&aacute;&Ccedil;&Eacute;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W=~=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
Figure 40.
=
288. &Uuml; =
O
~=
P
=
58
CHAPTER 3. GEOMETRY
289. p_ =
P~ O
=
Q
=
290. p = P~ =
=
N
~P
291. s = p_ &Uuml; =
K==
P
S O
=
=
=
O
3.26 Regular Pyramid
=
p&aacute;&Ccedil;&Eacute;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=~=
i~&iacute;&Eacute;&ecirc;~&auml;=&Eacute;&Ccedil;&Ouml;&Eacute;W=&Auml;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;==
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&euml;&aacute;&Ccedil;&Eacute;&euml;W=&aring;==
p&Eacute;&atilde;&aacute;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=&eacute;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=&ecirc;=
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= pi =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
59
CHAPTER 3. GEOMETRY
=
=
Figure 41.
=
292. &atilde; = &Auml;O −
~O
=
Q
=
293. &Uuml; =
π O
−~
&aring;
=
π
O &euml;&aacute;&aring;
&aring;
Q&Auml;O &euml;&aacute;&aring; O
=
N
N
294. p i = &aring;~&atilde; = &aring;~ Q&Auml;O − ~ O = &eacute;&atilde; =
O
Q
=
295. p_ = &eacute;&ecirc; =
=
296. p = p_ + p i =
=
N
N
297. s = p_ &Uuml; = &eacute;&ecirc;&Uuml; ==
P
P
=
=
=
60
CHAPTER 3. GEOMETRY
3.27 Frustum of a Regular Pyramid
=
~N I ~ O I ~ P IKI ~ &aring;
=
_~&euml;&Eacute;=~&aring;&Ccedil;=&iacute;&ccedil;&eacute;=&euml;&aacute;&Ccedil;&Eacute;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;&euml;W= 
&Auml;N I &Auml;O I &Auml;P IKI &Auml;&aring;
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;==
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;&euml;W= pN I= pO =
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= p i =
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;&euml;W= mN I= mO =
p&Aring;~&auml;&Eacute;=&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;W=&acirc;=
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
Figure 42.
=
298.
&Auml;N &Auml;O &Auml;P
&Auml;
&Auml;
= = =K= &aring; = = &acirc; =
~N ~ O ~ P
~&aring; ~
=
61
CHAPTER 3. GEOMETRY
299.
pO
= &acirc;O =
pN
=
&atilde;(mN + mO )
=
300. p i =
O
=
301. p = p i + pN + pO =
=
&Uuml;
302. s = pN + pNpO + pO =
P
=
O
&Uuml;p  &Auml;  &Auml;   &Uuml;p
303. s = N N + +    = N N + &acirc; + &acirc; O =
P  ~  ~   P
=
=
=
(
)
[
]
3.28 Rectangular Right Wedge
=
p&aacute;&Ccedil;&Eacute;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=~I=&Auml;=
q&ccedil;&eacute;=&Eacute;&Ccedil;&Ouml;&Eacute;W=&Aring;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= p i =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
62
CHAPTER 3. GEOMETRY
=
=
Figure 43.
=
N
(~ + &Aring; ) Q&Uuml;O + &Auml;O + &Auml; &Uuml;O + (~ − &Aring; )O =
O
=
305. p_ = ~&Auml; =
=
306. p = p_ + p i =
=
&Auml;&Uuml;
(O~ + &Aring; ) =
307. s =
S
=
=
=
304. p i =
3.29 Platonic Solids
=
b&Ccedil;&Ouml;&Eacute;W=~=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=&ecirc;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&igrave;&atilde;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W=o=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
63
CHAPTER 3. GEOMETRY
308. c&aacute;&icirc;&Eacute;=m&auml;~&iacute;&ccedil;&aring;&aacute;&Aring;=p&ccedil;&auml;&aacute;&Ccedil;&euml;=
q&Uuml;&Eacute;= &eacute;&auml;~&iacute;&ccedil;&aring;&aacute;&Aring;= &euml;&ccedil;&auml;&aacute;&Ccedil;&euml;= ~&ecirc;&Eacute;= &Aring;&ccedil;&aring;&icirc;&Eacute;&ntilde;= &eacute;&ccedil;&auml;&oacute;&Uuml;&Eacute;&Ccedil;&ecirc;~= &iuml;&aacute;&iacute;&Uuml;= &Eacute;&egrave;&igrave;&aacute;&icirc;~&auml;&Eacute;&aring;&iacute;=
&Ntilde;~&Aring;&Eacute;&euml;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&euml;&Eacute;&Ccedil;=&ccedil;&Ntilde;=&Aring;&ccedil;&aring;&Ouml;&ecirc;&igrave;&Eacute;&aring;&iacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ntilde;=&ecirc;&Eacute;&Ouml;&igrave;&auml;~&ecirc;=&eacute;&ccedil;&auml;&oacute;&Ouml;&ccedil;&aring;&euml;K==
=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
p&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=
p&ccedil;&auml;&aacute;&Ccedil;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=
&ccedil;&Ntilde;=s&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml; &ccedil;&Ntilde;=b&Ccedil;&Ouml;&Eacute;&euml;=
&ccedil;&Ntilde;=c~&Aring;&Eacute;&euml;=
q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;==
Q=
S=
Q=
PKOR=
`&igrave;&Auml;&Eacute;=
U=
NO=
S=
PKOO=
l&Aring;&iacute;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;=
S=
NO=
U=
PKOT=
f&Aring;&ccedil;&euml;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;=
NO=
PM=
OM=
PKOT=
a&ccedil;&Ccedil;&Eacute;&Aring;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;=
OM=
PM=
NO=
PKOT=
=
=
Octahedron
=
=
=
Figure 44.
=
309. &ecirc; =
~ S
=
S
=
310. o =
~ O
=
O
=
64
CHAPTER 3. GEOMETRY
311. p = O~ O P =
=
~P O
312. s =
=
P
=
=
Icosahedron
=
=
=
Figure 45.
=
313. &ecirc; =
(
=
314. o =
)
~ P P+ R
=
NO
(
)
~
O R+ R =
Q
=
315. p = R~ O P =
=
R~ P P + R
316. s =
=
NO
=
=
(
)
65
CHAPTER 3. GEOMETRY
Dodecahedron
=
=
=
Figure 46.
317. &ecirc; =
(
~ NM OR + NN R
=
O
=
318. o =
)
=
(
)
~ P N+ R
=
Q
=
(
)
319. p = P~ O R R + O R =
=
~ P NR + T R
320. s =
=
Q
=
=
=
(
)
3.30 Right Circular Cylinder
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=o=
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=&Ccedil;=
66
CHAPTER 3. GEOMETRY
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=e=
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= p i =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=====
=
Figure 47.
=
321. p i = Oπoe =
=
&Ccedil;

322. p = p i + Op_ = Oπo(e + o ) = π&Ccedil; e +  =
O

=
323. s = p_ e = πo O e =
=
=
=
67
CHAPTER 3. GEOMETRY
3.31 Right Circular Cylinder with
an Oblique Plane Face
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=o=
q&Uuml;&Eacute;=&Ouml;&ecirc;&Eacute;~&iacute;&Eacute;&euml;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;=&ccedil;&Ntilde;=~=&euml;&aacute;&Ccedil;&Eacute;W= &Uuml;N =
q&Uuml;&Eacute;=&euml;&Uuml;&ccedil;&ecirc;&iacute;&Eacute;&euml;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;=&ccedil;&Ntilde;=~=&euml;&aacute;&Ccedil;&Eacute;W= &Uuml; O =
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= p i =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&eacute;&auml;~&aring;&Eacute;=&Eacute;&aring;&Ccedil;=&Ntilde;~&Aring;&Eacute;&euml;W= p_ =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
Figure 48.
=
324. p i = πo(&Uuml;N + &Uuml; O ) =
=
O
 &Uuml; − &Uuml;O 
325. p_ = πo + πo o +  N
 =
 O 
=
O
O
68
CHAPTER 3. GEOMETRY
O

&Uuml;N − &Uuml; O  

O
326. p = p i + p_ = πo &Uuml;N + &Uuml; O + o + o + 
 =
 O  

=
πo O
(&Uuml;N + &Uuml;O ) =
327. s =
O
=
=
=
3.32 Right Circular Cone
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=o=
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=&Ccedil;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=e=
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;=
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= pi =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W= p_ =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
=
Figure 49.
69
CHAPTER 3. GEOMETRY
328. e = &atilde; O − o O =
=
π&atilde;&Ccedil;
329. p i = πo&atilde; =
=
O
=
330. p_ = πo O =
=
N 
&Ccedil;
331. p = p i + p_ = πo (&atilde; + o ) = π&Ccedil; &atilde; +  =
O 
O
=
N
N
332. s = p_ e = πo O e =
P
P
=
=
=
3.33 Frustum of a Right Circular Cone
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;&euml;W=oI=&ecirc;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=e=
p&auml;~&aring;&iacute;=&Uuml;&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&atilde;=
p&Aring;~&auml;&Eacute;=&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;W=&acirc;=
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;&euml;W= pN I= pO =
i~&iacute;&Eacute;&ecirc;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W= p i =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
70
CHAPTER 3. GEOMETRY
=
=
Figure 50.
=
333. e = &atilde; O − (o − &ecirc; ) =
=
o
334.
=&acirc;=
&ecirc;
=
p oO
335. O = O = &acirc; O =
pN &ecirc;
=
336. p i = π&atilde;(o + &ecirc; ) =
=
337. p = pN + pO + p i = π o O + &ecirc; O + &atilde;(o + &ecirc; ) =
=
&Uuml;
338. s = pN + pNpO + pO =
P
=
O
&Uuml;pN  o  o   &Uuml;pN
339. s =
N+ &acirc; + &acirc;O =
N + +    =
P  &ecirc;  &ecirc;   P
=
=
=
O
[
(
]
)
[
71
]
CHAPTER 3. GEOMETRY
3.34 Sphere
=
o~&Ccedil;&aacute;&igrave;&euml;W=o=
a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&Ccedil;=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
=
Figure 51.
=
340. p = Qπo O =
=
Q
N
N
341. s = πo P e = π&Ccedil; P = po =
P
S
P
=
=
=
3.35 Spherical Cap
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;W=&ecirc;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&eacute;&auml;~&aring;&Eacute;=&Ntilde;~&Aring;&Eacute;W= p_ =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&Aring;~&eacute;W= p` =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
72
CHAPTER 3. GEOMETRY
=
=
Figure 52.
=
342. o =
&ecirc; O + &Uuml;O
=
O&Uuml;
=
343. p_ = π&ecirc; O =
=
344. p` = π(&Uuml; O + &ecirc; O )=
=
345. p = p_ + p` = π(&Uuml; O + O&ecirc; O ) = π(Oo&Uuml; + &ecirc; O ) =
=
π
π
346. s = &Uuml; O (Po − &Uuml; ) = &Uuml;(P&ecirc; O + &Uuml; O ) =
S
S
=
=
=
3.36 Spherical Sector
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&Aring;~&eacute;W=&ecirc;=
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
73
CHAPTER 3. GEOMETRY
======
=
===
=
Figure 53.
=
347. p = πo(O&Uuml; + &ecirc; ) =
=
O
348. s = πo O &Uuml; =
P
=
k&ccedil;&iacute;&Eacute;W= q&Uuml;&Eacute;= &Ouml;&aacute;&icirc;&Eacute;&aring;= &Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;= ~&ecirc;&Eacute;= &Aring;&ccedil;&ecirc;&ecirc;&Eacute;&Aring;&iacute;= &Auml;&ccedil;&iacute;&Uuml;= &Ntilde;&ccedil;&ecirc;= &plusmn;&ccedil;&eacute;&Eacute;&aring;≤= ~&aring;&Ccedil;=
&plusmn;&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;≤=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;K=
=
=
=
3.37 Spherical Segment
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=o=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Auml;~&euml;&Eacute;&euml;W= &ecirc;N I= &ecirc;O =
e&Eacute;&aacute;&Ouml;&Uuml;&iacute;W=&Uuml;=
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W= pp =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&eacute;&auml;~&aring;&Eacute;=&Eacute;&aring;&Ccedil;=&Ntilde;~&Aring;&Eacute;&euml;W= pN I= pO =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
74
CHAPTER 3. GEOMETRY
=
=====
=
Figure 54.
=
349. pp = Oπo&Uuml; =
=
350. p = pp + pN + pO = π(Oo&Uuml; + &ecirc;NO + &ecirc;OO ) =
=
N
351. s = π&Uuml;(P&ecirc;NO + P&ecirc;OO + &Uuml; O )=
S
=
=
=
3.38 Spherical Wedge
=
o~&Ccedil;&aacute;&igrave;&euml;W=o=
a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=&aacute;&aring;=&Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;W=&ntilde;=
a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=&aacute;&aring;=&ecirc;~&Ccedil;&aacute;~&aring;&euml;W= α =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&auml;&igrave;&aring;&Eacute;W= p i =
q&ccedil;&iacute;~&auml;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
75
CHAPTER 3. GEOMETRY
=
=
Figure 55.
=
352. p i =
πo O
α = Oo O &ntilde; =
VM
=
353. p = πo O +
πo O
α = πo O + Oo O &ntilde; =
VM
=
354. s =
πoP
O
α = oP &ntilde; =
OTM
P
=
=
=
3.39 Ellipsoid
=
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W=~I=&Auml;I=&Aring;=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
76
CHAPTER 3. GEOMETRY
=
=======
=
Figure 56.
=
Q
355. s = π~&Auml;&Aring; =
P
=
=
=
Prolate Spheroid
=
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W=~I=&Auml;I=&Auml;=E ~ &gt; &Auml; F=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=
~ ~&ecirc;&Aring;&euml;&aacute;&aring; &Eacute; 

356. p = Oπ&Auml; &Auml; +
 I==
&Eacute;


&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &Eacute; =
~ O − &Auml;O
K=
~
=
Q
357. s = π&Auml;O~ =
P
=
77
CHAPTER 3. GEOMETRY
Oblate Spheroid
=
p&Eacute;&atilde;&aacute;-~&ntilde;&Eacute;&euml;W=~I=&Auml;I=&Auml;=E ~ &lt; &Auml; F=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
=

 &Auml;&Eacute;  
~ ~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml;   

 ~   I==
358. p = Oπ&Auml; &Auml; +


&Auml;&Eacute; L ~




&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &Eacute; =
&Auml;O − ~ O
K=
&Auml;
=
Q
359. s = π&Auml;O~ =
P
=
=
=
3.40 Circular Torus
=
j~&agrave;&ccedil;&ecirc;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;W=o=
j&aacute;&aring;&ccedil;&ecirc;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;W=&ecirc;=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
78
CHAPTER 3. GEOMETRY
==
Picture 57.
=
360. p = QπOo&ecirc; =
=
361. s = OπOo&ecirc; O =
=
=
79
=
Chapter 4
Trigonometry
=
=
=
=
^&aring;&Ouml;&auml;&Eacute;&euml;W= α I= β =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=E&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=~=&eacute;&ccedil;&aacute;&aring;&iacute;FW=&ntilde;I=&oacute;==
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&acirc;=
=
=
4.1 Radian and Degree Measures of Angles
=
362. N &ecirc;~&Ccedil; =
=
363. N&deg; =
=
364. N D =
=
365. N ? =
=
366. =
=
=
=
=
NUM&deg;
≈ RT&deg;NT DQR? =
π
π
&ecirc;~&Ccedil; ≈ MKMNTQRP &ecirc;~&Ccedil; =
NUM
π
&ecirc;~&Ccedil; ≈ MKMMMOVN &ecirc;~&Ccedil; =
NUM ⋅ SM
π
&ecirc;~&Ccedil; ≈ MKMMMMMR &ecirc;~&Ccedil; =
NUM ⋅ PSMM
^&aring;&Ouml;&auml;&Eacute;=
E&Ccedil;&Eacute;&Ouml;&ecirc;&Eacute;&Eacute;&euml;F=
^&aring;&Ouml;&auml;&Eacute;=
E&ecirc;~&Ccedil;&aacute;~&aring;&euml;F=
M=
PM= QR= SM= VM= NUM= OTM= PSM=
M=
π
=
S
π
=
Q
80
π
=
P
π
=
O
π=
Pπ
=
O
Oπ =
CHAPTER 4. TRIGONOMETRY
4.2 Definitions and Graphs of Trigonometric
Functions
=
=
=
=
Figure 58.
=
367. &euml;&aacute;&aring; α =
&oacute;
=
&ecirc;
=
368. &Aring;&ccedil;&euml; α =
&ntilde;
=
&ecirc;
=
369. &iacute;~&aring; α =
&oacute;
=
&ntilde;
=
370. &Aring;&ccedil;&iacute; α =
&ntilde;
=
&oacute;
=
81
CHAPTER 4. TRIGONOMETRY
371. &euml;&Eacute;&Aring; α =
&ecirc;
=
&ntilde;
=
372. &Aring;&ccedil;&euml;&Eacute;&Aring; α =
&ecirc;
=
&oacute;
=
373. p&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&oacute; = &euml;&aacute;&aring; &ntilde; I= − N ≤ &euml;&aacute;&aring; &ntilde; ≤ N K=
=
=
Figure 59.
=
374. `&ccedil;&euml;&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &Aring;&ccedil;&euml; &ntilde; I= − N ≤ &Aring;&ccedil;&euml; &ntilde; ≤ N K=
82
CHAPTER 4. TRIGONOMETRY
=
=
Figure 60.
=
375. q~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
π
&oacute; = &iacute;~&aring; &ntilde; I= &ntilde; ≠ (O&acirc; + N) I= − ∞ ≤ &iacute;~&aring; &ntilde; ≤ ∞K =
O
=
=
=
Figure 61.
=
83
CHAPTER 4. TRIGONOMETRY
376. `&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &Aring;&ccedil;&iacute; &ntilde; I= &ntilde; ≠ &acirc;π I== − ∞ ≤ &Aring;&ccedil;&iacute; &ntilde; ≤ ∞ K=
=
=
=
Figure 62.
=
377. p&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
π
&oacute; = &euml;&Eacute;&Aring; &ntilde; I= &ntilde; ≠ (O&acirc; + N) K=
O
==
84
CHAPTER 4. TRIGONOMETRY
=
=
Figure 63.
=
378. `&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &Aring;&ccedil;&euml; &Eacute;&Aring; &ntilde; I= &ntilde; ≠ &acirc;π K=
=
Figure 64.
85
CHAPTER 4. TRIGONOMETRY
4.3. Signs of Trigonometric Functions
379. =
=
=
=
380. =
n&igrave;~&Ccedil;&ecirc;~&aring;&iacute;=
=
f=
ff=
fff=
fs=
p&aacute;&aring;
α=
H=
H=
=
=
`&ccedil;&euml;
α=
H=
=
=
H=
q~&aring;
α=
H=
=
H=
=
`&ccedil;&iacute;
α=
H=
=
H=
=
p&Eacute;&Aring;
α=
H=
=
=
H=
`&ccedil;&euml;&Eacute;&Aring;=
α=
H=
H=
=
=
=
=
Figure 65.
=
=
=
=
=
=
=
=
=
=
86
CHAPTER 4. TRIGONOMETRY
4.4 Trigonometric Functions of Common
Angles
381. =
α&deg; = α &ecirc;~&Ccedil; =
M=
M=
π
=
PM=
S
π
=
QR=
Q
π
=
SM=
P
π
=
VM=
O
Oπ
=
NOM=
P
NUM=
π=
Pπ
=
OTM=
O
PSM= Oπ =
=
=
=
=
=
=
=
=
=
=
=
=
=
O
=
O
P
=
O
&Aring;&ccedil;&euml; α =
N=
P
=
O
O
=
O
N
=
O
N=
M=
P
=
O
M=
N
− =
O
− N=
− N=
M=
&euml;&aacute;&aring; α =
M=
N
=
O
&iacute;~&aring; α = &Aring;&ccedil;&iacute; α
M=
∞=
N
=
P=
P
&euml;&Eacute;&Aring; α =
N=
O
=
P
&Aring;&ccedil;&euml;&Eacute;&Aring; α =
∞=
O=
N=
N=
P=
N
=
P
O=
O
=
P
M=
∞=
N=
∞=
O=
O=
M=
N
P
∞=
− N=
O
=
P
∞=
M=
∞=
M=
∞=
− N=
N=
M=
∞=
N=
∞=
− P=
87
−
−O=
CHAPTER 4. TRIGONOMETRY
382. =
α&deg; = α &ecirc;~&Ccedil; =
π
=
NR=
NO
&euml;&aacute;&aring; α =
&Aring;&ccedil;&euml; α =
&iacute;~&aring; α =
&Aring;&ccedil;&iacute; α =
S− O
=
Q
S+ O
=
Q
O− P =
O+ P =
R−O R
=
R
R+O R =
NU=
π
=
NM
R −N
=
Q
NM + O R
Q
PS=
π
=
R
NM − O R
Q
R +N
=
Q
RQ=
Pπ
=
NM
R +N
=
Q
NM − O R
Q
TO=
Oπ
=
R
NM + O R
Q
R −N
=
Q
TR=
Rπ
=
NO
S+ O
=
Q
S− O
=
Q
=
=
=
4.5 Most Important Formulas
=
383. &euml;&aacute;&aring; O α + &Aring;&ccedil;&euml; O α = N =
=
384. &euml;&Eacute;&Aring; O α − &iacute;~&aring; O α = N =
=
385. &Aring;&euml;&Aring; O α − &Aring;&ccedil;&iacute; O α = N =
=
&euml;&aacute;&aring; α
=
386. &iacute;~&aring; α =
&Aring;&ccedil;&euml; α
88
NM − O R
R +N
R +N
NM − O R
R +N
NM − O R
=
NM − O R
R +N
=
R+O R =
R−O R
R
=
O+ P =
O− P =
CHAPTER 4. TRIGONOMETRY
387. &Aring;&ccedil;&iacute; α =
&Aring;&ccedil;&euml; α
=
&euml;&aacute;&aring; α
=
388. &iacute;~&aring; α ⋅ &Aring;&ccedil;&iacute; α = N =
=
N
389. &euml;&Eacute;&Aring; α =
=
&Aring;&ccedil;&euml; α
=
N
390. &Aring;&ccedil;&euml;&Eacute;&Aring; α =
=
&euml;&aacute;&aring; α
=
=
=
4.6 Reduction Formulas
=
391. =
=
=
=
=
=
=
β=
−α=
VM&deg; − α =
VM&deg; + α =
NUM&deg; − α
NUM&deg; + α
OTM&deg; − α
OTM&deg; + α
PSM&deg; − α
= PSM&deg; + α
&euml;&aacute;&aring; β =
− &euml;&aacute;&aring; α =
+ &Aring;&ccedil;&euml; α =
+ &Aring;&ccedil;&euml; α =
+ &euml;&aacute;&aring; α =
− &euml;&aacute;&aring; α =
− &Aring;&ccedil;&euml; α =
− &Aring;&ccedil;&euml; α =
− &euml;&aacute;&aring; α =
+ &euml;&aacute;&aring; α =
89
&Aring;&ccedil;&euml; β =
+ &Aring;&ccedil;&euml; α =
+ &euml;&aacute;&aring; α =
− &euml;&aacute;&aring; α =
− &Aring;&ccedil;&euml; α =
− &Aring;&ccedil;&euml; α =
− &euml;&aacute;&aring; α =
+ &euml;&aacute;&aring; α =
+ &Aring;&ccedil;&euml; α =
+ &Aring;&ccedil;&euml; α =
&iacute;~&aring; β =
− &iacute;~&aring; α =
+ &Aring;&ccedil;&iacute; α =
− &Aring;&ccedil;&iacute; α =
− &iacute;~&aring; α =
+ &iacute;~&aring; α =
+ &Aring;&ccedil;&iacute; α =
− &Aring;&ccedil;&iacute; α =
− &iacute;~&aring; α =
+ &iacute;~&aring; α =
&Aring;&ccedil;&iacute; β =
− &Aring;&ccedil;&iacute; α =
+ &iacute;~&aring; α =
− &iacute;~&aring; α =
− &Aring;&ccedil;&iacute; α =
+ &Aring;&ccedil;&iacute; α =
+ &iacute;~&aring; α =
− &iacute;~&aring; α =
− &Aring;&ccedil;&iacute; α =
+ &Aring;&ccedil;&iacute; α =
CHAPTER 4. TRIGONOMETRY
4.7 Periodicity of Trigonometric Functions
=
392. &euml;&aacute;&aring;(α &plusmn; Oπ&aring; ) = &euml;&aacute;&aring; α I=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;= Oπ =&ccedil;&ecirc;= PSM&deg; K=
=
393. &Aring;&ccedil;&euml;(α &plusmn; Oπ&aring; ) = &Aring;&ccedil;&euml; α I=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;= Oπ =&ccedil;&ecirc;= PSM&deg; K=
=
394. &iacute;~&aring;(α &plusmn; π&aring; ) = &iacute;~&aring; α I=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;= π =&ccedil;&ecirc;= NUM&deg; K=
=
395. &Aring;&ccedil;&iacute;(α &plusmn; π&aring; ) = &Aring;&ccedil;&iacute; α I=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;= π =&ccedil;&ecirc;= NUM&deg; K=
=
=
=
4.8 Relations between Trigonometric
Functions
=
396. &euml;&aacute;&aring; α = &plusmn; N − &Aring;&ccedil;&euml; O α = &plusmn;
α
O =
=
α
N + &iacute;~&aring; O
O
N
(N − &Aring;&ccedil;&euml; Oα ) = O &Aring;&ccedil;&euml; O  α − π  − N =
O
 O Q
O &iacute;~&aring;
=
=
397. &Aring;&ccedil;&euml; α = &plusmn; N − &euml;&aacute;&aring; O α = &plusmn;
α
O=
=
α
N + &iacute;~&aring; O
O
N
(N + &Aring;&ccedil;&euml; Oα ) = O &Aring;&ccedil;&euml; O α − N =
O
O
N − &iacute;~&aring; O
=
=
398. &iacute;~&aring; α =
&euml;&aacute;&aring; α
&euml;&aacute;&aring; Oα
N − &Aring;&ccedil;&euml; Oα
= &plusmn; &euml;&Eacute;&Aring; O α − N =
=
=
&Aring;&ccedil;&euml; α
N + &Aring;&ccedil;&euml; Oα
&euml;&aacute;&aring; Oα
90
CHAPTER 4. TRIGONOMETRY
α
N − &Aring;&ccedil;&euml; Oα
O =
=&plusmn;
=
N + &Aring;&ccedil;&euml; Oα
O α
N + &iacute;~&aring;
O
O &iacute;~&aring;
=
=
&Aring;&ccedil;&euml; α
N + &Aring;&ccedil;&euml; Oα
&euml;&aacute;&aring; Oα
= &plusmn; &Aring;&euml;&Aring; O α − N =
=
=
&euml;&aacute;&aring; α
&euml;&aacute;&aring; Oα
N − &Aring;&ccedil;&euml; Oα
α
N − &iacute;~&aring; O
N + &Aring;&ccedil;&euml; Oα
O=
=
= =&plusmn;
α
N − &Aring;&ccedil;&euml; Oα
O &iacute;~&aring;
O
399. &Aring;&ccedil;&iacute; α =
=
α
N
O=
400. &euml;&Eacute;&Aring; α =
= &plusmn; N + &iacute;~&aring; O α =
α
&Aring;&ccedil;&euml; α
N − &iacute;~&aring; O
O
=
α
N + &iacute;~&aring; O
N
O=
401. &Aring;&euml;&Aring; α =
= &plusmn; N + &Aring;&ccedil;&iacute; O α =
α
&euml;&aacute;&aring; α
O &iacute;~&aring;
O
=
=
=
N + &iacute;~&aring; O
4.9 Addition and Subtraction Formulas
=
402. &euml;&aacute;&aring;(α + β) = &euml;&aacute;&aring; α &Aring;&ccedil;&euml; β + &euml;&aacute;&aring; β &Aring;&ccedil;&euml; α =
=
403. &euml;&aacute;&aring;(α − &oacute; ) = &euml;&aacute;&aring; α &Aring;&ccedil;&euml; β − &euml;&aacute;&aring; β &Aring;&ccedil;&euml; α =
=
404. &Aring;&ccedil;&euml;(α + β ) = &Aring;&ccedil;&euml; α &Aring;&ccedil;&euml; β − &euml;&aacute;&aring; α &euml;&aacute;&aring; β =
=
405. &Aring;&ccedil;&euml;(α − β ) = &Aring;&ccedil;&euml; α &Aring;&ccedil;&euml; β + &euml;&aacute;&aring; α &euml;&aacute;&aring; β =
91
CHAPTER 4. TRIGONOMETRY
406. &iacute;~&aring;(α + β ) =
=
407. &iacute;~&aring;(α − β ) =
=
408. &Aring;&ccedil;&iacute;(α + β) =
=
409. &Aring;&ccedil;&iacute;(α − β) =
&iacute;~&aring; α + &iacute;~&aring; β
=
N − &iacute;~&aring; α &iacute;~&aring; β
&iacute;~&aring; α − &iacute;~&aring; β
=
N + &iacute;~&aring; α &iacute;~&aring; β
N − &iacute;~&aring; α &iacute;~&aring; β
=
&iacute;~&aring; α + &iacute;~&aring; β
N + &iacute;~&aring; α &iacute;~&aring; β
=
&iacute;~&aring; α − &iacute;~&aring; β
=
=
=
4.10 Double Angle Formulas
=
410. &euml;&aacute;&aring; Oα = O &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; α =
=
411. &Aring;&ccedil;&euml; Oα = &Aring;&ccedil;&euml; O α − &euml;&aacute;&aring; O α = N − O &euml;&aacute;&aring; O α = O &Aring;&ccedil;&euml; O α − N =
=
O &iacute;~&aring; α
O
412. &iacute;~&aring; Oα =
=
=
O
N − &iacute;~&aring; α &Aring;&ccedil;&iacute; α − &iacute;~&aring; α
=
&Aring;&ccedil;&iacute; O α − N &Aring;&ccedil;&iacute; α − &iacute;~&aring; α
=
=
413. &Aring;&ccedil;&iacute; Oα =
O &Aring;&ccedil;&iacute; α
O
=
=
=
=
=
=
92
CHAPTER 4. TRIGONOMETRY
4.11 Multiple Angle Formulas
=
414. &euml;&aacute;&aring; Pα = P &euml;&aacute;&aring; α − Q &euml;&aacute;&aring; P α = P &Aring;&ccedil;&euml; O α ⋅ &euml;&aacute;&aring; α − &euml;&aacute;&aring;P α =
=
415. &euml;&aacute;&aring; Qα = Q &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; α − U &euml;&aacute;&aring; P α ⋅ &Aring;&ccedil;&euml; α =
=
416. &euml;&aacute;&aring; Rα = R &euml;&aacute;&aring; α − OM &euml;&aacute;&aring; P α + NS &euml;&aacute;&aring; R α =
=
417. &Aring;&ccedil;&euml; Pα = Q &Aring;&ccedil;&euml;P α − P &Aring;&ccedil;&euml; α = &Aring;&ccedil;&euml; P α − P &Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; O α =
=
418. &Aring;&ccedil;&euml; Qα = U &Aring;&ccedil;&euml; Q α − U &Aring;&ccedil;&euml; O α + N =
=
419. &Aring;&ccedil;&euml; Rα = NS &Aring;&ccedil;&euml; R α − OM &Aring;&ccedil;&euml; P α + R &Aring;&ccedil;&euml; α =
=
P &iacute;~&aring; α − &iacute;~&aring; P α
420. &iacute;~&aring; Pα =
=
N − P &iacute;~&aring; O α
=
Q &iacute;~&aring; α − Q &iacute;~&aring; P α
=
421. &iacute;~&aring; Qα =
N − S &iacute;~&aring; O α + &iacute;~&aring; Q α
=
&iacute;~&aring; R α − NM &iacute;~&aring; P α + R &iacute;~&aring; α
=
422. &iacute;~&aring; Rα =
N − NM &iacute;~&aring; O α + R &iacute;~&aring; Q α
=
&Aring;&ccedil;&iacute; P α − P &Aring;&ccedil;&iacute; α
423. &Aring;&ccedil;&iacute; Pα =
=
P &Aring;&ccedil;&iacute; O α − N
=
N − S &iacute;~&aring; O α + &iacute;~&aring; Q α
==
424. &Aring;&ccedil;&iacute; Qα =
Q &iacute;~&aring; α − Q &iacute;~&aring; P α
=
93
CHAPTER 4. TRIGONOMETRY
425. &Aring;&ccedil;&iacute; Rα =
N − NM &iacute;~&aring; O α + R &iacute;~&aring; Q α
=
&iacute;~&aring; R α − NM &iacute;~&aring; P α + R &iacute;~&aring; α
=
=
=
4.12 Half Angle Formulas
=
426. &euml;&aacute;&aring;
α
N − &Aring;&ccedil;&euml; α
=
=&plusmn;
O
O
=
427. &Aring;&ccedil;&euml;
α
N + &Aring;&ccedil;&euml; α
=
=&plusmn;
O
O
=
428. &iacute;~&aring;
α
N − &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
N − &Aring;&ccedil;&euml; α
=&plusmn;
=
=
= &Aring;&euml;&Aring; α − &Aring;&ccedil;&iacute; α =
O
N + &Aring;&ccedil;&euml; α N + &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
=
429. &Aring;&ccedil;&iacute;
α
N + &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
N + &Aring;&ccedil;&euml; α
=&plusmn;
=
=
= &Aring;&euml;&Aring; α + &Aring;&ccedil;&iacute; α =
O
N − &Aring;&ccedil;&euml; α N − &Aring;&ccedil;&euml; α
&euml;&aacute;&aring; α
=
=
=
4.13 Half Angle Tangent Identities
=
α
O =
430. &euml;&aacute;&aring; α =
α
N + &iacute;~&aring; O
O
=
O &iacute;~&aring;
94
CHAPTER 4. TRIGONOMETRY
α
O=
431. &Aring;&ccedil;&euml; α =
O α
N + &iacute;~&aring;
O
=
α
O &iacute;~&aring;
O =
432. &iacute;~&aring; α =
α
N − &iacute;~&aring; O
O
=
α
N − &iacute;~&aring; O
O=
433. &Aring;&ccedil;&iacute; α =
α
O &iacute;~&aring;
O
=
=
=
N − &iacute;~&aring; O
4.14 Transforming of Trigonometric
Expressions to Product
=
434. &euml;&aacute;&aring; α + &euml;&aacute;&aring; β = O &euml;&aacute;&aring;
=
435. &euml;&aacute;&aring; α − &euml;&aacute;&aring; β = O &Aring;&ccedil;&euml;
α+β
α −β
=
&Aring;&ccedil;&euml;
O
O
α +β
α −β
=
&euml;&aacute;&aring;
O
O
=
436. &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; β = O &Aring;&ccedil;&euml;
α+β
α −β
=
&Aring;&ccedil;&euml;
O
O
=
437. &Aring;&ccedil;&euml; α − &Aring;&ccedil;&euml; β = −O &euml;&aacute;&aring;
α +β
α −β
=
&euml;&aacute;&aring;
O
O
=
95
CHAPTER 4. TRIGONOMETRY
438. &iacute;~&aring; α + &iacute;~&aring; β =
=
439. &iacute;~&aring; α − &iacute;~&aring; β =
=
440. &Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β =
=
441. &Aring;&ccedil;&iacute; α − &Aring;&ccedil;&iacute; β =
&euml;&aacute;&aring;(α + β )
=
&Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β
&euml;&aacute;&aring;(α − β )
=
&Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β
&euml;&aacute;&aring;(β + α )
=
&euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β
&euml;&aacute;&aring;(β − α )
=
&euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β
=
π

π

442. &Aring;&ccedil;&euml; α + &euml;&aacute;&aring; α = O &Aring;&ccedil;&euml; − α  = O &euml;&aacute;&aring; + α  =
Q

Q

=
π

π

443. &Aring;&ccedil;&euml; α − &euml;&aacute;&aring; α = O &euml;&aacute;&aring; − α  = O &Aring;&ccedil;&euml; + α  =
Q

Q

=
&Aring;&ccedil;&euml;(α − β)
=
444. &iacute;~&aring; α + &Aring;&ccedil;&iacute; β =
&Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; β
=
&Aring;&ccedil;&euml;(α + β )
=
445. &iacute;~&aring; α − &Aring;&ccedil;&iacute; β = −
&Aring;&ccedil;&euml; α ⋅ &euml;&aacute;&aring; β
=
α
446. N + &Aring;&ccedil;&euml; α = O &Aring;&ccedil;&euml; O =
O
=
α
447. N − &Aring;&ccedil;&euml; α = O &euml;&aacute;&aring; O =
O
=
96
CHAPTER 4. TRIGONOMETRY
π α
448. N + &euml;&aacute;&aring; α = O &Aring;&ccedil;&euml; O  −  =
Q O
=
π α
449. N − &euml;&aacute;&aring; α = O &euml;&aacute;&aring; O  −  =
Q O
=
=
=
4.15 Transforming of Trigonometric
Expressions to Sum
=
450. &euml;&aacute;&aring; α ⋅ &euml;&aacute;&aring; β =
&Aring;&ccedil;&euml;(α − β) − &Aring;&ccedil;&euml;(α + β )
=
O
=
451. &Aring;&ccedil;&euml; α ⋅ &Aring;&ccedil;&euml; β =
=
452. &euml;&aacute;&aring; α ⋅ &Aring;&ccedil;&euml; β =
=
453. &iacute;~&aring; α ⋅ &iacute;~&aring; β =
=
454. &Aring;&ccedil;&iacute; α ⋅ &Aring;&ccedil;&iacute; β =
=
455. &iacute;~&aring; α ⋅ &Aring;&ccedil;&iacute; β =
&Aring;&ccedil;&euml;(α − β ) + &Aring;&ccedil;&euml;(α + β )
=
O
&euml;&aacute;&aring;(α − β ) + &euml;&aacute;&aring;(α + β )
=
O
&iacute;~&aring; α + &iacute;~&aring; β
=
&Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β
&Aring;&ccedil;&iacute; α + &Aring;&ccedil;&iacute; β
=
&iacute;~&aring; α + &iacute;~&aring; β
&iacute;~&aring; α + &Aring;&ccedil;&iacute; β
=
&Aring;&ccedil;&iacute; α + &iacute;~&aring; β
=
=
=
97
CHAPTER 4. TRIGONOMETRY
4.16 Powers of Trigonometric Functions
=
456. &euml;&aacute;&aring; O α =
=
457. &euml;&aacute;&aring; P α =
=
458. &euml;&aacute;&aring; Q α =
=
459. &euml;&aacute;&aring; R α =
=
460. &euml;&aacute;&aring; S α =
=
461. &Aring;&ccedil;&euml; O α =
=
462. &Aring;&ccedil;&euml; P α =
=
463. &Aring;&ccedil;&euml; Q α =
=
464. &Aring;&ccedil;&euml; R α =
=
465. &Aring;&ccedil;&euml; S α =
N − &Aring;&ccedil;&euml; Oα
=
O
P &euml;&aacute;&aring; α − &euml;&aacute;&aring; Pα
=
Q
&Aring;&ccedil;&euml; Qα − Q &Aring;&ccedil;&euml; Oα + P
=
U
NM &euml;&aacute;&aring; α − R &euml;&aacute;&aring; Pα + &euml;&aacute;&aring; Rα
=
NS
NM − NR &Aring;&ccedil;&euml; Oα + S &Aring;&ccedil;&euml; Qα − &Aring;&ccedil;&euml; Sα
=
PO
N + &Aring;&ccedil;&euml; Oα
=
O
P &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; Pα
=
Q
&Aring;&ccedil;&euml; Qα + Q &Aring;&ccedil;&euml; Oα + P
=
U
NM &Aring;&ccedil;&euml; α + R &euml;&aacute;&aring; Pα + &Aring;&ccedil;&euml; Rα
=
NS
NM + NR &Aring;&ccedil;&euml; Oα + S &Aring;&ccedil;&euml; Qα + &Aring;&ccedil;&euml; Sα
=
PO
=
98
CHAPTER 4. TRIGONOMETRY
4.17 Graphs of Inverse Trigonometric
Functions
=
466. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=p&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; I= − N ≤ &ntilde; ≤ N I= −
π
π
≤ ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; ≤ K=
O
O
=
=
=
Figure 66.
=
467. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=`&ccedil;&euml;&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; I= − N ≤ &ntilde; ≤ N I= M ≤ ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; ≤ π K=
=
99
CHAPTER 4. TRIGONOMETRY
=
=
Figure 67.
=
468. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=q~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&iacute;~&aring; &ntilde; I= − ∞ ≤ &ntilde; ≤ ∞ I= −
π
π
&lt; ~&ecirc;&Aring;&iacute;~&aring; &ntilde; &lt; K=
O
O
=
=
=
=====
Figure 68.
100
CHAPTER 4. TRIGONOMETRY
469. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=`&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; I= − ∞ ≤ &ntilde; ≤ ∞ I= M &lt; ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; &lt; π K=
=====
=
Figure 69.
=
470. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=p&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
 π  π 
&oacute; = ~&ecirc;&Aring;&euml;&Eacute;&Aring;=&ntilde; I &ntilde; ∈ (− ∞I − N] ∪ [NI ∞ )I ~&ecirc;&Aring; &euml;&Eacute;&Aring; &ntilde; ∈ MI  ∪  I πK
 O  O 
=
Figure 70.
101
CHAPTER 4. TRIGONOMETRY
471. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=`&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
 π   π
&oacute; = ~&ecirc;&Aring;&Aring;&euml;&Aring; &ntilde; I &ntilde; ∈ (− ∞I − N] ∪ [NI ∞ )I ~&ecirc;&Aring; &Aring;&euml;&Aring; &ntilde; ∈ − I M  ∪  MI K
 O   O
=
=
Figure 71.
=
=
4.18 Principal Values of Inverse
Trigonometric Functions
472.
&ntilde;=
M=
N
=
O
PM&deg; =
SM&deg; =
O
−
O
~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = M&deg; =
~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = VM&deg;
N
−
&ntilde;=
O
− PM&deg;
~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; =
− QR&deg;
=
NOM&deg;
~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
NPR&deg; =
=
O
=
O
QR&deg; =
QR&deg; =
P
−
O
P
O
SM&deg;
PM&deg;
VM&deg;
M&deg; =
− N=
=
− VM&deg;
=
NUM&deg;
NRM&deg; =
=
− SM&deg;
102
N=
=
=
CHAPTER 4. TRIGONOMETRY
473.
&ntilde;=
M=
P
P
N=
~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
M&deg; =
PM&deg;
QR&deg;
SM&deg;
~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = VM&deg;
SM&deg;
QR&deg;
PM&deg;
P= −
P
P
4.19 Relations between Inverse
Trigonometric Functions
=
474. ~&ecirc;&Aring;&euml;&aacute;&aring;(− &ntilde; ) = − ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; =
=
π
475. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = − ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
O
=
476. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml; N − &ntilde; O I= M ≤ &ntilde; ≤ N K=
=
477. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = − ~&ecirc;&Aring;&Aring;&ccedil;&euml; N − &ntilde; O I= − N ≤ &ntilde; ≤ M K=
=
&ntilde;
O
I= &ntilde; &lt; N K=
478. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring;
O
N− &ntilde;
=
N− &ntilde;O
I= M &lt; &ntilde; ≤ N K=
&ntilde;
=
480. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
N− &ntilde;O
− π I= − N ≤ &ntilde; &lt; M K=
&ntilde;
=
481. ~&ecirc;&Aring;&Aring;&ccedil;&euml;(− &ntilde; ) = π − ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
103
− P=
− QR&deg;
− SM&deg; =
=
NPR&deg;
NOM&deg; =
NRM&deg; =
=
− PM&deg;
=
=
=
479. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
− N=
CHAPTER 4. TRIGONOMETRY
482. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; =
π
− ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; =
O
=
483. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring; N − &ntilde; O I= M ≤ &ntilde; ≤ N K=
=
484. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = π − ~&ecirc;&Aring;&euml;&aacute;&aring; N − &ntilde; O I= − N ≤ &ntilde; ≤ M K=
=
485. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring;
N− &ntilde;O
I= M &lt; &ntilde; ≤ N K=
&ntilde;
=
N− &ntilde;O
I= − N ≤ &ntilde; &lt; M K=
&ntilde;
486. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = π + ~&ecirc;&Aring;&iacute;~&aring;
=
487. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute;
&ntilde;
N− &ntilde;O
I= − N ≤ &ntilde; ≤ N K=
=
488. ~&ecirc;&Aring;&iacute;~&aring;(− &ntilde; ) = − ~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
=
π
489. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = − ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; =
O
=
&ntilde;
=
490. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring;
N+ &ntilde;O
=
N
I= &ntilde; ≥ M K=
491. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;
N+ &ntilde;O
=
N
I= &ntilde; ≤ M K=
492. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = − ~&ecirc;&Aring;&Aring;&ccedil;&euml;
N+ &ntilde;O
=
104
CHAPTER 4. TRIGONOMETRY
493. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
π
N
− ~&ecirc;&Aring;&iacute;~&aring; I= &ntilde; &gt; M K=
O
&ntilde;
=
π
N
494. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = − − ~&ecirc;&Aring;&iacute;~&aring; I= &ntilde; &lt; M K=
O
&ntilde;
=
N
495. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; I= &ntilde; &gt; M K=
&ntilde;
=
N
496. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; − π I= &ntilde; &lt; M K=
&ntilde;
=
497. ~&ecirc;&Aring; &Aring;&ccedil;&iacute;(− &ntilde; ) = π − ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; =
=
π
498. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = − ~&ecirc;&Aring;&iacute;~&aring; &ntilde; =
O
=
N
I= &ntilde; &gt; M K=
499. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&euml;&aacute;&aring;
N+ &ntilde;O
=
N
I= &ntilde; &lt; M K=
500. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = π − ~&ecirc;&Aring;&euml;&aacute;&aring;
N+ &ntilde;O
=
&ntilde;
=
501. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;
N+ &ntilde;O
=
N
502. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = ~&ecirc;&Aring;&iacute;~&aring; I= &ntilde; &gt; M K=
&ntilde;
=
N
503. ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; = π + ~&ecirc;&Aring;&iacute;~&aring; I= &ntilde; &lt; M K=
&ntilde;
=
=
105
CHAPTER 4. TRIGONOMETRY
4.20 Trigonometric Equations
504.
505.
506.
507.
=
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
=
=
&aring;
&euml;&aacute;&aring; &ntilde; = ~ I= &ntilde; = (− N) ~&ecirc;&Aring;&euml;&aacute;&aring; ~ + π&aring; =
=
&Aring;&ccedil;&euml; &ntilde; = ~ I= &ntilde; = &plusmn; ~&ecirc;&Aring;&Aring;&ccedil;&euml; ~ + Oπ&aring; =
=
&iacute;~&aring; &ntilde; = ~ I= &ntilde; = ~&ecirc;&Aring;&iacute;~&aring; ~ + π&aring; =
=
&Aring;&ccedil;&iacute; &ntilde; = ~ I= &ntilde; = ~&ecirc;&Aring; &Aring;&ccedil;&iacute; ~ + π&aring; =
=
=
=
4.21 Relations to Hyperbolic Functions
508.
509.
510.
511.
512.
=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=&igrave;&aring;&aacute;&iacute;W=&aacute;=
=
=
&euml;&aacute;&aring;(&aacute;&ntilde; ) = &aacute; &euml;&aacute;&aring;&Uuml; &ntilde; =
=
&iacute;~&aring;(&aacute;&ntilde; ) = &aacute; &iacute;~&aring;&Uuml; &ntilde; =
=
&Aring;&ccedil;&iacute;(&aacute;&ntilde; ) = −&aacute; &Aring;&ccedil;&iacute;&Uuml; &ntilde; =
=
&euml;&Eacute;&Aring;(&aacute;&ntilde; ) = &euml;&Eacute;&Aring;&Uuml; &ntilde; =
=
&Aring;&euml;&Aring;(&aacute;&ntilde; ) = −&aacute; &Aring;&euml;&Aring;&Uuml; &ntilde; =
=
=
=
106
Chapter 5
Matrices and Determinants
=
=
=
=
j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;W=^I=_I=`=
b&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= ~ &aacute; I= &Auml;&aacute; I= ~ &aacute;&agrave; I= &Auml;&aacute;&agrave; I= &Aring; &aacute;&agrave; =
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= &Ccedil;&Eacute;&iacute; ^ =
j&aacute;&aring;&ccedil;&ecirc;=&ccedil;&Ntilde;=~&aring;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;= ~ &aacute;&agrave; W= j &aacute;&agrave; =
`&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=~&aring;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;= ~ &aacute;&agrave; W= ` &aacute;&agrave; =
&uacute;
q&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= ^ q I= ^ =
^&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= ~&Ccedil;&agrave; ^ =
q&ecirc;~&Aring;&Eacute;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= &iacute;&ecirc; ^ =
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;W= ^ −N =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&acirc;=
o&Eacute;~&auml;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W= &ntilde; &aacute; =
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&atilde;I=&aring;===
=
=
5.1 Determinants
=
513. p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=l&ecirc;&Ccedil;&Eacute;&ecirc;=a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
~ &Auml;N
&Ccedil;&Eacute;&iacute; ^ = N
= ~ N &Auml; O − ~ O &Auml;N =
~ O &Auml;O
=
=
=
=
=
107
CHAPTER 5. MATRICES AND DETERMINANTS
514. q&Uuml;&aacute;&ecirc;&Ccedil;=l&ecirc;&Ccedil;&Eacute;&ecirc;=a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
~NN ~NO ~NP
&Ccedil;&Eacute;&iacute; ^ = ~ ON ~ OO
~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO − =
~ PN ~ PO ~ PP
− ~NN~ OP~ PO − ~NO~ ON~ PP − ~ NP~ OO~ PN =
=
515. p~&ecirc;&ecirc;&igrave;&euml;=o&igrave;&auml;&Eacute;=E^&ecirc;&ecirc;&ccedil;&iuml;=o&igrave;&auml;&Eacute;F=
=
=
Figure 72.
=
516. k-&iacute;&Uuml;=l&ecirc;&Ccedil;&Eacute;&ecirc;=a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
~NN ~NO K ~N&agrave;
~ ON ~ OO K ~ O &agrave;
K K K K
&Ccedil;&Eacute;&iacute; ^ =
~ &aacute;N ~ &aacute; O K ~ &aacute;&agrave;
K K K K
~ &aring;N ~ &aring; O K ~ &aring;&agrave;
K ~N&aring;
K ~ O&aring;
K K
K ~ &aacute;&aring;
=
K K
K ~ &aring;&aring;
=
517. j&aacute;&aring;&ccedil;&ecirc;=
q&Uuml;&Eacute;=&atilde;&aacute;&aring;&ccedil;&ecirc;= j &aacute;&agrave; =~&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&Eacute;&Ccedil;=&iuml;&aacute;&iacute;&Uuml;=&iacute;&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;= ~ &aacute;&agrave; =&ccedil;&Ntilde;=&aring;-&iacute;&Uuml;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=
&atilde;~&iacute;&ecirc;&aacute;&ntilde;= ^= &aacute;&euml;= &iacute;&Uuml;&Eacute;= (&aring; − N) -&iacute;&Uuml;= &ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;= &Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;= &Ccedil;&Eacute;&ecirc;&aacute;&icirc;&Eacute;&Ccedil;= &Ntilde;&ecirc;&ccedil;&atilde;=
&iacute;&Uuml;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=^=&Auml;&oacute;=&Ccedil;&Eacute;&auml;&Eacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&aacute;&iacute;&euml;=&aacute;-&iacute;&Uuml;=&ecirc;&ccedil;&iuml;=~&aring;&Ccedil;=&agrave;-&iacute;&Uuml;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;K===
=
108
CHAPTER 5. MATRICES AND DETERMINANTS
518. `&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;=
&aacute; +&agrave;
` &aacute;&agrave; = (− N) j &aacute;&agrave; =
=
519. i~&eacute;&auml;~&Aring;&Eacute;=b&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&aring;-&iacute;&Uuml;=l&ecirc;&Ccedil;&Eacute;&ecirc;=a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
i~&eacute;&auml;~&Aring;&Eacute;=&Eacute;&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring;=&Auml;&oacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&aacute;-&iacute;&Uuml;=&ecirc;&ccedil;&iuml;=
&aring;
&Ccedil;&Eacute;&iacute; ^ = ∑ ~ &aacute;&agrave;` &aacute;&agrave; I= &aacute; = NI OI KI &aring; K=
&agrave;=N
i~&eacute;&auml;~&Aring;&Eacute;=&Eacute;&ntilde;&eacute;~&aring;&euml;&aacute;&ccedil;&aring;=&Auml;&oacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&agrave;-&iacute;&Uuml;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;=
&aring;
&Ccedil;&Eacute;&iacute; ^ = ∑ ~ &aacute;&agrave;` &aacute;&agrave; I= &agrave; = NI OI KI &aring; K==
&aacute; =N
=
=
=
5.2 Properties of Determinants
=
520. q&Uuml;&Eacute;==&icirc;~&auml;&igrave;&Eacute;==&ccedil;&Ntilde;=~=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=&ecirc;&Eacute;&atilde;~&aacute;&aring;&euml;==&igrave;&aring;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;=&aacute;&Ntilde;=&ecirc;&ccedil;&iuml;&euml;=~&ecirc;&Eacute;=
&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;=&iacute;&ccedil;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;=~&aring;&Ccedil;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;=&iacute;&ccedil;=&ecirc;&ccedil;&iuml;&euml;K=
~ ~ O ~N &Auml;N
=
==
= N
&Auml;N &Auml;O ~ O &Auml;O
=
521. f&Ntilde;=&iacute;&iuml;&ccedil;==&ecirc;&ccedil;&iuml;&euml;==E&ccedil;&ecirc;=&iacute;&iuml;&ccedil;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;F=~&ecirc;&Eacute;==&aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;I=&iacute;&Uuml;&Eacute;=&euml;&aacute;&Ouml;&aring;=&ccedil;&Ntilde;=
&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=&aacute;&euml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;K=
~N &Auml;N
~ &Auml;O
=− O
=
~ O &Auml;O
~N &Auml;N
=
522. f&Ntilde;=&iacute;&iuml;&ccedil;=&ecirc;&ccedil;&iuml;&euml;==E&ccedil;&ecirc;=&iacute;&iuml;&ccedil;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;F=~&ecirc;&Eacute;==&aacute;&Ccedil;&Eacute;&aring;&iacute;&aacute;&Aring;~&auml;I=&iacute;&Uuml;&Eacute;=&icirc;~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=
&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=&aacute;&euml;=&ograve;&Eacute;&ecirc;&ccedil;K=
~N ~N
= M=
~O ~O
=
109
CHAPTER 5. MATRICES AND DETERMINANTS
523. f&Ntilde;==&iacute;&Uuml;&Eacute;===&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;==&ccedil;&Ntilde;==~&aring;&oacute;=&ecirc;&ccedil;&iuml;==E&ccedil;&ecirc;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;F=~&ecirc;&Eacute;=&atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil;=&Auml;&oacute;=====
~==&Aring;&ccedil;&atilde;&atilde;&ccedil;&aring;==&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;I==&iacute;&Uuml;&Eacute;==&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;==&aacute;&euml;==&atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil;==&Auml;&oacute;==&iacute;&Uuml;~&iacute;=
&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;K=
&acirc;~ N &acirc;&Auml;N
~ &Auml;N
=&acirc; N
=
~ O &Auml;O
~ O &Auml;O
=
524. f&Ntilde;==&iacute;&Uuml;&Eacute;==&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;==&ccedil;&Ntilde;==~&aring;&oacute;==&ecirc;&ccedil;&iuml;==E&ccedil;&ecirc;==&Aring;&ccedil;&auml;&igrave;&atilde;&aring;F=~&ecirc;&Eacute;=&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&Eacute;&Ccedil;=E&ccedil;&ecirc;=
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&Eacute;&Ccedil;F&Auml;&oacute;=&Eacute;&egrave;&igrave;~&auml;=&atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ecirc;&ecirc;&Eacute;&euml;&eacute;&ccedil;&aring;&Ccedil;&aacute;&aring;&Ouml;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=
&ccedil;&Ntilde;=~&aring;&oacute;=&ccedil;&iacute;&Uuml;&Eacute;&ecirc;=&ecirc;&ccedil;&iuml;==E&ccedil;&ecirc;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;FI==&iacute;&Uuml;&Eacute;=&icirc;~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
&aacute;&euml;=&igrave;&aring;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;K=
~N + &acirc;&Auml;N &Auml;N ~N &Auml;N
=
=
~ O + &acirc;&Auml;O &Auml;O ~ O &Auml;O
=
=
=
5.3 Matrices
=
525. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=
^&aring;= &atilde; &times; &aring; =&atilde;~&iacute;&ecirc;&aacute;&ntilde;=^=&aacute;&euml;=~=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=~&ecirc;&ecirc;~&oacute;=&ccedil;&Ntilde;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=E&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;F=&iuml;&aacute;&iacute;&Uuml;=&atilde;=&ecirc;&ccedil;&iuml;&euml;=~&aring;&Ccedil;=&aring;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;K==
 ~ NN ~ NO K ~ N&aring; 
~
~ OO K ~ O&aring; 
ON
 ==

^ = ~ &aacute;&agrave; =
 M
M
M 


~ &atilde;N ~ &atilde; O K ~ &atilde;&aring; 
=
526. p&egrave;&igrave;~&ecirc;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&aacute;&euml;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&ccedil;&Ntilde;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;= &aring;&times; &aring; K==
=
527. ^=&euml;&egrave;&igrave;~&ecirc;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;== ~ &aacute;&agrave; ==&aacute;&euml;==&euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;==&aacute;&Ntilde;== ~ &aacute;&agrave; = ~ &agrave;&aacute; I==&aacute;K&Eacute;K==&aacute;&iacute;==&aacute;&euml;=
[ ]
[ ]
&euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;K==
=
528. ^=&euml;&egrave;&igrave;~&ecirc;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;= ~ &aacute;&agrave; =&aacute;&euml;=&euml;&acirc;&Eacute;&iuml;-&euml;&oacute;&atilde;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=&aacute;&Ntilde;= ~ &aacute;&agrave; = −~ &agrave;&aacute; K==
=
[ ]
110
CHAPTER 5. MATRICES AND DETERMINANTS
529. a&aacute;~&Ouml;&ccedil;&aring;~&auml;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;==&aacute;&euml;==~=&euml;&egrave;&igrave;~&ecirc;&Eacute;==&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&iuml;&aacute;&iacute;&Uuml;=~&auml;&auml;==&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;==&ograve;&Eacute;&ecirc;&ccedil;=
&Eacute;&ntilde;&Aring;&Eacute;&eacute;&iacute;=&iacute;&Uuml;&ccedil;&euml;&Eacute;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;K==
=
530. r&aring;&aacute;&iacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;==&aacute;&euml;==~=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;==&atilde;~&iacute;&ecirc;&aacute;&ntilde;==&aacute;&aring;=&iuml;&Uuml;&aacute;&Aring;&Uuml;=&iacute;&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&aring;=
&iacute;&Uuml;&Eacute;=&auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;=~&ecirc;&Eacute;=~&auml;&auml;=&igrave;&aring;&aacute;&iacute;&oacute;K=q&Uuml;&Eacute;=&igrave;&aring;&aacute;&iacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&aacute;&euml;===========
&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;=&Auml;&oacute;=fK==
=
531. ^=&aring;&igrave;&auml;&auml;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&aacute;&euml;=&ccedil;&aring;&Eacute;=&iuml;&Uuml;&ccedil;&euml;&Eacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=~&ecirc;&Eacute;=~&auml;&auml;=&ograve;&Eacute;&ecirc;&ccedil;K=
=
=
=
5.4 Operations with Matrices
=
532. q&iuml;&ccedil;=&atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;=^=~&aring;&Ccedil;=_=~&ecirc;&Eacute;=&Eacute;&egrave;&igrave;~&auml;=&aacute;&Ntilde;I=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;I=&iacute;&Uuml;&Eacute;&oacute;=~&ecirc;&Eacute;=&Auml;&ccedil;&iacute;&Uuml;=
&ccedil;&Ntilde;==&iacute;&Uuml;&Eacute;==&euml;~&atilde;&Eacute;==&euml;&Uuml;~&eacute;&Eacute;== &atilde; &times; &aring; ==~&aring;&Ccedil;=&Aring;&ccedil;&ecirc;&ecirc;&Eacute;&euml;&eacute;&ccedil;&aring;&Ccedil;&aacute;&aring;&Ouml;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;&euml;=~&ecirc;&Eacute;=
&Eacute;&egrave;&igrave;~&auml;K=
=
533. q&iuml;&ccedil;=&atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;==^=~&aring;&Ccedil;=_==&Aring;~&aring;=&Auml;&Eacute;=~&Ccedil;&Ccedil;&Eacute;&Ccedil;=E&ccedil;&ecirc;=&euml;&igrave;&Auml;&iacute;&ecirc;~&Aring;&iacute;&Eacute;&Ccedil;F=&ccedil;&Ntilde;I=~&aring;&Ccedil;=
&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;I=&iacute;&Uuml;&Eacute;&oacute;=&Uuml;~&icirc;&Eacute;=&iacute;&Uuml;&Eacute;=&euml;~&atilde;&Eacute;=&euml;&Uuml;~&eacute;&Eacute;= &atilde; &times; &aring; K=f&Ntilde;==
 ~NN ~NO K ~N&aring; 
~
~ OO K ~ O&aring; 
 I==
^ = ~ &aacute;&agrave; =  ON
 M
M
M 


~ &atilde;N ~ &atilde; O K ~ &atilde;&aring; 
 &Auml;NN &Auml;NO K &Auml;N&aring; 
&Auml;
&Auml;OO K &Auml;O&aring; 
 I==
_ = &Auml;&aacute;&agrave; =  ON
 M
M
M 


&Auml;&atilde;N &Auml;&atilde; O K &Auml;&atilde;&aring; 
=
=
=
=
=
[ ]
[ ]
111
CHAPTER 5. MATRICES AND DETERMINANTS
&iacute;&Uuml;&Eacute;&aring;==
~NO + &Auml;NO K ~N&aring; + &Auml;N&aring; 
 ~NN + &Auml;NN
~ +&Auml;
~ OO + &Auml;OO K ~ O&aring; + &Auml;O&aring; 
ON
ON
 K=

^+_=


M
M
M


~ &atilde;N + &Auml;&atilde;N ~ &atilde; O + &Auml;&atilde; O K ~ &atilde;&aring; + &Auml;&atilde;&aring; 
=
534. f&Ntilde;=&acirc;=&aacute;&euml;=~=&euml;&Aring;~&auml;~&ecirc;I=~&aring;&Ccedil;= ^ = ~ &aacute;&agrave; =&aacute;&euml;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;I=&iacute;&Uuml;&Eacute;&aring;=
[ ]
 &acirc;~NN &acirc;~NO K &acirc;~N&aring; 
 &acirc;~
&acirc;~ OO K &acirc;~ O&aring; 
ON
 K=

&acirc;^ = &acirc;~ &aacute;&agrave; =
 M
M
M 


&acirc;~ &atilde;N &acirc;~ &atilde; O K &acirc;~ &atilde;&aring; 
=
535. j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&iuml;&ccedil;=j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;=
q&iuml;&ccedil;= &atilde;~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;= &Aring;~&aring;= &Auml;&Eacute;= &atilde;&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Eacute;&Ccedil;= &iacute;&ccedil;&Ouml;&Eacute;&iacute;&Uuml;&Eacute;&ecirc;= &ccedil;&aring;&auml;&oacute;= &iuml;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;=
&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;= &ccedil;&Ntilde;= &Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;= &aacute;&aring;= &iacute;&Uuml;&Eacute;= &Ntilde;&aacute;&ecirc;&euml;&iacute;= &aacute;&euml;= &Eacute;&egrave;&igrave;~&auml;= &iacute;&ccedil;= &iacute;&Uuml;&Eacute;= &aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;= &ccedil;&Ntilde;=
&ecirc;&ccedil;&iuml;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&Aring;&ccedil;&aring;&Ccedil;K==
=
f&Ntilde;=
 ~NN ~NO K ~N&aring; 
~
~ OO K ~ O&aring; 
 I==
^ = ~ &aacute;&agrave; =  ON
 M
M
M 


~ &atilde;N ~ &atilde; O K ~ &atilde;&aring; 
 &Auml;NN &Auml;NO K &Auml;N&acirc; 
&Auml;
&Auml;OO K &Auml;O &acirc; 
 I=
_ = &Auml;&aacute;&agrave; =  ON
 M
M
M 


&Auml;&aring;N &Auml;&aring; O K &Auml;&aring;&acirc; 
=
=
=
=
=
[ ]
[ ]
[ ]
112
CHAPTER 5. MATRICES AND DETERMINANTS
&iacute;&Uuml;&Eacute;&aring;==
 &Aring;NN &Aring;NO K &Aring;N&acirc; 
&Aring;
&Aring; OO K &Aring; O &acirc; 
ON
 I==

^_ = ` =
 M
M
M 


&Auml; &atilde;N &Aring; &atilde; O K &Aring; &atilde;&acirc; 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&aring;
&Aring; &aacute;&agrave; = ~ &aacute;N&Auml;N&agrave; + ~ &aacute; O &Auml;O &agrave; + K + ~ &aacute;&aring; &Auml;&aring;&agrave; = ∑ ~ &aacute; λ &Auml;λ &agrave; =
E &aacute; = NI OI KI &atilde; X &agrave; = NI OI KI &acirc; FK==
=
q&Uuml;&igrave;&euml;=&aacute;&Ntilde;=
[ ]
~ NN
^ = ~ &aacute;&agrave; = 
~ ON
~ NO
~ OO
λ =N
 &Auml;N 
~ NP 
I= _ = [&Auml; &aacute; ] = &Auml; O  I==

~ OP 
&Auml;P 
&iacute;&Uuml;&Eacute;&aring;==
~ NN ~ NO
^_ = 
~ ON ~ OO
&Auml; 
~ NP   N  ~ NN&Auml;N
⋅ &Auml; =
~ OP   O  ~ ON&Auml;N
&Auml;P 
~ NO &Auml; O
~ OO &Auml; O
~ NP &Auml;P 
K==
~ OP &Auml;P 
=
536. q&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute;=&ccedil;&Ntilde;=~=j~&iacute;&ecirc;&aacute;&ntilde;=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&ecirc;&ccedil;&iuml;&euml;=~&aring;&Ccedil;=&Aring;&ccedil;&auml;&igrave;&atilde;&aring;&euml;=&ccedil;&Ntilde;=~=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=~&ecirc;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Uuml;~&aring;&Ouml;&Eacute;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;=
&iacute;&Uuml;&Eacute;=&aring;&Eacute;&iuml;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=&iacute;&Uuml;&Eacute;=&iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;~&auml;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;K===
f&Ntilde;= ^= &aacute;&euml;= &iacute;&Uuml;&Eacute;= &ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;~&auml;= &atilde;~&iacute;&ecirc;&aacute;&ntilde;I= &aacute;&iacute;&euml;= &iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute;= &aacute;&euml;= &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;= ^ q = &ccedil;&ecirc;=
&uacute;
^ K==
=
537. q&Uuml;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=^=&aacute;&euml;=&ccedil;&ecirc;&iacute;&Uuml;&ccedil;&Ouml;&ccedil;&aring;~&auml;=&aacute;&Ntilde;= ^^ q = f K==
=
538. f&Ntilde;=&iacute;&Uuml;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=^_=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;==
(^_ )q = _ q ^ q K=
=
=
113
CHAPTER 5. MATRICES AND DETERMINANTS
539. ^&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=j~&iacute;&ecirc;&aacute;&ntilde;=
f&Ntilde;=^=&aacute;&euml;=~=&euml;&egrave;&igrave;~&ecirc;&Eacute;= &aring;&times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I=&aacute;&iacute;&euml;=~&Ccedil;&agrave;&ccedil;&aacute;&aring;&iacute;I=&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;=&Auml;&oacute;= ~&Ccedil;&agrave; ^ I=
&aacute;&euml;=&iacute;&Uuml;&Eacute;=&iacute;&ecirc;~&aring;&euml;&eacute;&ccedil;&euml;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&ccedil;&Ntilde;=&Aring;&ccedil;&Ntilde;~&Aring;&iacute;&ccedil;&ecirc;&euml;= ` &aacute;&agrave; =&ccedil;&Ntilde;=^W=
[ ]
~&Ccedil;&agrave; ^ = ` &aacute;&agrave; K==
=
540. q&ecirc;~&Aring;&Eacute;=&ccedil;&Ntilde;=~=j~&iacute;&ecirc;&aacute;&ntilde;=
f&Ntilde;= ^= &aacute;&euml;= ~= &euml;&egrave;&igrave;~&ecirc;&Eacute;= &aring; &times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;I= &aacute;&iacute;&euml;= &iacute;&ecirc;~&Aring;&Eacute;I= &Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;= &Auml;&oacute;= &iacute;&ecirc; ^ I= &aacute;&euml;=
&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&iacute;&ccedil;=&Auml;&Eacute;==&iacute;&Uuml;&Eacute;=&euml;&igrave;&atilde;=&ccedil;&Ntilde;==&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&ecirc;&atilde;&euml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&Eacute;~&Ccedil;&aacute;&aring;&Ouml;=&Ccedil;&aacute;~&Ouml;&ccedil;&aring;~&auml;W=
&iacute;&ecirc; ^ = ~NN + ~ OO + K + ~ &aring;&aring; K=
=
541. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&ccedil;&Ntilde;=~=j~&iacute;&ecirc;&aacute;&ntilde;=
f&Ntilde;=^=&aacute;&euml;=~=&euml;&egrave;&igrave;~&ecirc;&Eacute;= &aring;&times; &aring; &atilde;~&iacute;&ecirc;&aacute;&ntilde;=&iuml;&aacute;&iacute;&Uuml;=~=&aring;&ccedil;&aring;&euml;&aacute;&aring;&Ouml;&igrave;&auml;~&ecirc;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;=
&Ccedil;&Eacute;&iacute; ^ I=&iacute;&Uuml;&Eacute;&aring;=&aacute;&iacute;&euml;=&aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;= ^ −N =&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=
~&Ccedil;&agrave; ^
^ −N =
K=
&Ccedil;&Eacute;&iacute; ^
=
542. f&Ntilde;=&iacute;&Uuml;&Eacute;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=^_=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;==
(^_)−N = _ −N^ −N K=
=
543. f&Ntilde;==^==&aacute;&euml;=~=&euml;&egrave;&igrave;~&ecirc;&Eacute;=== &aring; &times; &aring; ==&atilde;~&iacute;&ecirc;&aacute;&ntilde;I==&iacute;&Uuml;&Eacute;==&Eacute;&aacute;&Ouml;&Eacute;&aring;&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;==u===&euml;~&iacute;&aacute;&euml;&Ntilde;&oacute;=
&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
^u = λu I==
&iuml;&Uuml;&aacute;&auml;&Eacute;=&iacute;&Uuml;&Eacute;=&Eacute;&aacute;&Ouml;&Eacute;&aring;&icirc;~&auml;&igrave;&Eacute;&euml;= λ =&euml;~&iacute;&aacute;&euml;&Ntilde;&oacute;=&iacute;&Uuml;&Eacute;=&Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
^ − λf = M K===
=
=
=
q
5.5 Systems of Linear Equations
=
=
s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;I= &ntilde; N I= &ntilde; O I K =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W= ~ N I ~ O I ~ P I &Auml;N I ~ NN I ~ NO I K =
114
CHAPTER 5. MATRICES AND DETERMINANTS
a&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;~&aring;&iacute;&euml;W=aI= a&ntilde; I= a&oacute; I= a&ograve; ==
j~&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;W=^I=_I=u=
=
=
~ &ntilde; + &Auml;N&oacute; = &Ccedil;N
I==
544.  N
~ O &ntilde; + &Auml;O &oacute; = &Ccedil; O
a&oacute;
a
=E`&ecirc;~&atilde;&Eacute;&ecirc;∞&euml;=&ecirc;&igrave;&auml;&Eacute;FI==
&ntilde; = &ntilde; I= &oacute; =
a
a
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
~ &Auml;N
a= N
= ~N&Auml;O − ~ O &Auml;N I==
~ O &Auml;O
&Ccedil; &Auml;N
a&ntilde; = N
= &Ccedil;N&Auml;O − &Ccedil; O &Auml;N I==
&Ccedil; O &Auml;O
~ &Ccedil;N
a&oacute; = N
= ~N&Ccedil; O − ~ O&Ccedil;N K==
~ O &Ccedil;O
=
545. f&Ntilde;= a ≠ M I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&oacute;&euml;&iacute;&Eacute;&atilde;=&Uuml;~&euml;=~=&euml;&aacute;&aring;&Ouml;&auml;&Eacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W==
a&oacute;
a
K=
&ntilde; = &ntilde; I= &oacute; =
a
a
f&Ntilde;= a = M = ~&aring;&Ccedil;= a&ntilde; ≠ M E&ccedil;&ecirc;= a&oacute; ≠ M FI= &iacute;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= &euml;&oacute;&euml;&iacute;&Eacute;&atilde;= &Uuml;~&euml;= = &aring;&ccedil;==
&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;K=
f&Ntilde;= a = a&ntilde; = a&oacute; = M I= &iacute;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= &euml;&oacute;&euml;&iacute;&Eacute;&atilde;= &Uuml;~&euml;= = &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;&auml;&oacute;= = &atilde;~&aring;&oacute;==
&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;K=
=
~N&ntilde; + &Auml;N&oacute; + &Aring;N&ograve; = &Ccedil;N=

546. ~ O &ntilde; + &Auml;O &oacute; + &Aring; O&ograve; = &Ccedil; O I==
~ &ntilde; + &Auml; &oacute; + &Aring; &ograve; = &Ccedil;
P
P
P
 P
&ntilde;=
a&oacute;
a&ntilde;
a
I= &oacute; =
I= &ograve; = &ograve; =E`&ecirc;~&atilde;&Eacute;&ecirc;∞&euml;=&ecirc;&igrave;&auml;&Eacute;FI==
a
a
a
=
115
CHAPTER 5. MATRICES AND DETERMINANTS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
~N &Auml;N
a = ~ O &Auml;O
~ P &Auml;P
&Aring;N
&Ccedil;N
&Auml;N
&Aring;N
&Aring; O I= a&ntilde; = &Ccedil; O
&Auml;O
&Aring; O I=
&Aring;P
&Auml;P
&Aring;P
&Ccedil;P
~N
&Ccedil;N
&Aring;N
~N
&Auml;N
&Ccedil;N
a&oacute; = ~ O
~P
&Ccedil;O
&Ccedil;P
&Aring; O I= a&ograve; = ~ O
&Aring;P
~P
&Auml;O
&Auml;P
&Ccedil; O K==
&Ccedil;P
=
547. f&Ntilde;= a ≠ M I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&oacute;&euml;&iacute;&Eacute;&atilde;=&Uuml;~&euml;=~=&euml;&aacute;&aring;&Ouml;&auml;&Eacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W==
a&oacute;
a
a
I= &ograve; = &ograve; K=
&ntilde; = &ntilde; I= &oacute; =
a
a
a
f&Ntilde;= a = M =~&aring;&Ccedil;= a&ntilde; ≠ M E&ccedil;&ecirc;= a&oacute; ≠ M =&ccedil;&ecirc;= a&ograve; ≠ M FI=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&oacute;&euml;&iacute;&Eacute;&atilde;=
&Uuml;~&euml;=&aring;&ccedil;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;K=
f&Ntilde;= a = a&ntilde; = a&oacute; = a&ograve; = M I= &iacute;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= &euml;&oacute;&euml;&iacute;&Eacute;&atilde;= &Uuml;~&euml;= &aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;&auml;&oacute;=
&atilde;~&aring;&oacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;K=
=
548. j~&iacute;&ecirc;&aacute;&ntilde;=c&ccedil;&ecirc;&atilde;=&ccedil;&Ntilde;=~=p&oacute;&euml;&iacute;&Eacute;&atilde;=&ccedil;&Ntilde;=&aring;=i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&aacute;&aring;=================
&aring;=r&aring;&acirc;&aring;&ccedil;&iuml;&aring;&euml;=
q&Uuml;&Eacute;=&euml;&Eacute;&iacute;=&ccedil;&Ntilde;=&auml;&aacute;&aring;&Eacute;~&ecirc;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;==
~NN&ntilde; N + ~ NO &ntilde; O + K + ~ N&aring; &ntilde; &aring; = &Auml;N
~ &ntilde; + ~ &ntilde; + K + ~ &ntilde; = &Auml;
 ON N OO O
O&aring; &aring;
O
=

K
K
K
K
K
K
K
K
K
K
K
K

~ &aring;N&ntilde; N + ~ &aring; O &ntilde; O + K + ~ &aring;&aring; &ntilde; &aring; = &Auml;&aring;
&Aring;~&aring;=&Auml;&Eacute;=&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring;=&aacute;&aring;=&atilde;~&iacute;&ecirc;&aacute;&ntilde;=&Ntilde;&ccedil;&ecirc;&atilde;=
 ~ NN ~ NO K ~ N&aring;   &ntilde; N   &Auml;N 
    

 ~ ON ~ OO K ~ O&aring;   &ntilde; O   &Auml; O 
I==
=
⋅
 M
M
M   M   M 
    

    
~
 &aring;N ~ &aring; O K ~ &aring;&aring;   &ntilde; &aring;   &Auml; &aring; 
&aacute;K&Eacute;K==
^ ⋅ u = _ I==
116
CHAPTER 5. MATRICES AND DETERMINANTS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
 ~ NN

~
^ =  ON
M

~
 &aring;N
~ NO K ~ N&aring; 
 &ntilde;N 
 &Auml;N 

 
 
~ OO K ~ O&aring; 
 &ntilde;O 
 &Auml;O 
u
_
I=
I=
=
=
 M 
 M  K==
M
M 

 
 
&ntilde; 
&Auml; 
~ &aring; O K ~ &aring;&aring; 
 &aring;
 &aring;
=
549. p&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=p&Eacute;&iacute;=&ccedil;&Ntilde;=i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;= &aring; &times; &aring; =
u = ^ −N ⋅ _ I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ^ −N =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&ccedil;&Ntilde;=^K=
=
=
117
Chapter 6
Vectors
=
=
=
=
r r r r →
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &igrave; I= &icirc; I= &iuml; I= &ecirc; I= ^_ I=&pound;=
r r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W= &igrave; I= &icirc; I=&pound;=
r r r
r&aring;&aacute;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &aacute; I= &agrave; I= &acirc; =
r
k&igrave;&auml;&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W= M =
r
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &igrave; W= uN I vN I wN =
r
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &icirc; W= u O I vO I wO =
p&Aring;~&auml;~&ecirc;&euml;W= λ I &micro; =
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W= &Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= θ =
=
=
6.1 Vector Coordinates
=
550. r&aring;&aacute;&iacute;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=
r
&aacute; = (NI MI M) I=
r
&agrave; = (MI NI M) I=
r
&acirc; = (MI MI N) I=
r r r
&aacute; = &agrave; = &acirc; = N K=
=
r
r
r
r →
551. &ecirc; = ^_ = (&ntilde; N − &ntilde; M ) &aacute; + (&oacute; N − &oacute; M ) &agrave; + (&ograve; N − &ograve; M ) &acirc; =
=
118
CHAPTER 6. VECTORS
=======
=
=
Figure 73.
=
→
r
&ecirc; = ^_ =
552.
(&ntilde; N − &ntilde; M )O + (&oacute;N − &oacute; M )O + (&ograve;N − &ograve; M )O =
=
→
→
r
r
553. f&Ntilde;= ^_ = &ecirc; I=&iacute;&Uuml;&Eacute;&aring;= _^ = − &ecirc; K=
=
=
=
Figure 74.
r
554. u = &ecirc; &Aring;&ccedil;&euml; α I=
r
v = &ecirc; &Aring;&ccedil;&euml; β I=
r
w = &ecirc; &Aring;&ccedil;&euml; γ K=
=
119
CHAPTER 6. VECTORS
=
=====
=
Figure 75.
=
r
r
555. f&Ntilde;= &ecirc; (uI v I w ) = &ecirc;N (uN I vN I wN ) I=&iacute;&Uuml;&Eacute;&aring;==
u = uN I= v = vN I= w = wN K==
==
=
=
r r r
556. &iuml; = &igrave; + &icirc; =
=
=
==
=
Figure 76.
120
CHAPTER 6. VECTORS
=
==
=
Figure 77.
=
r
r r r r
557. &iuml; = &igrave;N + &igrave; O + &igrave;P + K + &igrave; &aring; =
=
=
=
==
Figure 78.
=
558. `&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute;=i~&iuml;=
r r r r
&igrave;+ &icirc; =&icirc;+&igrave;=
=
559. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute;=i~&iuml;=
(&igrave;r + &icirc;r ) + &iuml;r = &igrave;r + (&icirc;r + &iuml;r ) =
=
r r
560. &igrave; + &icirc; = (uN + u O I vN + vO I wN + wO ) =
=
=
=
=
=
=
121
CHAPTER 6. VECTORS
6.3 Vector Subtraction
=
r r r r r r
561. &iuml; = &igrave; − &icirc; =&aacute;&Ntilde;= &icirc; + &iuml; = &igrave; K=
=
=
=
Figure 79.
=
=
==
=
Figure 80.
=
r r r
r
562. &igrave; − &icirc; = &igrave; + (− &icirc; ) =
=
r r r
563. &igrave; − &igrave; = M = (MI MI M ) =
=
r
564. M = M =
=
r r
565. &igrave; − &icirc; = (uN − u O I vN − vO I wN − w O ) I==
=
=
=
6.4 Scaling Vectors
=
r
r
566. &iuml; = λ&igrave; =
122
CHAPTER 6. VECTORS
=
=
Figure 81.
=
567.
r
r
&iuml; = λ⋅&igrave;=
=
r
568. λ&igrave; = (λuI λv I λw ) =
=
r r
569. λ&igrave; = &igrave;λ =
=
r
r
r
570. (λ + &micro; ) &igrave; = λ&igrave; + &micro;&igrave; =
=
r
r
r
571. λ(&micro;&igrave; ) = &micro;(λ&igrave; ) = (λ&micro; )&igrave; =
=
r r
r
r
572. λ(&igrave; + &icirc; ) = λ&igrave; + λ&icirc; =
=
=
=
6.5 Scalar Product
=
r
r
573. p&Aring;~&auml;~&ecirc;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=&ccedil;&Ntilde;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;= &igrave; =~&aring;&Ccedil; &icirc; =
r r r r
&igrave; ⋅ &icirc; = &igrave; ⋅ &icirc; ⋅ &Aring;&ccedil;&euml; θ I==
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= θ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;= &igrave; =~&aring;&Ccedil; &icirc; K====
=
123
CHAPTER 6. VECTORS
=
=
=
Figure 82.
=
574. p&Aring;~&auml;~&ecirc;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=&aacute;&aring;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=c&ccedil;&ecirc;&atilde;=
r
r
f&Ntilde;= &igrave; = (uN I vN I wN ) I= &icirc; = (u O I vO I w O ) I=&iacute;&Uuml;&Eacute;&aring;==
r r
&igrave; ⋅ &icirc; = uNu O + vNvO + wNwO K=
=
575. ^&aring;&Ouml;&auml;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;==
r
r
f&Ntilde;= &igrave; = (uN I vN I wN ) I= &icirc; = (u O I vO I w O ) I=&iacute;&Uuml;&Eacute;&aring;==
uNu O + vNvO + wNw O
K=
&Aring;&ccedil;&euml; θ =
O
O
O
O
O
O
uN + vN + wN u O + vO + w O
=
576. `&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
r r r r
&igrave;⋅&icirc; = &icirc; ⋅&igrave;=
=
577. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
(λ&igrave;r ) ⋅ (&micro;&icirc;r ) = λ&micro;&igrave;r ⋅ &icirc;r =
=
578. a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
r r r r r r r
&igrave; ⋅ (&icirc; + &iuml; ) = &igrave; ⋅ &icirc; + &igrave; ⋅ &iuml; =
=
π
r r
r r
579. &igrave; ⋅ &icirc; = M =&aacute;&Ntilde;= &igrave; I &icirc; =~&ecirc;&Eacute;=&ccedil;&ecirc;&iacute;&Uuml;&ccedil;&Ouml;&ccedil;&aring;~&auml;=E θ = FK=
O
=
π
r r
580. &igrave; ⋅ &icirc; &gt; M =&aacute;&Ntilde;= M &lt; θ &lt; K=
O
=
124
CHAPTER 6. VECTORS
π
r r
581. &igrave; ⋅ &icirc; &lt; M =&aacute;&Ntilde;= &lt; θ &lt; π K=
O
=
r r r r
582. &igrave; ⋅ &icirc; ≤ &igrave; ⋅ &icirc; =
=
r r r r
r r
583. &igrave; ⋅ &icirc; = &igrave; ⋅ &icirc; =&aacute;&Ntilde;= &igrave; I &icirc; =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=E θ = M FK=
=
r
584. f&Ntilde;= &igrave; = (uN I vN I wN ) I=&iacute;&Uuml;&Eacute;&aring;==
r r r
rO
O
O
O
&igrave; ⋅ &igrave; = &igrave; O = &igrave; = uN + vN + wN K=
=
r r r r r r
585. &aacute; ⋅ &aacute; = &agrave; ⋅ &agrave; = &acirc; ⋅ &acirc; = N =
=
r r r r r r
586. &aacute; ⋅ &agrave; = &agrave; ⋅ &acirc; = &acirc; ⋅ &aacute; = M =
=
=
=
6.6 Vector Product
=
r
r
587. s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=&ccedil;&Ntilde;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;= &igrave; =~&aring;&Ccedil; &icirc; =
r r r
&igrave; &times; &icirc; = &iuml; I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
π
r r r
•
&iuml; = &igrave; ⋅ &icirc; ⋅ &euml;&aacute;&aring; θ I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= M ≤ θ ≤ X=
O
r r
r r
•
&iuml; ⊥&igrave;=
~&aring;&Ccedil;= &iuml; ⊥ &icirc; X=
r r r
• =s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;= &igrave; I= &icirc; I= &iuml; =&Ntilde;&ccedil;&ecirc;&atilde;=~=&ecirc;&aacute;&Ouml;&Uuml;&iacute;-&Uuml;~&aring;&Ccedil;&Eacute;&Ccedil;=&euml;&Aring;&ecirc;&Eacute;&iuml;K=
=
125
CHAPTER 6. VECTORS
=
=======
=
Figure 83.
=
r
&aacute;
r r r
588. &iuml; = &igrave; &times; &icirc; = u N
uO
r
&agrave;
vN
vO
r
&acirc;
wN =
wO
=
uN wN uN vN 
r r r  v wN
=
589. &iuml; = &igrave; &times; &icirc; =  N
I−
I

v
w
u
w
u
v
O
O
O
O
O
O


=
r r r r
590. p = &igrave; &times; &icirc; = &igrave; ⋅ &icirc; ⋅ &euml;&aacute;&aring; θ =Ec&aacute;&Ouml;KUPF=
=
591. ^&aring;&Ouml;&auml;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=Ec&aacute;&Ouml;KUPF=
r r
&igrave;&times; &icirc;
&euml;&aacute;&aring; θ = r r =
&igrave;⋅&icirc;
=
592. k&ccedil;&aring;&Aring;&ccedil;&atilde;&atilde;&igrave;&iacute;~&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
r r
r r
&igrave; &times; &icirc; = −(&icirc; &times; &igrave; ) ==
=
593. ^&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
(λ&igrave;r )&times; (&micro;&icirc;r ) = λ&micro;&igrave;r &times; &icirc;r =
=
=
126
CHAPTER 6. VECTORS
594. a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&icirc;&Eacute;=m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&oacute;=
r r r r r r r
&igrave; &times; (&icirc; + &iuml; ) = &igrave; &times; &icirc; + &igrave; &times; &iuml; =
=
r r r r
r
595. &igrave; &times; &icirc; = M =&aacute;&Ntilde;= &igrave; =~&aring;&Ccedil;= &icirc; =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=E θ = M FK=
=
r r r r r r r
596. &aacute; &times; &aacute; = &agrave; &times; &agrave; = &acirc; &times; &acirc; = M =
=
r r r r r r r r r
597. &aacute; &times; &agrave; = &acirc; I= &agrave; &times; &acirc; = &aacute; I= &acirc; &times; &aacute; = &agrave; =
=
=
=
6.7 Triple Product
598.
599.
600.
601.
=
p&Aring;~&auml;~&ecirc;=q&ecirc;&aacute;&eacute;&auml;&Eacute;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=
[&igrave;r &icirc;r&iuml;r ] = &igrave;r ⋅ (&icirc;r &times; &iuml;r ) = &icirc;r ⋅ (&iuml;r &times; &igrave;r ) = &iuml;r ⋅ (&igrave;r &times; &icirc;r ) =
=
[&igrave;r &icirc;r&iuml;r ] = [&iuml;r &igrave;r &icirc;r ] = [&icirc;r&iuml;r &igrave;r ] = −[&icirc;r&igrave;r &iuml;r ] = −[&iuml;r &icirc;r&igrave;r ] = −[&igrave;r &iuml;r &icirc;r ] =
=
r r r
rr r
&acirc;&igrave; ⋅ (&icirc; &times; &iuml; ) = &acirc;[&igrave;&icirc;&iuml; ] =
=
p&Aring;~&auml;~&ecirc;=q&ecirc;&aacute;&eacute;&auml;&Eacute;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=&aacute;&aring;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=c&ccedil;&ecirc;&atilde;=
uN vN wN
r r r
&igrave; ⋅ (&icirc; &times; &iuml; ) = u O vO w O I==
uP vP wP
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
r
r
r
&igrave; = (uN I vN I wN ) I= &icirc; = (u O I vO I w O ) I= &iuml; = (uP I vP I wP ) K==
=
602. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;=
r r r
s = &igrave; ⋅ (&icirc; &times; &iuml; ) =
=
127
CHAPTER 6. VECTORS
=
============
=
Figure 84.
=
603. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=m&oacute;&ecirc;~&atilde;&aacute;&Ccedil;=
Nr r r
s = &igrave; ⋅ (&icirc; &times; &iuml; ) =
S
=
=
=
Figure 85.
=
r r r
r r
r
604. f&Ntilde;== &igrave; ⋅ (&icirc; &times; &iuml; ) = M I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;== &igrave; I= &icirc; I=~&aring;&Ccedil;= &iuml; =~&ecirc;&Eacute;=&auml;&aacute;&aring;&Eacute;~&ecirc;&auml;&oacute;=
r
r
r
&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=I=&euml;&ccedil;= &iuml; = λ&igrave; + &micro;&icirc; =&Ntilde;&ccedil;&ecirc;=&euml;&ccedil;&atilde;&Eacute;=&euml;&Aring;~&auml;~&ecirc;&euml;= λ =~&aring;&Ccedil;= &micro; K==
=
r r r
r r
r
605. f&Ntilde;== &igrave; ⋅ (&icirc; &times; &iuml; ) ≠ M I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;== &igrave; I= &icirc; I=~&aring;&Ccedil;= &iuml; =~&ecirc;&Eacute;=&auml;&aacute;&aring;&Eacute;~&ecirc;&auml;&oacute;=
&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;K=
=
128
CHAPTER 6. VECTORS
606. s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=q&ecirc;&aacute;&eacute;&auml;&Eacute;=m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=
r r r
r r r r r r
&igrave; &times; (&icirc; &times; &iuml; ) = (&igrave; ⋅ &iuml; )&icirc; − (&igrave; ⋅ &icirc; )&iuml; ==
=
=
=
=
=
=
=
=
129
Chapter 7
Analytic Geometry
=
=
=
=
7.1 One-Dimensional Coordinate System
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ntilde; M I= &ntilde; N I= &ntilde; O I= &oacute; M I= &oacute; N I= &oacute; O =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W= λ ==
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;W=&Ccedil;=
=
=
607. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;&euml;=
&Ccedil; = ^_ = &ntilde; O − &ntilde; N = &ntilde; N − &ntilde; O =
=
=
=
Figure 86.
=
608. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=o~&iacute;&aacute;&ccedil;= λ =
&ntilde; + λ&ntilde; O
^`
I= λ =
&ntilde;M = N
I= λ ≠ −N K=
N+ λ
`_
=
=
========
Figure 87.
130
=
CHAPTER 7. ANALYTIC GEOMETRY
609. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=
&ntilde; + &ntilde;O
&ntilde;M = N
I= λ = N K=
O
=
=
=
7.2 Two-Dimensional Coordinate System
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ntilde; M I= &ntilde; N I= &ntilde; O I= &oacute; M I= &oacute; N I= &oacute; O =
m&ccedil;&auml;~&ecirc;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ecirc;I ϕ =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W= λ ==
m&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;I==
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;W=&Ccedil;=
^&ecirc;&Eacute;~W=p=
=
=
610. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;&euml;=
&Ccedil; = ^_ =
=
(&ntilde; O − &ntilde; N )O + (&oacute; O − &oacute;N )O =
=
=
Figure 88.
131
CHAPTER 7. ANALYTIC GEOMETRY
611. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=o~&iacute;&aacute;&ccedil;= λ =
&ntilde; + λ&ntilde; O
&oacute; + λ&oacute; O
&ntilde;M = N
I= &oacute; M = N
I==
N+ λ
N+ λ
^`
λ=
I= λ ≠ −N K=
`_
=
=======
=
=
Figure 89.
=
=
132
CHAPTER 7. ANALYTIC GEOMETRY
=======
=
=
Figure 90.
=
612. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=
&ntilde; + &ntilde;O
&oacute; + &oacute;O
I= &oacute; M = N
I= λ = N K=
&ntilde;M = N
O
O
=
613. `&Eacute;&aring;&iacute;&ecirc;&ccedil;&aacute;&Ccedil;=Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=j&Eacute;&Ccedil;&aacute;~&aring;&euml;F=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
&ntilde; + &ntilde; O + &ntilde;P
&oacute; + &oacute;O + &oacute;P
I= &oacute; M = N
&ntilde;M = N
I==
P
P
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;== ^(&ntilde; N I &oacute; N ) I== _(&ntilde; O I &oacute; O ) I==~&aring;&Ccedil;== `(&ntilde; P I &oacute; P ) ==~&ecirc;&Eacute;=&icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml;=&ccedil;&Ntilde;=
&iacute;&Uuml;&Eacute;=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;= ^_` K=
=
=
133
CHAPTER 7. ANALYTIC GEOMETRY
=========
=
=
Figure 91.
=
614. f&aring;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=^&aring;&Ouml;&auml;&Eacute;=_&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;F=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
~&ntilde; + &Auml;&ntilde; O + &Aring;&ntilde; P
~&oacute; + &Auml;&oacute; O + &Aring;&oacute; P
I= &oacute; M = N
&ntilde;M = N
I==
~ +&Auml;+&Aring;
~ +&Auml;+&Aring;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ~ = _` I= &Auml; = `^ I= &Aring; = ^_ K==
=
========
=
=
Figure 92.
134
CHAPTER 7. ANALYTIC GEOMETRY
615. `&aacute;&ecirc;&Aring;&igrave;&atilde;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=p&aacute;&Ccedil;&Eacute;=m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;======================
_&aacute;&euml;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;F=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
&ntilde; NO + &oacute; NO &oacute; N N
&ntilde; N &ntilde; NO + &oacute; NO N
&ntilde; OO + &oacute; OO &oacute; O N
&ntilde; O &ntilde; OO + &oacute; OO N
&ntilde; PO + &oacute; PO &oacute; P N
&ntilde; P &ntilde; PO + &oacute; PO N
&ntilde;M =
I= &oacute; M =
=
&ntilde;N &oacute;N N
&ntilde;N &oacute;N N
O &ntilde;O
&ntilde;P
&oacute;O N
&oacute;P N
O &ntilde;O
&ntilde;P
&oacute;O N
&oacute;P N
=
=
========
==
Figure 93.
=
=
=
=
=
=
=
135
CHAPTER 7. ANALYTIC GEOMETRY
616. l&ecirc;&iacute;&Uuml;&ccedil;&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=Ef&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=^&auml;&iacute;&aacute;&iacute;&igrave;&Ccedil;&Eacute;&euml;F=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
&oacute; N &ntilde; O &ntilde; P + &oacute; NO N
&ntilde; NO + &oacute; O &oacute; P &ntilde; N N
&oacute; O &ntilde; P &ntilde; N + &oacute; OO N
&ntilde; OO + &oacute; P &oacute; N &ntilde; O N
&oacute; P &ntilde; N&ntilde; O + &oacute; PO N
&ntilde; PO + &oacute; N&oacute; O &ntilde; P N
I= &oacute; M =
=
&ntilde;M =
&ntilde;N &oacute;N N
&ntilde;N &oacute;N N
&ntilde;O &oacute;O N
&ntilde;O &oacute;O N
&ntilde;P &oacute;P N
&ntilde;P &oacute;P N
=
=
======
=
Figure 94.
=
617. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
&ntilde; N &oacute;N N
N
N &ntilde; O − &ntilde;N
p = (&plusmn; ) &ntilde; O &oacute; O N = (&plusmn; )
O
O &ntilde; P − &ntilde;N
&ntilde;P &oacute;P N
=
=
=
136
&oacute; O − &oacute;N
&oacute; P − &oacute;N
=
CHAPTER 7. ANALYTIC GEOMETRY
618. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=n&igrave;~&Ccedil;&ecirc;&aacute;&auml;~&iacute;&Eacute;&ecirc;~&auml;=
N
p = (&plusmn; ) [(&ntilde; N − &ntilde; O )(&oacute; N + &oacute; O ) + (&ntilde; O − &ntilde; P )(&oacute; O + &oacute; P ) + =
O
+ (&ntilde; P − &ntilde; Q )(&oacute; P + &oacute; Q ) + (&ntilde; Q − &ntilde; N )(&oacute; Q + &oacute; N )] =
=
===
=
=
Figure 95.
=
k&ccedil;&iacute;&Eacute;W=f&aring;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;=SNTI=SNU=&iuml;&Eacute;=&Aring;&Uuml;&ccedil;&ccedil;&euml;&Eacute;=&iacute;&Uuml;&Eacute;=&euml;&aacute;&Ouml;&aring;=EHF=&ccedil;&ecirc;=E&yen;F=&euml;&ccedil;=
&iacute;&Uuml;~&iacute;=&iacute;&ccedil;=&Ouml;&Eacute;&iacute;=~=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=~&aring;&euml;&iuml;&Eacute;&ecirc;=&Ntilde;&ccedil;&ecirc;=~&ecirc;&Eacute;~K==
=
619. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;&euml;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
&Ccedil; = ^_ = &ecirc;NO + &ecirc;OO − O&ecirc;N&ecirc;O &Aring;&ccedil;&euml;(ϕ O − ϕN ) =
=
137
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 96.
=
620. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&iacute;&aacute;&aring;&Ouml;=o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&iacute;&ccedil;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
&ntilde; = &ecirc; &Aring;&ccedil;&euml; ϕ I= &oacute; = &ecirc; &euml;&aacute;&aring; ϕ K=
=
=
=
Figure 97.
=
621. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&iacute;&aacute;&aring;&Ouml;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&iacute;&ccedil;=o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
&oacute;
&ecirc; = &ntilde; O + &oacute; O I= &iacute;~&aring; ϕ = K=
&ntilde;
138
CHAPTER 7. ANALYTIC GEOMETRY
7.3 Straight Line in Plane
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W=uI=vI=&ntilde;I= &ntilde; M I= &ntilde; N I== &oacute; M I= &oacute; N I= ~N I= ~ O I=&pound;==
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&acirc;I=~I=&Auml;I=&eacute;I=&iacute;I=^I=_I=`I= ^N I= ^ O I=&pound;=
^&aring;&Ouml;&auml;&Eacute;&euml;W= α I= β =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;W= ϕ =
r
k&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W= &aring; =
r r r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &ecirc; I= ~ I= &Auml; =
=
=
622. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=
^&ntilde; + _&oacute; + ` = M =
=
623. k&ccedil;&ecirc;&atilde;~&auml;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&iacute;&ccedil;=~=p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=
r
q&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &aring;(^I _ ) =&aacute;&euml;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;= ^&ntilde; + _&oacute; + ` = M K=
=
=
=
Figure 98.
=
624. b&ntilde;&eacute;&auml;&aacute;&Aring;&aacute;&iacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=Ep&auml;&ccedil;&eacute;&Eacute;-f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute;=c&ccedil;&ecirc;&atilde;F=
&oacute; = &acirc;&ntilde; + &Auml; K==
139
CHAPTER 7. ANALYTIC GEOMETRY
q&Uuml;&Eacute;=&Ouml;&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&euml;= &acirc; = &iacute;~&aring; α K=
=
=
=
Figure 99.
=
625. d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;==
&oacute; − &oacute;N
&acirc; = &iacute;~&aring; α = O
=
&ntilde; O − &ntilde;N
=
=
=
Figure 100.
140
CHAPTER 7. ANALYTIC GEOMETRY
626. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;=d&aacute;&icirc;&Eacute;&aring;=~=m&ccedil;&aacute;&aring;&iacute;=~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;=d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;=
&oacute; = &oacute; M + &acirc; (&ntilde; − &ntilde; M ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&acirc;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ouml;&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;I= m(&ntilde; M I &oacute; M ) =&aacute;&euml;=~=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;K=
=
=
=
Figure 101.
=
627. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;=q&Uuml;~&iacute;=m~&euml;&euml;&Eacute;&euml;=q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;=q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;&euml;=
&oacute; − &oacute;N
&ntilde; − &ntilde;N
=
==
&oacute; O − &oacute;N &ntilde; O − &ntilde;N
&ccedil;&ecirc;=
&ntilde; &oacute; N
&ntilde; N &oacute; N N = M K=
&ntilde;O &oacute;O N
=
141
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 102.
=
628. f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute;=c&ccedil;&ecirc;&atilde;=
&ntilde; &oacute;
+ =N=
~ &Auml;
=
=
=
Figure 103.
=
=
142
CHAPTER 7. ANALYTIC GEOMETRY
629. k&ccedil;&ecirc;&atilde;~&auml;=c&ccedil;&ecirc;&atilde;=
&ntilde; &Aring;&ccedil;&euml; β + &oacute; &euml;&aacute;&aring; β − &eacute; = M =
=
=
=
Figure 104.
=
630. m&ccedil;&aacute;&aring;&iacute;=a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;=
&ntilde; − &ntilde; N &oacute; − &oacute;N
=
I==
u
v
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= (uI v ) = &aacute;&euml;= &iacute;&Uuml;&Eacute;= &Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &auml;&aacute;&aring;&Eacute;= ~&aring;&Ccedil;= mN (&ntilde; N I &oacute; N ) = &auml;&aacute;&Eacute;&euml;=
&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;K=
=
143
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 105.
=
631. s&Eacute;&ecirc;&iacute;&aacute;&Aring;~&auml;=i&aacute;&aring;&Eacute;=
&ntilde; =~=
=
632. e&ccedil;&ecirc;&aacute;&ograve;&ccedil;&aring;&iacute;~&auml;=i&aacute;&aring;&Eacute;=
&oacute;=&Auml;=
=
633. s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=
r r r
&ecirc; = ~ + &iacute;&Auml; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
l=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;I=
u=&aacute;&euml;=~&aring;&oacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==
r
~ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=~=&acirc;&aring;&ccedil;&iuml;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;=^=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=I=
r
&Auml; =&aacute;&euml;=~=&acirc;&aring;&ccedil;&iuml;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;I=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==
&iacute;=&aacute;&euml;=~=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;I==
r →
&ecirc; = lu =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=~&aring;&oacute;=&eacute;&ccedil;&aacute;&aring;&iacute;=u=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;K==
=
144
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 106.
=
634. p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;=&aacute;&aring;=m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;=
&ntilde; = ~N + &iacute;&Auml;N
I==

&oacute; = ~ O + &iacute;&Auml;O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
(&ntilde; I &oacute; ) ~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=~&aring;&oacute;=&igrave;&aring;&acirc;&aring;&ccedil;&iuml;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==
(~N I ~ O ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=~=&acirc;&aring;&ccedil;&iuml;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==
(&Auml;N I &Auml;O ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==
&iacute;=&aacute;&euml;=~=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;K=
=
145
CHAPTER 7. ANALYTIC GEOMETRY
=
Figure 107.
=
635. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=c&ecirc;&ccedil;&atilde;=~=m&ccedil;&aacute;&aring;&iacute;=q&ccedil;=~=i&aacute;&aring;&Eacute;=
q&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Ntilde;&ecirc;&ccedil;&atilde;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= m(~ I &Auml;) =&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=
^&ntilde; + _&oacute; + ` = M =&aacute;&euml;==
^~ + _&Auml; + `
K=
&Ccedil;=
^ O + _O
=
=
=
Figure 108.
146
CHAPTER 7. ANALYTIC GEOMETRY
636. m~&ecirc;~&auml;&auml;&Eacute;&auml;=i&aacute;&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;= &oacute; = &acirc; N&ntilde; + &Auml;N =~&aring;&Ccedil;= &oacute; = &acirc; O &ntilde; + &Auml;O =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;==
&acirc; N = &acirc; O K=
q&iuml;&ccedil;= &auml;&aacute;&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N = M = ~&aring;&Ccedil;= ^ O &ntilde; + _O &oacute; + ` O = M = ~&ecirc;&Eacute;=
&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;=
^N _N
=
K=
^ O _O
=
=
=
Figure 109.
=
637. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=i&aacute;&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;= &oacute; = &acirc; N&ntilde; + &Auml;N =~&aring;&Ccedil;= &oacute; = &acirc; O &ntilde; + &Auml;O =~&ecirc;&Eacute;=&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=&aacute;&Ntilde;==
N
&acirc; O = − =&ccedil;&ecirc;I=&Eacute;&egrave;&igrave;&aacute;&icirc;~&auml;&Eacute;&aring;&iacute;&auml;&oacute;I= &acirc; N&acirc; O = −N K=
&acirc;N
q&iuml;&ccedil;= &auml;&aacute;&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N = M = ~&aring;&Ccedil;= ^ O &ntilde; + _ O &oacute; + ` O = M = ~&ecirc;&Eacute;=
&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=&aacute;&Ntilde;=
^N^ O + _N_ O = M K=
=
147
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 110.
=
638. ^&aring;&Ouml;&auml;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=i&aacute;&aring;&Eacute;&euml;=
&acirc; − &acirc;N
&iacute;~&aring; ϕ = O
I==
N + &acirc; N&acirc; O
^N^ O + _N_ O
&Aring;&ccedil;&euml; ϕ =
K=
^NO + _NO ⋅ ^ OO + _ OO
=
148
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 111.
=
639. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&iuml;&ccedil;=i&aacute;&aring;&Eacute;&euml;=
f&Ntilde;=&iacute;&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N = M =~&aring;&Ccedil;= ^ O &ntilde; + _ O &oacute; + ` O = M =&aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;I=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;=&Uuml;~&euml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
− `N_ O + ` O_N
− ^N` O + ^ O`N
&ntilde;M =
I= &oacute; M =
K=
^N_ O − ^ O_N
^N_ O − ^ O_N
=
=
=
7.4 Circle
=
o~&Ccedil;&aacute;&igrave;&euml;W=o=
`&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;W= (~ I &Auml;) =
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W=&ntilde;I=&oacute;I= &ntilde; N I= &oacute; N I=&pound;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI=bI=cI=&iacute;=
149
CHAPTER 7. ANALYTIC GEOMETRY
640. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=`&aacute;&ecirc;&Aring;&auml;&Eacute;=`&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil;=~&iacute;=&iacute;&Uuml;&Eacute;=l&ecirc;&aacute;&Ouml;&aacute;&aring;=Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=
c&ccedil;&ecirc;&atilde;F=
&ntilde; O + &oacute; O = oO =
======
=
=
Figure 112.
=
641. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=`&aacute;&ecirc;&Aring;&auml;&Eacute;=`&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil;=~&iacute;=^&aring;&oacute;=m&ccedil;&aacute;&aring;&iacute;= (~I &Auml;)
(&ntilde; − ~ )O + (&oacute; − &Auml;)O = o O
Figure 113.
150
CHAPTER 7. ANALYTIC GEOMETRY
642. q&Uuml;&ecirc;&Eacute;&Eacute;=m&ccedil;&aacute;&aring;&iacute;=c&ccedil;&ecirc;&atilde;
&ntilde;O + &oacute;O &ntilde; &oacute; N
&ntilde; NO + &oacute; NO &ntilde; N &oacute; N N
=M
&ntilde; OO + &oacute; OO &ntilde; O &oacute; O N
&ntilde; PO + &oacute; PO &ntilde; P &oacute; P N
=
=
=
Figure 114.
=
643. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;
&ntilde; = o &Aring;&ccedil;&euml; &iacute;
I= M ≤ &iacute; ≤ Oπ K

&oacute; = o &euml;&aacute;&aring; &iacute;
=
644. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;
^&ntilde; O + ^&oacute; O + a&ntilde; + b&oacute; + c = M =E^=&aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;I= aO + b O &gt; Q ^c FK==
q&Uuml;&Eacute;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;=&Uuml;~&euml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;= (~ I &Auml;) I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
a
b
~=−
I= &Auml; = −
K=
O^
O^
q&Uuml;&Eacute;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&aacute;&ecirc;&Aring;&auml;&Eacute;=&aacute;&euml;
151
CHAPTER 7. ANALYTIC GEOMETRY
o=
aO + b O − Q ^c
K
O^
=
=
=
7.5 Ellipse
=
p&Eacute;&atilde;&aacute;&atilde;~&agrave;&ccedil;&ecirc;=~&ntilde;&aacute;&euml;W=~=
p&Eacute;&atilde;&aacute;&atilde;&aacute;&aring;&ccedil;&ecirc;=~&ntilde;&aacute;&euml;W=&Auml;=
c&ccedil;&Aring;&aacute;W= cN (− &Aring;I M) I= cO (&Aring;I M) =
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&Aring;&aacute;W=O&Aring;= =
b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;W=&Eacute;==
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI=bI=cI=&iacute;=
m&Eacute;&ecirc;&aacute;&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=i=
^&ecirc;&Eacute;~W=p=
=
=
645. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~&aring;=b&auml;&auml;&aacute;&eacute;&euml;&Eacute;=Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=c&ccedil;&ecirc;&atilde;F
&ntilde;O &oacute;O
+ =N
~ O &Auml;O
=
=
Figure 115.
152
CHAPTER 7. ANALYTIC GEOMETRY
646. &ecirc;N + &ecirc;O = O~ I=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;== &ecirc;N I== &ecirc;O ==~&ecirc;&Eacute;==&Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;&euml;==&Ntilde;&ecirc;&ccedil;&atilde;==~&aring;&oacute;==&eacute;&ccedil;&aacute;&aring;&iacute;== m(&ntilde; I &oacute; ) ==&ccedil;&aring;=
&iacute;&Uuml;&Eacute;=&Eacute;&auml;&auml;&aacute;&eacute;&euml;&Eacute;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&iacute;&iuml;&ccedil;=&Ntilde;&ccedil;&Aring;&aacute;K=
=
=
=
Figure 116.
=
647. ~ O = &Auml;O + &Aring; O
=
648. b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;
&Aring;
&Eacute; = &lt;N=
~
=
649. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=a&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
~
~O
&ntilde;=&plusmn; =&plusmn; =
&Eacute;
&Aring;
=
650. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;
&ntilde; = ~ &Aring;&ccedil;&euml; &iacute;
I= M ≤ &iacute; ≤ Oπ K

&oacute; = &Auml; &euml;&aacute;&aring; &iacute;
=
=
153
CHAPTER 7. ANALYTIC GEOMETRY
651. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= _ O − Q ^` &lt; M K=
=
652. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;=&iuml;&aacute;&iacute;&Uuml;=^&ntilde;&Eacute;&euml;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=^&ntilde;&Eacute;&euml;
^&ntilde; O + `&oacute; O + a&ntilde; + b&oacute; + c = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ^` &gt; M K
=
653. `&aacute;&ecirc;&Aring;&igrave;&atilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;
i = Q~b(&Eacute; ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==&iacute;&Uuml;&Eacute;==&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=b==&aacute;&euml;==&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&iacute;&Eacute;==&Eacute;&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==&ccedil;&Ntilde;=
&iacute;&Uuml;&Eacute;=&euml;&Eacute;&Aring;&ccedil;&aring;&Ccedil;=&acirc;&aacute;&aring;&Ccedil;K==
=
654. ^&eacute;&eacute;&ecirc;&ccedil;&ntilde;&aacute;&atilde;~&iacute;&Eacute;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=`&aacute;&ecirc;&Aring;&igrave;&atilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;
i = π NKR(~ + &Auml;) − ~&Auml; I==
(
i = π O(~ O + &Auml;O ) K=
=
655. p = π~&Auml; =
=
=
=
)
7.6 Hyperbola
=
q&ecirc;~&aring;&euml;&icirc;&Eacute;&ecirc;&euml;&Eacute;=~&ntilde;&aacute;&euml;W=~=
`&ccedil;&aring;&agrave;&igrave;&Ouml;~&iacute;&Eacute;=~&ntilde;&aacute;&euml;W=&Auml;=
c&ccedil;&Aring;&aacute;W= cN (− &Aring;I M) I= cO (&Aring;I M) =
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&Aring;&aacute;W=O&Aring;= =
b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;W=&Eacute;==
^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml;W=&euml;I=&iacute;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI=bI=cI=&iacute;I=&acirc;=
=
=
=
154
CHAPTER 7. ANALYTIC GEOMETRY
656. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~=Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=c&ccedil;&ecirc;&atilde;F=
&ntilde;O &oacute;O
− = N=
~ O &Auml;O
=
=
=
Figure 117.
=
657.
&ecirc;N − &ecirc;O = O~ I=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;== &ecirc;N I== &ecirc;O ==~&ecirc;&Eacute;==&Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;&euml;==&Ntilde;&ecirc;&ccedil;&atilde;==~&aring;&oacute;=&eacute;&ccedil;&aacute;&aring;&iacute;== m(&ntilde; I &oacute; ) ==&ccedil;&aring;=
&iacute;&Uuml;&Eacute;=&Uuml;&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&iacute;&iuml;&ccedil;=&Ntilde;&ccedil;&Aring;&aacute;K=
=
155
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 118.
658.
659.
660.
661.
=
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml;=
&Auml;
&oacute;=&plusmn; &ntilde;=
~
=
&Aring; O = ~ O + &Auml;O =
=
b&Aring;&Aring;&Eacute;&aring;&iacute;&ecirc;&aacute;&Aring;&aacute;&iacute;&oacute;
&Aring;
&Eacute; = &gt; N=
~
=
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=a&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;&Eacute;&euml;
~
~O
&ntilde;=&plusmn; =&plusmn; =
&Eacute;
&Aring;
=
=
=
156
CHAPTER 7. ANALYTIC GEOMETRY
662. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=o&aacute;&Ouml;&Uuml;&iacute;=_&ecirc;~&aring;&Aring;&Uuml;=&ccedil;&Ntilde;=~=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;~=
&ntilde; = ~ &Aring;&ccedil;&euml;&Uuml; &iacute;
I= M ≤ &iacute; ≤ Oπ K

&oacute; = &Auml; &euml;&aacute;&aring;&Uuml; &iacute;
=
663. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= _ O − Q ^` &gt; M K=
=
664. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;=&iuml;&aacute;&iacute;&Uuml;=^&ntilde;&Eacute;&euml;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=^&ntilde;&Eacute;&euml;
^&ntilde; O + `&oacute; O + a&ntilde; + b&oacute; + c = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ^` &lt; M K=
665. ^&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;=
&Eacute;O
&ntilde;&oacute; = I==
Q
&ccedil;&ecirc;==
&Eacute;O
&acirc;
&oacute; = I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &acirc; = K=
&ntilde;
Q
f&aring;= &iacute;&Uuml;&aacute;&euml;= &Aring;~&euml;&Eacute;= I= &iacute;&Uuml;&Eacute;= ~&euml;&oacute;&atilde;&eacute;&iacute;&ccedil;&iacute;&Eacute;&euml;= &Uuml;~&icirc;&Eacute;= &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;= &ntilde; = M = ~&aring;&Ccedil;=
&oacute; = M K==
=
157
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 119.
=
=
=
7.7 Parabola
=
c&ccedil;&Aring;~&auml;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;W=&eacute;=
c&ccedil;&Aring;&igrave;&euml;W=c=
s&Eacute;&ecirc;&iacute;&Eacute;&ntilde;W= j(&ntilde; M I &oacute; M ) =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI=bI=cI=&eacute;I=~I=&Auml;I=&Aring;=
=
=
666. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=m~&ecirc;~&Auml;&ccedil;&auml;~=Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=c&ccedil;&ecirc;&atilde;F
&oacute; O = O&eacute;&ntilde;
=
158
CHAPTER 7. ANALYTIC GEOMETRY
=
=
Figure 120.
=
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
&eacute;
&ntilde; = − I=
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&Aring;&igrave;&euml;=
&eacute; 
c I M  I=
O 
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;=
j(MI M) K=
=
667. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;
^&ntilde; O + _&ntilde;&oacute; + `&oacute; O + a&ntilde; + b&oacute; + c = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= _ O − Q ^` = M K=
=
N
668. &oacute; = ~&ntilde; O I= &eacute; = K=
O~
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
159
CHAPTER 7. ANALYTIC GEOMETRY
&eacute;
&oacute; = − I=
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&Aring;&igrave;&euml;=
 &eacute;
c MI  I=
 O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;=
j(MI M) K=
=
=
=
Figure 121.
=
669. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;I=^&ntilde;&aacute;&euml;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&oacute;-~&ntilde;&aacute;&euml;==
^&ntilde; O + a&ntilde; + b&oacute; + c = M =E^I=b=&aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;FI==
N
&oacute; = ~&ntilde; O + &Auml;&ntilde; + &Aring; I= &eacute; = K==
O~
b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&ecirc;&aacute;&ntilde;
&eacute;
&oacute; = &oacute; M − I=
O
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&Aring;&igrave;&euml;=
160
CHAPTER 7. ANALYTIC GEOMETRY
&eacute;

c &ntilde; M I &oacute; M +  I=
O

`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&ecirc;&iacute;&Eacute;&ntilde;=
&Auml;
Q~&Aring; − &Auml;O
K=
&ntilde; M = − I= &oacute; M = ~&ntilde; OM + &Auml;&ntilde; M + &Aring; =
O~
Q~
=
=
=
Figure 122.
=
=
=
7.8 Three-Dimensional Coordinate System
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ntilde; M I= &oacute; M I= &ograve; M I= &ntilde; N I= &oacute; N I= &ograve; N I=&pound;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W= λ ==
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;W=&Ccedil;=
^&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;W=s=
=
161
CHAPTER 7. ANALYTIC GEOMETRY
670. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;&euml;=
&Ccedil; = ^_ =
=
=
(&ntilde; O − &ntilde; N )O + (&oacute; O − &oacute;N )O + (&ograve; O − &ograve;N )O =
=
===
Figure 123.
=
671. a&aacute;&icirc;&aacute;&Ccedil;&aacute;&aring;&Ouml;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=o~&iacute;&aacute;&ccedil;= λ =
&ntilde; + λ&ntilde; O
&oacute; + λ&oacute; O
&ograve; + λ&ograve; O
&ntilde;M = N
I= &oacute; M = N
I= &ograve; M = N
I==
N+ λ
N+ λ
N+ λ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
^`
λ=
I= λ ≠ −N K=
`_
=
162
CHAPTER 7. ANALYTIC GEOMETRY
========
=
=
Figure 124.
=
=
Figure 125.
163
CHAPTER 7. ANALYTIC GEOMETRY
672. j&aacute;&Ccedil;&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;=p&Eacute;&Ouml;&atilde;&Eacute;&aring;&iacute;=
&ntilde; + &ntilde;O
&oacute; + &oacute;O
&ograve; +&ograve;
&ntilde;M = N
I= &oacute; M = N
I= &ograve; M = N O I= λ = N K=
O
O
O
=
673. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
q&Uuml;&Eacute;=~&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=&iacute;&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=&iuml;&aacute;&iacute;&Uuml;=&icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml;= mN (&ntilde; N I &oacute; N I &ograve; N ) I=
mO (&ntilde; O I &oacute; O I &ograve; O ) I=~&aring;&Ccedil;= mP (&ntilde; P I &oacute; P I &ograve; P ) =&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
p=
N
O
O
&oacute;O
&ograve;O N + &ograve;O
&ntilde;O N + &ntilde;O
&oacute; O N K=
&oacute;P
&ograve;P
&ntilde;P
&oacute;P
&ograve;P
&ntilde;N N
N
&ntilde;N
&ntilde;P
&oacute;N N
O
&ograve;N N
N
&ograve;N
O
&oacute;N
N
=
674. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=q&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;=
q&Uuml;&Eacute;=&icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=&iacute;&Eacute;&iacute;&ecirc;~&Uuml;&Eacute;&Ccedil;&ecirc;&ccedil;&aring;=&iuml;&aacute;&iacute;&Uuml;=&icirc;&Eacute;&ecirc;&iacute;&aacute;&Aring;&Eacute;&euml;= mN (&ntilde; N I &oacute; N I &ograve; N ) I=
mO (&ntilde; O I &oacute; O I &ograve; O ) I= mP (&ntilde; P I &oacute; P I &ograve; P ) I=~&aring;&Ccedil;= mQ (&ntilde; Q I &oacute; Q I &ograve; Q ) =&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
&ntilde; N &oacute;N &ograve;N N
N &ntilde;O &oacute;O &ograve;O N
s=&plusmn;
I==
S &ntilde; P &oacute;P &ograve;P N
&ntilde;Q &oacute;Q &ograve;Q N
&ccedil;&ecirc;=
&ntilde; N − &ntilde; Q &oacute;N − &oacute; Q &ograve;N − &ograve; Q
N
s = &plusmn; &ntilde; O − &ntilde; Q &oacute; O − &oacute; Q &ograve; O − &ograve; Q K=
S
&ntilde;P − &ntilde; Q &oacute;P − &oacute; Q &ograve;P − &ograve;Q
k&ccedil;&iacute;&Eacute;W=t&Eacute;=&Aring;&Uuml;&ccedil;&ccedil;&euml;&Eacute;=&iacute;&Uuml;&Eacute;=&euml;&aacute;&Ouml;&aring;=EHF=&ccedil;&ecirc;=E&yen;F=&euml;&ccedil;=&iacute;&Uuml;~&iacute;=&iacute;&ccedil;=&Ouml;&Eacute;&iacute;=~=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=
~&aring;&euml;&iuml;&Eacute;&ecirc;=&Ntilde;&ccedil;&ecirc;=&icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;K==
=
164
CHAPTER 7. ANALYTIC GEOMETRY
====
=
=
Figure 126.
=
=
=
7.9 Plane
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;I= &ntilde; M I= &oacute; M I= &ograve; M I= &ntilde; N I= &oacute; N I= &ograve; N I=&pound;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI= ^N I= ^ O I=~I=&Auml;I=&Aring;I= ~N I= ~ O I= λ I=&eacute;I=&iacute;I=&pound;==
r r r
k&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &aring; I= &aring;N I= &aring; O =
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W= &Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ =
a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Ntilde;&ecirc;&ccedil;&atilde;=&eacute;&ccedil;&aacute;&aring;&iacute;=&iacute;&ccedil;=&eacute;&auml;~&aring;&Eacute;W=&Ccedil;=
=
=
675. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=m&auml;~&aring;&Eacute;=
^&ntilde; + _&oacute; + `&ograve; + a = M =
=
=
165
CHAPTER 7. ANALYTIC GEOMETRY
676. k&ccedil;&ecirc;&atilde;~&auml;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&iacute;&ccedil;=~=m&auml;~&aring;&Eacute;=
r
q&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &aring; (^I _I ` ) =&aacute;&euml;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=
^&ntilde; + _&oacute; + `&ograve; + a = M K=
=
=
=
===
Figure 127.
=
677. m~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=`~&euml;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=m&auml;~&aring;&Eacute;=
^&ntilde; + _&oacute; + `&ograve; + a = M =
=
f&Ntilde;= ^ = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;K=
f&Ntilde;= _ = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&oacute;-~&ntilde;&aacute;&euml;K=
f&Ntilde;= ` = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ograve;-~&ntilde;&aacute;&euml;K=
f&Ntilde;= a = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&auml;&aacute;&Eacute;&euml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;K==
=
f&Ntilde;= ^ = _ = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K=
f&Ntilde;= _ = ` = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;K=
f&Ntilde;= ^ = ` = M I=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=&aacute;&euml;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute;K=
=
166
CHAPTER 7. ANALYTIC GEOMETRY
678. m&ccedil;&aacute;&aring;&iacute;=a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;=
^(&ntilde; − &ntilde; M ) + _(&oacute; − &oacute; M ) + `(&ograve; − &ograve; M ) = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= m(&ntilde; M I &oacute; M I &ograve; M ) =&auml;&aacute;&Eacute;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;I=~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= (^I _I ` ) =&aacute;&euml;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;K===
=
=
=
====
Figure 128.
=
679. f&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute;=c&ccedil;&ecirc;&atilde;=
&ntilde; &oacute; &ograve;
+ + = N=
~ &Auml; &Aring;
=
167
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 129.
=
680. q&Uuml;&ecirc;&Eacute;&Eacute;=m&ccedil;&aacute;&aring;&iacute;=c&ccedil;&ecirc;&atilde;=
&ntilde; − &ntilde;P &oacute; − &oacute;P &ograve; − &ograve;P
&ntilde; N − &ntilde; P &oacute; N − &oacute; P &ograve; N − &ograve; P = M I==
&ntilde; O − &ntilde; P &oacute; O − &oacute;P &ograve; O − &ograve;P
&ccedil;&ecirc;==
&ntilde; &oacute; &ograve; N
&ntilde;N &oacute;N &ograve;N N
= M K=
&ntilde;O &oacute;O &ograve;O N
&ntilde;P &oacute;P &ograve;P N
=
168
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 130.
=
681. k&ccedil;&ecirc;&atilde;~&auml;=c&ccedil;&ecirc;&atilde;=
&ntilde; &Aring;&ccedil;&euml; α + &oacute; &Aring;&ccedil;&euml; β + &ograve; &Aring;&ccedil;&euml; γ − &eacute; = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==&eacute;==&aacute;&euml;==&iacute;&Uuml;&Eacute;==&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;==&Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;==&Ntilde;&ecirc;&ccedil;&atilde;==&iacute;&Uuml;&Eacute;=&ccedil;&ecirc;&aacute;&Ouml;&aacute;&aring;=&iacute;&ccedil;=
&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=I=~&aring;&Ccedil;= &Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;=
&ccedil;&Ntilde;=~&aring;&oacute;=&auml;&aacute;&aring;&Eacute;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;K==
=
169
CHAPTER 7. ANALYTIC GEOMETRY
=
======
=
Figure 131.
=
682. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;=
&ntilde; = &ntilde; N + ~N&euml; + ~ O &iacute;

&oacute; = &oacute; N + &Auml;N&euml; + &Auml;O &iacute; I==
&ograve; = &ograve; + &Aring; &euml; + &Aring; &iacute;
N
N
O

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= (&ntilde; I &oacute; I &ograve; ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=~&aring;&oacute;=&igrave;&aring;&acirc;&aring;&ccedil;&iuml;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=
&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=I=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= m(&ntilde; N I &oacute; N I &ograve; N ) =&auml;&aacute;&Eacute;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;I=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=
(~N I &Auml;N I &Aring;N ) =~&aring;&Ccedil;= (~ O I &Auml;O I &Aring; O ) =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;K=
=
170
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 132.
=
683. a&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml;=^&aring;&Ouml;&auml;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=m&auml;~&aring;&Eacute;&euml;=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;&euml;=~&ecirc;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
^N&ntilde; + _N&oacute; + `N&ograve; + aN = M I==
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M I==
&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&Uuml;&Eacute;&Ccedil;&ecirc;~&auml;=~&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;&atilde;=&aacute;&euml;==
r r
&aring;N ⋅ &aring; O
^N^ O + _N_ O + `N` O
&Aring;&ccedil;&euml; ϕ = r r =
K=
&aring;N ⋅ &aring; O
^NO + _NO + `NO ⋅ ^ OO + _ OO + ` OO
=
171
CHAPTER 7. ANALYTIC GEOMETRY
=
======
=
Figure 133.
=
684. m~&ecirc;~&auml;&auml;&Eacute;&auml;=m&auml;~&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&eacute;&auml;~&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M =~&aring;&Ccedil;=
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;==
^N _N `N
=
=
K=
^ O _O `O
=
685. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=m&auml;~&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&eacute;&auml;~&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M =~&aring;&Ccedil;=
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M =~&ecirc;&Eacute;=&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=&aacute;&Ntilde;==
^N^ O + _N_ O + `N` O = M K=
=
686. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=m&auml;~&aring;&Eacute;=q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;= m(&ntilde; N I &oacute; N I &ograve; N ) =~&aring;&Ccedil;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=q&ccedil;=
&iacute;&Uuml;&Eacute;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;= (~N I &Auml;N I &Aring;N ) =~&aring;&Ccedil;= (~ O I &Auml;O I &Aring; O ) =Ec&aacute;&Ouml;KNPOF=
172
CHAPTER 7. ANALYTIC GEOMETRY
&ntilde; − &ntilde;N
~N
~O
&oacute; − &oacute;N &ograve; − &ograve;N
&Auml;N
&Aring;N = M =
&Auml;O
&Aring;O
=
687. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=m&auml;~&aring;&Eacute;=q&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;= mN (&ntilde; N I &oacute; N I &ograve; N ) =~&aring;&Ccedil;= mO (&ntilde; O I &oacute; O I &ograve; O ) I=
~&aring;&Ccedil;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=q&ccedil;=&iacute;&Uuml;&Eacute;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;= (~ I &Auml;I &Aring; ) =
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
&ntilde; O − &ntilde;N
~
&oacute; O − &oacute;N &ograve; O − &ograve;N = M =
&Auml;
&Aring;
=
=
Figure 134.
=
688. a&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=c&ecirc;&ccedil;&atilde;=~=m&ccedil;&aacute;&aring;&iacute;=q&ccedil;=~=m&auml;~&aring;&Eacute;=
q&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;=&Ntilde;&ecirc;&ccedil;&atilde;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= mN (&ntilde; N I &oacute; N I &ograve; N ) =&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=
^&ntilde; + _&oacute; + `&ograve; + a = M =&aacute;&euml;==
173
CHAPTER 7. ANALYTIC GEOMETRY
&Ccedil;=
^&ntilde; N + _&oacute; N + `&ograve;N + a
^O + _O + `O
K=
=
=
=
======
Figure 135.
=
689. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&iuml;&ccedil;=m&auml;~&aring;&Eacute;&euml;=
f&Ntilde;=&iacute;&iuml;&ccedil;=&eacute;&auml;~&aring;&Eacute;&euml;= ^N&ntilde; + _N&oacute; + `N&ograve; + aN = M =~&aring;&Ccedil;=
^ O &ntilde; + _ O &oacute; + ` O&ograve; + aO = M = &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;I= &iacute;&Uuml;&Eacute;= &aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;= &euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=
&auml;&aacute;&aring;&Eacute;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=
&ntilde; = &ntilde; N + ~&iacute;

&oacute; = &oacute; N + &Auml;&iacute; I==
&ograve; = &ograve; + &Aring;&iacute;
N

&ccedil;&ecirc;==
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
I==
&Auml;
&Aring;
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
174
CHAPTER 7. ANALYTIC GEOMETRY
~=
_N
_O
&Auml;
aN
aO
&Aring;
aN
aO
~
aN
aO
&ntilde;N =
&oacute;N =
&ograve;N =
`N
`
I= &Auml; = N
`O
`O
^N
^
I= &Aring; = N
^O
^O
_N
I==
_O
`N
a _N
−&Aring; N
`O
aO _ O
I==
O
O
~ + &Auml; + &Aring;O
^N
a `N
−~ N
^O
aO ` O
I==
O
O
~ + &Auml; + &Aring;O
a ^N
_N
−&Auml; N
_O
aO ^ O
K==
O
O
~ + &Auml; + &Aring;O
=
=
=
7.10 Straight Line in Space
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;I= &ntilde; N I= &oacute; N I= &ograve; N I=&pound;=
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;W= &Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=aI=~I=&Auml;I=&Aring;I= ~N I= ~ O I=&iacute;I=&pound;==
r r r
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=&ccedil;&Ntilde;=~=&auml;&aacute;&aring;&Eacute;W= &euml; I= &euml;N I= &euml;O =
r
k&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&iacute;&ccedil;=~=&eacute;&auml;~&aring;&Eacute;W= &aring; =
^&aring;&Ouml;&auml;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;W= ϕ =
=
690. m&ccedil;&aacute;&aring;&iacute;=a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=c&ccedil;&ecirc;&atilde;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=i&aacute;&aring;&Eacute;==
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
I==
&Auml;
&Aring;
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= mN (&ntilde; N I &oacute; N I &ograve; N ) =&auml;&aacute;&Eacute;&euml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I=~&aring;&Ccedil;= (~ I &Auml;I &Aring; ) =&aacute;&euml;=
&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;K==
=
175
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 136.
=
691. q&iuml;&ccedil;=m&ccedil;&aacute;&aring;&iacute;=c&ccedil;&ecirc;&atilde;=
&ntilde; − &ntilde;N
&oacute; − &oacute;N
&ograve; − &ograve;N
=
=
==
&ntilde; O − &ntilde; N &oacute; O − &oacute;N &ograve; O − &ograve;N
=
=====
Figure 137.
176
=
CHAPTER 7. ANALYTIC GEOMETRY
692. m~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=c&ccedil;&ecirc;&atilde;==
&ntilde; = &ntilde; N + &iacute; &Aring;&ccedil;&euml; α

&oacute; = &oacute; N + &iacute; &Aring;&ccedil;&euml; β I==
&ograve; = &ograve; + &iacute; &Aring;&ccedil;&euml; γ
N

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;= mN (&ntilde; N I &oacute; N I &ograve; N ) =&auml;&aacute;&Eacute;&euml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=&auml;&aacute;&aring;&Eacute;I=
&Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&euml;&aacute;&aring;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=
&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I=&iacute;&Uuml;&Eacute;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;=&iacute;=&aacute;&euml;=~&aring;&oacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;K==
=
=
=====
=
Figure 138.
=
693. ^&aring;&Ouml;&auml;&Eacute;=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=p&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=i&aacute;&aring;&Eacute;&euml;=
r r
&euml; ⋅&euml;
~N~ O + &Auml;N&Auml;O + &Aring;N&Aring; O
&Aring;&ccedil;&euml; ϕ = r N rO =
=
O
&euml;N ⋅ &euml;O
~N + &Auml;NO + &Aring;NO ⋅ ~ OO + &Auml;OO + &Aring; OO
=
177
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 139.
=
694. m~&ecirc;~&auml;&auml;&Eacute;&auml;=i&aacute;&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;=~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;==
r r
&euml;N &ouml;&ouml; &euml;O I==
&ccedil;&ecirc;==
~ N &Auml;N &Aring;N
= = K=
~ O &Auml;O &Aring; O
=
695. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=i&aacute;&aring;&Eacute;&euml;=
q&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;=~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;==
r r
&euml;N ⋅ &euml;O = M I==
&ccedil;&ecirc;==
~N~ O + &Auml;N&Auml;O + &Aring;N&Aring; O = M K=
=
696. f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&iuml;&ccedil;=i&aacute;&aring;&Eacute;&euml;=
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
=~&aring;&Ccedil;=
q&iuml;&ccedil;=&auml;&aacute;&aring;&Eacute;&euml;=
~N
&Auml;N
&Aring;N
178
CHAPTER 7. ANALYTIC GEOMETRY
&ntilde; − &ntilde;O &oacute; − &oacute;O &ograve; − &ograve;O
=
=
=&aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;=&aacute;&Ntilde;==
~O
&Auml;O
&Aring;O
&ntilde; O − &ntilde; N &oacute; O − &oacute;N &ograve; O − &ograve;N
~N
&Auml;N
&Aring;N = M K=
~O
&Auml;O
&Aring;O
=
697. m~&ecirc;~&auml;&auml;&Eacute;&auml;=i&aacute;&aring;&Eacute;=~&aring;&Ccedil;=m&auml;~&aring;&Eacute;==
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
=~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=
q&Uuml;&Eacute;=&euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=&auml;&aacute;&aring;&Eacute;=
&Auml;
&Aring;
~
^&ntilde; + _&oacute; + `&ograve; + a = M =~&ecirc;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;=&aacute;&Ntilde;=
r r
&aring; ⋅ &euml; = M I==
&ccedil;&ecirc;==
^~ + _&Auml; + `&Aring; = M K=
=
=
=====
=
Figure 140.
=
=
179
CHAPTER 7. ANALYTIC GEOMETRY
698. m&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=i&aacute;&aring;&Eacute;=~&aring;&Ccedil;=m&auml;~&aring;&Eacute;==
&ntilde; − &ntilde; N &oacute; − &oacute;N &ograve; − &ograve;N
=
=
=~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;=&eacute;&auml;~&aring;&Eacute;=
q&Uuml;&Eacute;=&euml;&iacute;&ecirc;~&aacute;&Ouml;&Uuml;&iacute;=&auml;&aacute;&aring;&Eacute;=
&Auml;
&Aring;
~
^&ntilde; + _&oacute; + `&ograve; + a = M =~&ecirc;&Eacute;=&eacute;&Eacute;&ecirc;&eacute;&Eacute;&aring;&Ccedil;&aacute;&Aring;&igrave;&auml;~&ecirc;=&aacute;&Ntilde;=
r r
&aring; &ouml;&ouml; &euml; I==
&ccedil;&ecirc;==
^ _ `
= = K=
~ &Auml; &Aring;
=
====
=
=
Figure 141.
=
=
=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&egrave;&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=_I=`I=~I=&Auml;I=&Aring;I= &acirc; N I &acirc; O I &acirc; P I=&pound;=
180
CHAPTER 7. ANALYTIC GEOMETRY
699. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
^&ntilde; O + _&oacute; O + `&ograve; O + Oc&oacute;&ograve; + Od&ograve;&ntilde; + Oe&ntilde;&oacute; + Om&ntilde; + On&oacute; + Oo&ograve; + a = M
=
700. `&auml;~&euml;&euml;&aacute;&Ntilde;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;&euml;=
=
`~&euml;&Eacute;= o~&aring;&acirc;E&Eacute;F= o~&aring;&acirc;EbF=
&acirc;=&euml;&aacute;&Ouml;&aring;&euml;=
q&oacute;&eacute;&Eacute;=&ccedil;&Ntilde;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=
∆=
N=
P=
Q=
p~&atilde;&Eacute;=
o&Eacute;~&auml;=b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;=
&lt;M=
O=
P=
Q=
p~&atilde;&Eacute;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;=
&gt; M=
P=
P=
Q=
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=&ccedil;&Ntilde;=N=p&Uuml;&Eacute;&Eacute;&iacute;=
&gt; M = a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
Q=
P=
Q=
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=&ccedil;&Ntilde;=O=p&Uuml;&Eacute;&Eacute;&iacute;&euml;=
&lt; M = a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
R=
S=
T=
U=
V=
NM=
NN=
NO=
NP=
NQ=
NR=
NS=
NT=
P=
P=
O=
O=
O=
O=
O=
O=
O=
N=
N=
N=
N=
P=
P=
Q=
Q=
P=
P=
P=
O=
O=
P=
O=
O=
N=
=
=
&lt;M=
&gt; M=
=
=
=
=
=
=
=
=
=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
p~&atilde;&Eacute;=
p~&atilde;&Eacute;=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
p~&atilde;&Eacute;=
p~&atilde;&Eacute;=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;=
p~&atilde;&Eacute;=
=
=
=
=
o&Eacute;~&auml;=n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=`&ccedil;&aring;&Eacute;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=`&ccedil;&aring;&Eacute;=
b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=
o&Eacute;~&auml;=b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=
e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=
o&Eacute;~&auml;=f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml;=m&auml;~&aring;&Eacute;&euml;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml;=m&auml;~&aring;&Eacute;&euml;=
m~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=
o&Eacute;~&auml;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=m&auml;~&aring;&Eacute;&euml;=
f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=m&auml;~&aring;&Eacute;&euml;=
`&ccedil;&aacute;&aring;&Aring;&aacute;&Ccedil;&Eacute;&aring;&iacute;=m&auml;~&aring;&Eacute;&euml;=
=
e&Eacute;&ecirc;&Eacute;==
^ e n m


 ^ e d


 e _ c n
&Eacute; =  e _ c  I= b = 
I= ∆ = &Ccedil;&Eacute;&iacute;(b ) I==
d c ` o
 d c `




 m n o a


&acirc; N I &acirc; O I &acirc; P =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&ecirc;&ccedil;&ccedil;&iacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I==
^−&ntilde;
e
d
e
_−&ntilde;
c = M K=
d
c
`−&ntilde;
=
=
181
CHAPTER 7. ANALYTIC GEOMETRY
701. o&Eacute;~&auml;=b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;=E`~&euml;&Eacute;=NF=
&ntilde;O &oacute;O &ograve;O
+ + = N=
~ O &Auml;O &Aring; O
=
=
=====
=
Figure 142.
=
702. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=b&auml;&auml;&aacute;&eacute;&euml;&ccedil;&aacute;&Ccedil;=E`~&euml;&Eacute;=OF=
&ntilde;O &oacute;O &ograve;O
+ + = −N =
~ O &Auml;O &Aring; O
=
703. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=&ccedil;&Ntilde;=N=p&Uuml;&Eacute;&Eacute;&iacute;=E`~&euml;&Eacute;=PF=
&ntilde;O &oacute;O &ograve;O
+ − = N=
~ O &Auml;O &Aring; O
=
182
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 143.
=
704. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=&ccedil;&Ntilde;=O=p&Uuml;&Eacute;&Eacute;&iacute;&euml;=E`~&euml;&Eacute;=QF=
&ntilde;O &oacute;O &ograve;O
+ − = −N =
~ O &Auml;O &Aring; O
=
=
=
=
Figure 144.
183
CHAPTER 7. ANALYTIC GEOMETRY
705. o&Eacute;~&auml;=n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=`&ccedil;&aring;&Eacute;=E`~&euml;&Eacute;=RF=
&ntilde;O &oacute;O &ograve;O
+ − =M=
~ O &Auml;O &Aring; O
=
=
==
=
Figure 145.
=
706. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=n&igrave;~&Ccedil;&ecirc;&aacute;&Aring;=`&ccedil;&aring;&Eacute;=E`~&euml;&Eacute;=SF=
&ntilde;O &oacute;O &ograve;O
+ + =M=
~ O &Auml;O &Aring; O
=
707. b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=E`~&euml;&Eacute;=TF=
&ntilde;O &oacute;O
+ −&ograve; = M=
~ O &Auml;O
=
184
CHAPTER 7. ANALYTIC GEOMETRY
======
Figure 146.
=
708. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=m~&ecirc;~&Auml;&ccedil;&auml;&ccedil;&aacute;&Ccedil;=E`~&euml;&Eacute;=UF=
&ntilde;O &oacute;O
− −&ograve; = M=
~ O &Auml;O
====
Figure 147.
185
=
CHAPTER 7. ANALYTIC GEOMETRY
709. o&Eacute;~&auml;=b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=E`~&euml;&Eacute;=VF=
&ntilde;O &oacute;O
+ = N=
~ O &Auml;O
=
=
=====
=
Figure 148.
=
710. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=b&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=E`~&euml;&Eacute;=NMF=
&ntilde;O &oacute;O
+ = −N =
~ O &Auml;O
=
711. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=E`~&euml;&Eacute;=NNF=
&ntilde;O &oacute;O
− = N=
~ O &Auml;O
=
186
CHAPTER 7. ANALYTIC GEOMETRY
=
======
=
Figure 149.
=
712. o&Eacute;~&auml;=f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml;=m&auml;~&aring;&Eacute;&euml;=E`~&euml;&Eacute;=NOF=
&ntilde;O &oacute;O
− =M=
~ O &Auml;O
=
713. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=f&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&aring;&Ouml;=m&auml;~&aring;&Eacute;&euml;=E`~&euml;&Eacute;=NPF=
&ntilde;O &oacute;O
+ = M=
~ O &Auml;O
=
714. m~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&Eacute;&ecirc;=E`~&euml;&Eacute;=NQF=
&ntilde;O
−&oacute; =M=
~O
=
187
CHAPTER 7. ANALYTIC GEOMETRY
=
=====
=
Figure 150.
=
715. o&Eacute;~&auml;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=m&auml;~&aring;&Eacute;&euml;=E`~&euml;&Eacute;=NRF=
&ntilde;O
= N=
~O
=
716. f&atilde;~&Ouml;&aacute;&aring;~&ecirc;&oacute;=m~&ecirc;~&auml;&auml;&Eacute;&auml;=m&auml;~&aring;&Eacute;&euml;=E`~&euml;&Eacute;=NSF=
&ntilde;O
= −N =
~O
=
717. `&ccedil;&aacute;&aring;&Aring;&aacute;&Ccedil;&Eacute;&aring;&iacute;=m&auml;~&aring;&Eacute;&euml;=E`~&euml;&Eacute;=NTF=
&ntilde;O = M =
=
=
=
=
=
188
CHAPTER 7. ANALYTIC GEOMETRY
7.12 Sphere
=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=~=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W=o=
m&ccedil;&aacute;&aring;&iacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;I= &ntilde; N I= &oacute; N I= &ograve; N I=&pound;=
`&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=~=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;W= (~ I &Auml;I &Aring; ) =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=^I=aI=bI=cI=j=
=
=
718. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=p&eacute;&Uuml;&Eacute;&ecirc;&Eacute;=`&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil;=~&iacute;=&iacute;&Uuml;&Eacute;=l&ecirc;&aacute;&Ouml;&aacute;&aring;=Ep&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=
c&ccedil;&ecirc;&atilde;F=
&ntilde; O + &oacute; O + &ograve; O = oO =
=
=
======
=
Figure 151.
=
719. b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=`&aacute;&ecirc;&Aring;&auml;&Eacute;=`&Eacute;&aring;&iacute;&Eacute;&ecirc;&Eacute;&Ccedil;=~&iacute;=^&aring;&oacute;=m&ccedil;&aacute;&aring;&iacute;= (~ I &Auml;I &Aring; )
(&ntilde; − ~ )O + (&oacute; − &Auml;)O + (&ograve; − &Aring; )O = o O
720. a&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;=c&ccedil;&ecirc;&atilde;
(&ntilde; − &ntilde; N )(&ntilde; − &ntilde; O ) + (&oacute; − &oacute;N )(&oacute; − &oacute; O ) + (&ograve; − &ograve;N )(&ograve; − &ograve; O ) = M I==
189
CHAPTER 7. ANALYTIC GEOMETRY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
mN (&ntilde; N I &oacute; N I &ograve; N ) I= mO (&ntilde; O I &oacute; O I &ograve; O ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Eacute;&aring;&Ccedil;&euml;=&ccedil;&Ntilde;=~=&Ccedil;&aacute;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;K==
=
721. c&ccedil;&igrave;&ecirc;=m&ccedil;&aacute;&aring;&iacute;=c&ccedil;&ecirc;&atilde;=
&ntilde;O + &oacute;O + &ograve;O &ntilde; &oacute; &ograve; N
&ntilde; NO + &oacute; NO + &ntilde; NO &ntilde; N &oacute; N &ograve; N N
&ntilde; OO + &oacute; OO + &ntilde; OO &ntilde; O &oacute; O &ograve; O N = M =
&ntilde; PO + &oacute; PO + &ntilde; PO &ntilde; P &oacute; P &ograve; P N
&ntilde; OQ + &oacute; OQ + &ntilde; OQ &ntilde; Q &oacute; Q &ograve; Q N
=
722. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=c&ccedil;&ecirc;&atilde;
^&ntilde; O + ^&oacute; O + ^&ograve; O + a&ntilde; + b&oacute; + c&ograve; + j = M =E^=&aacute;&euml;=&aring;&ccedil;&aring;&ograve;&Eacute;&ecirc;&ccedil;FK==
q&Uuml;&Eacute;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;=&Uuml;~&euml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;= (~ I &Auml;I &Aring; ) I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
a
b
c
~=−
I= &Auml; = −
I= &Aring; = −
K=
O^
O^
O^
q&Uuml;&Eacute;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&Eacute;=&aacute;&euml;
aO + b O + c O − Q ^ O j
K
o=
O^
=
=
190
Chapter 8
Differential Calculus
=
=
=
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&Ouml;I=&oacute;I=&igrave;I=&icirc;=
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;=E&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;FW=&ntilde;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=~I=&Auml;I=&Aring;I=&Ccedil;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
^&aring;&Ouml;&auml;&Eacute;W= α =
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &Ntilde; −N =
=
=
8.1 Functions and Their Graphs
723.
724.
725.
726.
=
b&icirc;&Eacute;&aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&Ntilde; (− &ntilde; ) = &Ntilde; (&ntilde; ) =
=
l&Ccedil;&Ccedil;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&Ntilde; (− &ntilde; ) = −&Ntilde; (&ntilde; ) =
=
m&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&Ntilde; (&ntilde; + &aring;q ) = &Ntilde; (&ntilde; ) =
=
f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&oacute; = &Ntilde; (&ntilde; ) =&aacute;&euml;=~&aring;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I= &ntilde; = &Ouml;(&oacute; ) =&ccedil;&ecirc;= &oacute; = &Ntilde; −N (&ntilde; ) =&aacute;&euml;=&aacute;&iacute;&euml;=&aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=
&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K==
=
191
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=====
=
Figure 152.
=
727. `&ccedil;&atilde;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&oacute; = &Ntilde; (&igrave; ) I= &igrave; = &Ouml;(&ntilde; ) I= &oacute; = &Ntilde; (&Ouml;(&ntilde; )) =&aacute;&euml;=~=&Aring;&ccedil;&atilde;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K=
=
728. i&aacute;&aring;&Eacute;~&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&oacute; = ~&ntilde; + &Auml; I== &ntilde; ∈ o I== ~ = &iacute;~&aring; α ==&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&auml;&ccedil;&eacute;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;I==&Auml;==&aacute;&euml;=
&iacute;&Uuml;&Eacute;=&oacute;-&aacute;&aring;&iacute;&Eacute;&ecirc;&Aring;&Eacute;&eacute;&iacute;K=
=
192
CHAPTER 8. DIFFERENTIAL CALCULUS
=
======
=
Figure 153.
=
729. n&igrave;~&Ccedil;&ecirc;~&iacute;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &ntilde; O I= &ntilde; ∈ o K=
=
=
======
=
Figure 154.
=
193
CHAPTER 8. DIFFERENTIAL CALCULUS
730. &oacute; = ~&ntilde; O + &Auml;&ntilde; + &Aring; I= &ntilde; ∈ o K=
=
=
=
===
Figure 155.
=
731. `&igrave;&Auml;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &ntilde; P I= &ntilde; ∈ o K=
=
194
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=
=
Figure 156.
=
732. &oacute; = ~&ntilde; P + &Auml;&ntilde; O + &Aring;&ntilde; + &Ccedil; I= &ntilde; ∈ o K=
=
195
CHAPTER 8. DIFFERENTIAL CALCULUS
=
==
=
Figure 157.
=
733. m&ccedil;&iuml;&Eacute;&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &ntilde; &aring; I= &aring;∈ k K=
=
196
CHAPTER 8. DIFFERENTIAL CALCULUS
======
Figure 158.
=
=
Figure 159.
=
197
=
CHAPTER 8. DIFFERENTIAL CALCULUS
734. p&egrave;&igrave;~&ecirc;&Eacute;=o&ccedil;&ccedil;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = &ntilde; I= &ntilde; ∈ [MI ∞ ) K=
=
=======
Figure 160.
=
=
735. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=
&oacute; = ~ &ntilde; I= ~ &gt; M I= ~ ≠ N I=
&oacute; = &Eacute; &ntilde; =&aacute;&Ntilde;= ~ = &Eacute; I= &Eacute; = OKTNUOUNUOUQSK =
=
=
=
=
Figure 161.
198
CHAPTER 8. DIFFERENTIAL CALCULUS
736. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=
&oacute; = &auml;&ccedil;&Ouml; ~ &ntilde; I= &ntilde; ∈ (MI ∞ ) I= ~ &gt; M I= ~ ≠ N I=
&oacute; = &auml;&aring; &ntilde; =&aacute;&Ntilde;= ~ = &Eacute; I= &ntilde; &gt; M K=
=
=
Figure 162.
=
737. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=p&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&Eacute;&ntilde; − &Eacute;−&ntilde;
&oacute; = &euml;&aacute;&aring;&Uuml; &ntilde; I= &euml;&aacute;&aring;&Uuml; &ntilde; =
I= &ntilde; ∈ o K=
O
=
199
=
CHAPTER 8. DIFFERENTIAL CALCULUS
=
====
=
Figure 163.
=
738. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&euml;&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&Eacute;&ntilde; + &Eacute;−&ntilde;
&oacute; = &Aring;&ccedil;&euml; &Uuml; &ntilde; I= &Aring;&ccedil;&euml; &Uuml; &ntilde; =
I= &ntilde; ∈ o K=
O
=
=
======
Figure 164.
200
CHAPTER 8. DIFFERENTIAL CALCULUS
739. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=q~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; &ntilde; − &Eacute; − &ntilde;
&oacute; = &iacute;~&aring;&Uuml; &ntilde; I= &oacute; = &iacute;~&aring;&Uuml; &ntilde; =
=
I= &ntilde; ∈ o K=
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; &ntilde; + &Eacute; − &ntilde;
=
=
======
=
Figure 165.
=
740. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; &ntilde; + &Eacute; − &ntilde;
&oacute; = &Aring;&ccedil;&iacute; &Uuml; &ntilde; I= &oacute; = &Aring;&ccedil;&iacute; &Uuml; &ntilde; =
=
I= &ntilde; ∈ o I= &ntilde; ≠ M K=
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; &ntilde; − &Eacute; − &ntilde;
=
201
CHAPTER 8. DIFFERENTIAL CALCULUS
=
======
=
Figure 166.
=
741. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=p&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
N
O
&oacute; = &euml;&Eacute;&Aring; &Uuml; &ntilde; I= &oacute; = &euml;&Eacute;&Aring; &Uuml; &ntilde; =
= &ntilde; − &ntilde; I= &ntilde; ∈ o K=
&Aring;&ccedil;&euml;&Uuml; &ntilde; &Eacute; + &Eacute;
Figure 167.
202
=
CHAPTER 8. DIFFERENTIAL CALCULUS
742. e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
N
O
= &ntilde; − &ntilde; I= &ntilde; ∈ o I= &ntilde; ≠ M K=
&oacute; = &Aring;&euml;&Aring;&Uuml; &ntilde; I= &oacute; = &Aring;&euml;&Aring;&Uuml; &ntilde; =
&euml;&aacute;&aring;&Uuml; &ntilde; &Eacute; − &Eacute;
=
=
======
=
Figure 168.
=
743. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=p&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml; &ntilde; I= &ntilde; ∈ o K=
=
203
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=====
=
Figure 169.
=
744. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&euml;&aacute;&aring;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&euml;&Uuml; &ntilde; I= &ntilde; ∈ [NI ∞ ) K=
=
=
=====
=
Figure 170.
=
745. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=q~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&iacute;~&aring;&Uuml; &ntilde; I= &ntilde; ∈ (− NI N) K=
=
204
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=====
=
Figure 171.
=
746. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&Aring;&ccedil;&iacute;&Uuml; &ntilde; I= &ntilde; ∈ (− ∞I − N) ∪ (NI ∞ ) K==
=
205
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=====
=
Figure 172.
=
747. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=p&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&euml;&Eacute;&Aring;&Uuml; &ntilde; I= &ntilde; ∈ (MI N] K=
=
206
CHAPTER 8. DIFFERENTIAL CALCULUS
Figure 173.
=
=
748. f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=e&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;=`&ccedil;&euml;&Eacute;&Aring;~&aring;&iacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&oacute; = ~&ecirc;&Aring;&Aring;&euml;&Aring;&Uuml; &ntilde; I= &ntilde; ∈ o I= &ntilde; ≠ M K==
Figure 174.
=
207
=
CHAPTER 8. DIFFERENTIAL CALCULUS
8.2 Limits of Functions
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W= &Ntilde; (&ntilde; ) I= &Ouml;(&ntilde; ) =
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;W=&ntilde;=
o&Eacute;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W=~I=&acirc;=
=
=
749. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) + &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =
&ntilde; →~
&ntilde; →~
&ntilde; →~
=
750. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) − &Ouml; (&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) − &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =
&ntilde; →~
&ntilde; →~
&ntilde; →~
=
751. &auml;&aacute;&atilde;[&Ntilde; (&ntilde; ) ⋅ &Ouml; (&ntilde; )] = &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) ⋅ &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =
&ntilde; →~
=
&ntilde; →~
&ntilde; →~
&Ntilde; (&ntilde; )
&Ntilde; (&ntilde; ) &auml;&aacute;&atilde;
= &ntilde; →~
I=&aacute;&Ntilde;= &auml;&aacute;&atilde; &Ouml;(&ntilde; ) ≠ M K=
&ntilde; →~
&ntilde; → ~ &Ouml; (&ntilde; )
&auml;&aacute;&atilde; &Ouml;(&ntilde; )
752. &auml;&aacute;&atilde;
&ntilde; →~
=
753. &auml;&aacute;&atilde;[&acirc;&Ntilde; (&ntilde; )] = &acirc; &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) =
&ntilde; →~
&ntilde; →~
(
=
)
754. &auml;&aacute;&atilde; &Ntilde; (&Ouml; (&ntilde; )) = &Ntilde; &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =
&ntilde; →~
&ntilde; →~
=
755. &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) = &Ntilde; (~ ) I=&aacute;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &Ntilde; (&ntilde; ) =&aacute;&euml;=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=~&iacute;= &ntilde; = ~ K=
&ntilde; →~
=
&euml;&aacute;&aring; &ntilde;
= N=
&ntilde; →M
&ntilde;
756. &auml;&aacute;&atilde;
=
757. &auml;&aacute;&atilde;
&ntilde; →M
&iacute;~&aring; &ntilde;
=N=
&ntilde;
=
&euml;&aacute;&aring; −N &ntilde;
= N=
&ntilde; →M
&ntilde;
758. &auml;&aacute;&atilde;
208
CHAPTER 8. DIFFERENTIAL CALCULUS
&iacute;~&aring; −N &ntilde;
= N=
&ntilde; →M
&ntilde;
759. &auml;&aacute;&atilde;
=
&auml;&aring;(N + &ntilde; )
= N=
&ntilde; →M
&ntilde;
760. &auml;&aacute;&atilde;
=
&ntilde;
 N
761. &auml;&aacute;&atilde; N +  = &Eacute; =
&ntilde; →∞
 &ntilde;
=
&ntilde;
 &acirc;
762. &auml;&aacute;&atilde; N +  = &Eacute; &acirc; =
&ntilde; →∞
 &ntilde;
=
763. &auml;&aacute;&atilde; ~ &ntilde; = N =
&ntilde; →M
=
=
=
8.3 Definition and Properties of the Derivative
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&Ouml;I=&oacute;I=&igrave;I=&icirc;=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&ntilde;=
o&Eacute;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W=&acirc;=
^&aring;&Ouml;&auml;&Eacute;W= α =
=
=
&Ntilde; (&ntilde; + ∆&ntilde; ) − &Ntilde; (&ntilde; )
∆&oacute; &Ccedil;&oacute;
764. &oacute;′(&ntilde; ) = &auml;&aacute;&atilde;
==
= &auml;&aacute;&atilde;
=
∆&ntilde; → M
∆&ntilde; →M ∆&ntilde;
∆&ntilde;
&Ccedil;&ntilde;
=
209
CHAPTER 8. DIFFERENTIAL CALCULUS
=
==
=
Figure 175.
=
&Ccedil;&oacute;
= &iacute;~&aring; α ==
&Ccedil;&ntilde;
765.
=
766.
=
767.
=
768.
&Ccedil;(&igrave; + &icirc; ) &Ccedil;&igrave; &Ccedil;&icirc;
=
+
=
&Ccedil;&ntilde;
&Ccedil;&ntilde; &Ccedil;&ntilde;
&Ccedil;(&igrave; − &icirc; ) &Ccedil;&igrave; &Ccedil;&icirc;
=
−
=
&Ccedil;&ntilde;
&Ccedil;&ntilde; &Ccedil;&ntilde;
&Ccedil;(&acirc;&igrave; )
&Ccedil;&igrave;
=&acirc;
=
&Ccedil;&ntilde;
&Ccedil;&ntilde;
=
769. m&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;=o&igrave;&auml;&Eacute;=
&Ccedil;(&igrave; ⋅ &icirc; ) &Ccedil;&igrave;
&Ccedil;&icirc;
=
⋅ &icirc; + &igrave; ⋅ ==
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
=
=
210
CHAPTER 8. DIFFERENTIAL CALCULUS
770. n&igrave;&ccedil;&iacute;&aacute;&Eacute;&aring;&iacute;=o&igrave;&auml;&Eacute;=
&Ccedil;&igrave;
&Ccedil;&icirc;
⋅&icirc; −&igrave;⋅
&Ccedil;  &igrave;  &Ccedil;&ntilde;
&Ccedil;&ntilde; =
 =
O
&Ccedil;&ntilde;  &icirc; 
&icirc;
=
771. `&Uuml;~&aacute;&aring;=o&igrave;&auml;&Eacute;=
&oacute; = &Ntilde; (&Ouml;(&ntilde; )) I= &igrave; = &Ouml;(&ntilde; ) I==
&Ccedil;&oacute; &Ccedil;&oacute; &Ccedil;&igrave;
=
⋅
K=
&Ccedil;&ntilde; &Ccedil;&igrave; &Ccedil;&ntilde;
=
772. a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=f&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&Ccedil;&oacute;
N
=
I==
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&oacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &ntilde; (&oacute; ) &aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&icirc;&Eacute;&ecirc;&euml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;= &oacute; (&ntilde; ) K==
=
773. o&Eacute;&Aring;&aacute;&eacute;&ecirc;&ccedil;&Aring;~&auml;=o&igrave;&auml;&Eacute;=
&Ccedil;&oacute;
&Ccedil;  N
  = − &Ccedil;&ntilde;O =
&Ccedil;&ntilde;  &oacute; 
&oacute;
=
774. i&ccedil;&Ouml;~&ecirc;&aacute;&iacute;&Uuml;&atilde;&aacute;&Aring;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring;=
&oacute; = &Ntilde; (&ntilde; ) I= &auml;&aring; &oacute; = &auml;&aring; &Ntilde; (&ntilde; ) I==
&Ccedil;&oacute;
&Ccedil;
= &Ntilde; (&ntilde; ) ⋅ [&auml;&aring; &Ntilde; (&ntilde; )] K=
&Ccedil;&ntilde;
&Ccedil;&ntilde;
=
=
8.4 Table of Derivatives
=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&ntilde;=
o&Eacute;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W=`I=~I=&Auml;I=&Aring;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
211
CHAPTER 8. DIFFERENTIAL CALCULUS
&Ccedil;
(` ) = M =
&Ccedil;&ntilde;
775.
=
&Ccedil;
(&ntilde; ) = N =
&Ccedil;&ntilde;
776.
=
&Ccedil;
(~&ntilde; + &Auml;) = ~ =
&Ccedil;&ntilde;
777.
=
778.
=
779.
=
780.
&Ccedil;
(~&ntilde; O + &Auml;&ntilde; + &Aring; ) = ~&ntilde; + &Auml; =
&Ccedil;&ntilde;
&Ccedil; &aring;
(
&ntilde; ) = &aring;&ntilde; &aring; −N =
&Ccedil;&ntilde;
&Ccedil; −&aring;
&aring;
(
&ntilde; ) = − &aring; +N =
&ntilde;
&Ccedil;&ntilde;
=
&Ccedil; N
N
 =− O =
&Ccedil;&ntilde;  &ntilde; 
&ntilde;
781.
=
782.
&Ccedil;
&Ccedil;&ntilde;
( &ntilde; ) = O N&ntilde; =
&Ccedil;
&Ccedil;&ntilde;
( &ntilde; )=
=
783.
N
&aring;
&aring; &aring; &ntilde; &aring; −N
=
=
&Ccedil;
(&auml;&aring; &ntilde; ) = N =
&ntilde;
&Ccedil;&ntilde;
784.
=
&Ccedil;
(&auml;&ccedil;&Ouml; ~ &ntilde; ) = N I= ~ &gt; M I= ~ ≠ N K=
&ntilde; &auml;&aring; ~
&Ccedil;&ntilde;
785.
=
212
CHAPTER 8. DIFFERENTIAL CALCULUS
&Ccedil; &ntilde;
(
~ ) = ~ &ntilde; &auml;&aring; ~ I= ~ &gt; M I= ~ ≠ N K=
&Ccedil;&ntilde;
786.
=
787.
&Ccedil; &ntilde;
(
&Eacute; ) = &Eacute;&ntilde; =
&Ccedil;&ntilde;
=
&Ccedil;
(&euml;&aacute;&aring; &ntilde; ) = &Aring;&ccedil;&euml; &ntilde; =
&Ccedil;&ntilde;
788.
=
&Ccedil;
(&Aring;&ccedil;&euml; &ntilde; ) = − &euml;&aacute;&aring; &ntilde; =
&Ccedil;&ntilde;
789.
=
&Ccedil;
(&iacute;~&aring; &ntilde; ) = NO = &euml;&Eacute;&Aring; O &ntilde; =
&Ccedil;&ntilde;
&Aring;&ccedil;&euml; &ntilde;
790.
=
&Ccedil;
(&Aring;&ccedil;&iacute; &ntilde; ) = − NO = − &Aring;&euml;&Aring; O &ntilde; =
&Ccedil;&ntilde;
&euml;&aacute;&aring; &ntilde;
791.
=
&Ccedil;
(&euml;&Eacute;&Aring; &ntilde; ) = &iacute;~&aring; &ntilde; ⋅ &euml;&Eacute;&Aring; &ntilde; =
&Ccedil;&ntilde;
792.
=
&Ccedil;
(&Aring;&euml;&Aring; &ntilde; ) = − &Aring;&ccedil;&iacute; &ntilde; ⋅ &Aring;&euml;&Aring; &ntilde; =
&Ccedil;&ntilde;
793.
=
&Ccedil;
(~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; ) = N O =
&Ccedil;&ntilde;
N− &ntilde;
794.
=
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; ) = − N O =
&Ccedil;&ntilde;
N− &ntilde;
795.
=
&Ccedil;
(~&ecirc;&Aring;&iacute;~&aring; &ntilde; ) = N O =
N+ &ntilde;
&Ccedil;&ntilde;
796.
=
213
CHAPTER 8. DIFFERENTIAL CALCULUS
&Ccedil;
(~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; ) = − N O =
&Ccedil;&ntilde;
N+ &ntilde;
797.
=
&Ccedil;
(~&ecirc;&Aring; &euml;&Eacute;&Aring; &ntilde; ) = NO =
&Ccedil;&ntilde;
&ntilde; &ntilde; −N
798.
=
&Ccedil;
(~&ecirc;&Aring; &Aring;&euml;&Aring; &ntilde; ) = − NO =
&Ccedil;&ntilde;
&ntilde; &ntilde; −N
799.
=
&Ccedil;
(&euml;&aacute;&aring;&Uuml; &ntilde; ) = &Aring;&ccedil;&euml;&Uuml; &ntilde; =
&Ccedil;&ntilde;
800.
=
&Ccedil;
(&Aring;&ccedil;&euml;&Uuml; &ntilde; ) = &euml;&aacute;&aring;&Uuml; &ntilde; =
&Ccedil;&ntilde;
801.
=
&Ccedil;
(&iacute;~&aring;&Uuml; &ntilde; ) = N O = &euml;&Eacute;&Aring;&Uuml; O &ntilde; =
&Aring;&ccedil;&euml;&Uuml; &ntilde;
&Ccedil;&ntilde;
802.
=
&Ccedil;
(&Aring;&ccedil;&iacute;&Uuml; &ntilde; ) = − N O = −&Aring;&euml;&Aring;&Uuml; O &ntilde; =
&Ccedil;&ntilde;
&euml;&aacute;&aring;&Uuml; &ntilde;
803.
=
&Ccedil;
(&euml;&Eacute;&Aring;&Uuml; &ntilde; ) = −&euml;&Eacute;&Aring;&Uuml; &ntilde; ⋅ &iacute;~&aring;&Uuml; &ntilde; =
&Ccedil;&ntilde;
804.
=
&Ccedil;
(&Aring;&euml;&Aring;&Uuml; &ntilde; ) = −&Aring;&euml;&Aring;&Uuml; &ntilde; ⋅ &Aring;&ccedil;&iacute;&Uuml; &ntilde; =
&Ccedil;&ntilde;
805.
=
&Ccedil;
(~&ecirc;&Aring;&euml;&aacute;&aring;&Uuml;=&ntilde; ) = NO =
&Ccedil;&ntilde;
&ntilde; +N
806.
=
807.
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&euml;&Uuml;=&ntilde; ) = ON =
&Ccedil;&ntilde;
&ntilde; −N
214
CHAPTER 8. DIFFERENTIAL CALCULUS
&Ccedil;
(~&ecirc;&Aring;&iacute;~&aring;&Uuml;=&ntilde; ) = N O I= &ntilde; &lt; N K=
&Ccedil;&ntilde;
N− &ntilde;
808.
=
&Ccedil;
(~&ecirc;&Aring;&Aring;&ccedil;&iacute;&Uuml;=&ntilde; ) = − ON I= &ntilde; &gt; N K=
&Ccedil;&ntilde;
&ntilde; −N
809.
=
810.
&Ccedil; &icirc;
&Ccedil;&igrave;
&Ccedil;&icirc;
(
+ &igrave; &icirc; &auml;&aring; &igrave; ⋅ =
&igrave; ) = &icirc;&igrave; &icirc; −N ⋅
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
=
=
=
8.5 Higher Order Derivatives
811.
812.
813.
814.
815.
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&oacute;I=&igrave;I=&icirc;=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&ntilde;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
=
=
p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=
O
 &Ccedil;&oacute;  ′ &Ccedil;  &Ccedil;&oacute;  &Ccedil; &oacute;
′
&Ntilde; ′′ = (&Ntilde; ′) =   =
 = O =
 &Ccedil;&ntilde;  &Ccedil;&ntilde;  &Ccedil;&ntilde;  &Ccedil;&ntilde;
=
e&aacute;&Ouml;&Uuml;&Eacute;&ecirc;-l&ecirc;&Ccedil;&Eacute;&ecirc;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=
&Ccedil;&aring; &oacute;
(&aring; )
&Ntilde; = &aring; = &oacute; (&aring; ) = (&Ntilde; (&aring; −N) ) ′ =
&Ccedil;&ntilde;
=
(&igrave; + &icirc; )(&aring; ) = &igrave; (&aring; ) + &icirc; (&aring; ) =
=
(&igrave; − &icirc; )(&aring; ) = &igrave;(&aring; ) − &icirc; (&aring; ) =
=
i&Eacute;&aacute;&Auml;&aring;&aacute;&iacute;&ograve;∞&euml;=c&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;=
(&igrave;&icirc; )′′ = &igrave;′′&icirc; + O&igrave;′&icirc;′ + &igrave;&icirc;′′ =
215
CHAPTER 8. DIFFERENTIAL CALCULUS
(&igrave;&icirc; )′′′ = &igrave;′′′&icirc; + P&igrave;′′&icirc;′ + P&igrave;′&icirc;′′ + &igrave;&icirc;′′′ =
(&igrave;&icirc; )(&aring; ) = &igrave; (&aring; ) &icirc; + &aring;&igrave; (&aring;−N) &icirc; ′ + &aring;(&aring; − N) &igrave; (&aring;−O) &icirc; ′′ + K + &igrave;&icirc; (&aring; ) =
N⋅ O
=
(&ntilde; )( ) = (&atilde;&atilde;−&gt;&aring;)&gt; &ntilde;
&atilde;
816.
=
=
&atilde; −&aring;
=
(&ntilde; )( ) = &aring;&gt; =
&aring;
817.
&aring;
&aring;
&aring; −N
(
− N) (&aring; − N)&gt;
(&auml;&ccedil;&Ouml; ~ &ntilde; ) =
=
&aring;
(&aring; )
818.
&ntilde; &auml;&aring; ~
=
819.
&aring; −N
(&auml;&aring; &ntilde; )(&aring; ) = (− N) (&aring;&aring; − N)&gt; =
&ntilde;
=
(~ )( ) = ~
&aring;
&ntilde;
&auml;&aring; &aring; ~ =
(&Eacute; )( ) = &Eacute;
&ntilde;
=
&ntilde;
820.
=
&ntilde;
821.
=
(~ )( ) = &atilde; ~
&atilde;&ntilde;
822.
&aring;
&aring;
&aring; &atilde;&ntilde;
&auml;&aring; &aring; ~ =
=
(&euml;&aacute;&aring; &ntilde; )(&aring; ) = &euml;&aacute;&aring; &ntilde; + &aring;π  =
823.

O 
=
(&Aring;&ccedil;&euml; &ntilde; )(&aring; ) = &Aring;&ccedil;&euml; &ntilde; + &aring;π  =
824.

O 
=
=
=
216
CHAPTER 8. DIFFERENTIAL CALCULUS
8.6 Applications of Derivative
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&Ouml;I=&oacute;=
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~&aring;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;W=&euml;==
s&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;W=&icirc;=
^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;W=&iuml;=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&ntilde;=
q&aacute;&atilde;&Eacute;W=&iacute;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
=
=
825. s&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;=~&aring;&Ccedil;=^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;=
&euml; = &Ntilde; (&iacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~&aring;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;=&ecirc;&Eacute;&auml;~&iacute;&aacute;&icirc;&Eacute;=&iacute;&ccedil;=~=&Ntilde;&aacute;&ntilde;&Eacute;&Ccedil;=
&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=&euml;&oacute;&euml;&iacute;&Eacute;&atilde;=~&iacute;=~=&iacute;&aacute;&atilde;&Eacute;=&iacute;I==
&icirc; = &euml;′ = &Ntilde; ′(&iacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&euml;&iacute;~&aring;&iacute;~&aring;&Eacute;&ccedil;&igrave;&euml;=&icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I=
&iuml; = &icirc;′ = &euml;′′ = &Ntilde; ′′(&iacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&euml;&iacute;~&aring;&iacute;~&aring;&Eacute;&ccedil;&igrave;&euml;=~&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=
&iacute;&Uuml;&Eacute;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;K==
=
826. q~&aring;&Ouml;&Eacute;&aring;&iacute;=i&aacute;&aring;&Eacute;=
&oacute; − &oacute; M = &Ntilde; ′(&ntilde; M )(&ntilde; − &ntilde; M ) =
=
217
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=
Figure 176.
=
827. k&ccedil;&ecirc;&atilde;~&auml;=i&aacute;&aring;&Eacute;=
N
(&ntilde; − &ntilde; M ) =Ec&aacute;&Ouml;=NTSF=
&oacute; − &oacute;M = −
&Ntilde; ′(&ntilde; M )
=
828. f&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=~&aring;&Ccedil;=a&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K==
f&Ntilde;= &Ntilde; ′(&ntilde; M ) &gt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&aacute;&euml;=&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=~&iacute;= &ntilde; M K=Ec&aacute;&Ouml;=NTTI= &ntilde; &lt; &ntilde; N I=
&ntilde; O &lt; &ntilde; FI=
f&Ntilde;= &Ntilde; ′(&ntilde; M ) &lt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&aacute;&euml;=&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=~&iacute;= &ntilde; M K=Ec&aacute;&Ouml;=NTTI=
&ntilde; N &lt; &ntilde; &lt; &ntilde; O FI=
f&Ntilde;= &Ntilde; ′ (&ntilde; M ) =&Ccedil;&ccedil;&Eacute;&euml;=&aring;&ccedil;&iacute;=&Eacute;&ntilde;&aacute;&euml;&iacute;=&ccedil;&ecirc;=&aacute;&euml;=&ograve;&Eacute;&ecirc;&ccedil;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&euml;&iacute;=&Ntilde;~&aacute;&auml;&euml;K==
=
218
CHAPTER 8. DIFFERENTIAL CALCULUS
=
=
Figure 177.
=
829. i&ccedil;&Aring;~&auml;=&Eacute;&ntilde;&iacute;&ecirc;&Eacute;&atilde;~=
^=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&Ntilde;E&ntilde;F=&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;=~&iacute;= &ntilde; N =&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;=
&iacute;&Uuml;&Eacute;&ecirc;&Eacute;=&Eacute;&ntilde;&aacute;&euml;&iacute;&euml;=&euml;&ccedil;&atilde;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&aacute;&aring;&Ouml;= &ntilde; N =&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
&Ntilde; (&ntilde; N ) ≥ &Ntilde; (&ntilde; ) =&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&ntilde;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=Ec&aacute;&Ouml;KNTTFK==
=
^=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&Ntilde;E&ntilde;F=&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;=~&iacute;= &ntilde; O =&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;=
&iacute;&Uuml;&Eacute;&ecirc;&Eacute;=&Eacute;&ntilde;&aacute;&euml;&iacute;&euml;=&euml;&ccedil;&atilde;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&aacute;&aring;&Ouml;= &ntilde; O =&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
&Ntilde; (&ntilde; O ) ≤ &Ntilde; (&ntilde; ) =&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&ntilde;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=Ec&aacute;&Ouml;KNTTFK=
=
830. `&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml;=m&ccedil;&aacute;&aring;&iacute;&euml;=
^=&Aring;&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml;=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&aring;=&Ntilde;E&ntilde;F=&ccedil;&Aring;&Aring;&igrave;&ecirc;&euml;=~&iacute;= &ntilde; M =&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;=&Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc;=
&Ntilde; ′ (&ntilde; M ) =&aacute;&euml;=&ograve;&Eacute;&ecirc;&ccedil;=&ccedil;&ecirc;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&Ccedil;&ccedil;&Eacute;&euml;&aring;∞&iacute;=&Eacute;&ntilde;&aacute;&euml;&iacute;K=
=
831. c&aacute;&ecirc;&euml;&iacute;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=q&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=i&ccedil;&Aring;~&auml;=b&ntilde;&iacute;&ecirc;&Eacute;&atilde;~K=
f&Ntilde;=&Ntilde;E&ntilde;F=&aacute;&euml;==&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;==E &Ntilde; ′ (&ntilde; ) &gt; M F=&Ntilde;&ccedil;&ecirc;==~&auml;&auml;==&ntilde;==&aacute;&aring;==&euml;&ccedil;&atilde;&Eacute;==&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=
(~I &ntilde; N ] ==~&aring;&Ccedil;==&Ntilde;E&ntilde;F==&aacute;&euml;==&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;==E &Ntilde; ′ (&ntilde; ) &lt; M F==&Ntilde;&ccedil;&ecirc;=~&auml;&auml;==&ntilde;=&aacute;&aring;=&euml;&ccedil;&atilde;&Eacute;=
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;= = [&ntilde; N I &Auml;) I= = &iacute;&Uuml;&Eacute;&aring;= &Ntilde;E&ntilde;F= &Uuml;~&euml;= ~= = &auml;&ccedil;&Aring;~&auml;= &atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;= = ~&iacute;= = &ntilde; N =
Ec&aacute;&Ouml;KNTTFK==
219
CHAPTER 8. DIFFERENTIAL CALCULUS
832. f&Ntilde;=&Ntilde;E&ntilde;F=&aacute;&euml;=&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=E &Ntilde; ′ (&ntilde; ) &lt; M F=&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&ntilde;=&aacute;&aring;=&euml;&ccedil;&atilde;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=
(~I &ntilde; O ]=~&aring;&Ccedil;=&Ntilde;E&ntilde;F=&aacute;&euml;=&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=E &Ntilde; ′ (&ntilde; ) &gt; M F=&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&ntilde;=&aacute;&aring;=&euml;&ccedil;&atilde;&Eacute;=
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;= [&ntilde; O I &Auml;) I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;=~&iacute;= &ntilde; O K==
Ec&aacute;&Ouml;KNTTFK=
=
833. p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=q&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=i&ccedil;&Aring;~&auml;=b&ntilde;&iacute;&ecirc;&Eacute;&atilde;~K=
f&Ntilde;= &Ntilde; ′ (&ntilde; N ) = M =~&aring;&Ccedil;= &Ntilde; ′′(&ntilde; N ) &lt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;=
~&iacute;== &ntilde; N K=
f&Ntilde;= &Ntilde; ′ (&ntilde; O ) = M =~&aring;&Ccedil;= &Ntilde; ′′(&ntilde; O ) &gt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;=
~&iacute;= &ntilde; O K=Ec&aacute;&Ouml;KNTTF=
=
834. `&ccedil;&aring;&Aring;~&icirc;&aacute;&iacute;&oacute;K==
&Ntilde;E&ntilde;F= &aacute;&euml;= = &Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute;= &igrave;&eacute;&iuml;~&ecirc;&Ccedil;= ~&iacute;= = &ntilde; M = = &aacute;&Ntilde;= = ~&aring;&Ccedil;= = &ccedil;&aring;&auml;&oacute;= = &aacute;&Ntilde;= = &Ntilde; ′ (&ntilde; ) = &aacute;&euml;============
&aacute;&aring;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=~&iacute;= &ntilde; M =Ec&aacute;&Ouml;KNTTI= &ntilde; P &lt; &ntilde; FK===
&Ntilde;E&ntilde;F=&aacute;&euml;==&Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute;==&Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil;=~&iacute;== &ntilde; M ==&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;== &Ntilde; ′ (&ntilde; ) ==&aacute;&euml;===============
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=~&iacute;= &ntilde; M K=Ec&aacute;&Ouml;KNTTI= &ntilde; &lt; &ntilde; P FK===
=
835. p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=q&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=`&ccedil;&aring;&Aring;~&icirc;&aacute;&iacute;&oacute;K==
f&Ntilde;= &Ntilde; ′′(&ntilde; M ) &gt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&aacute;&euml;=&Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute;=&igrave;&eacute;&iuml;~&ecirc;&Ccedil;=~&iacute;= &ntilde; M K==
f&Ntilde;= &Ntilde; ′′(&ntilde; M ) &lt; M I=&iacute;&Uuml;&Eacute;&aring;=&Ntilde;E&ntilde;F=&aacute;&euml;=&Aring;&ccedil;&aring;&Aring;~&icirc;&Eacute;=&Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil;=~&iacute;= &ntilde; M K=
f&Ntilde;= &Ntilde; ′′(&ntilde; ) =&Ccedil;&ccedil;&Eacute;&euml;=&aring;&ccedil;&iacute;=&Eacute;&ntilde;&aacute;&euml;&iacute;=&ccedil;&ecirc;=&aacute;&euml;=&ograve;&Eacute;&ecirc;&ccedil;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&euml;&iacute;=&Ntilde;~&aacute;&auml;&euml;K=
=
836. f&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=m&ccedil;&aacute;&aring;&iacute;&euml;=
f&Ntilde;== &Ntilde; ′ (&ntilde; P ) ==&Eacute;&ntilde;&aacute;&euml;&iacute;&euml;==~&aring;&Ccedil;== &Ntilde; ′′(&ntilde; ) ==&Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;=&euml;&aacute;&Ouml;&aring;=~&iacute;= &ntilde; = &ntilde; P I==&iacute;&Uuml;&Eacute;&aring;=
&iacute;&Uuml;&Eacute;= &eacute;&ccedil;&aacute;&aring;&iacute;= (&ntilde; P I &Ntilde; (&ntilde; P )) = &aacute;&euml;= ~&aring;= &aacute;&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;= &eacute;&ccedil;&aacute;&aring;&iacute;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &Ouml;&ecirc;~&eacute;&Uuml;= &ccedil;&Ntilde;=
&Ntilde; (&ntilde; ) K=f&Ntilde;= &Ntilde; ′′(&ntilde; P ) =&Eacute;&ntilde;&aacute;&euml;&iacute;&euml;=~&iacute;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&Ntilde;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&eacute;&ccedil;&aacute;&aring;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= &Ntilde; ′′(&ntilde; P ) = M =
Ec&aacute;&Ouml;KNTTFK=
=
837. i∞e&ccedil;&eacute;&aacute;&iacute;~&auml;∞&euml;=o&igrave;&auml;&Eacute;=
M
&Ntilde; (&ntilde; )
&Ntilde; ′(&ntilde; )
&auml;&aacute;&atilde;
= &auml;&aacute;&atilde;
=&aacute;&Ntilde;= &auml;&aacute;&atilde; &Ntilde; (&ntilde; ) = &auml;&aacute;&atilde; &Ouml; (&ntilde; ) =  K==
&ntilde; → &Aring; &Ouml; (&ntilde; )
&ntilde; →&Aring; &Ouml; ′(&ntilde; )
&ntilde; →&Aring;
&ntilde; →&Aring;
∞
=
220
CHAPTER 8. DIFFERENTIAL CALCULUS
8.7 Differential
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&igrave;I=&icirc;=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&ntilde;=
a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &oacute;′(&ntilde; ) I= &Ntilde; ′(&ntilde; ) =
o&Eacute;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W=`=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&ccedil;&Ntilde;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &oacute; = &Ntilde; (&ntilde; ) W=&Ccedil;&oacute;=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&ccedil;&Ntilde;=&ntilde;W=&Ccedil;&ntilde;=
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;=&aacute;&aring;=&ntilde;W= ∆&ntilde; =
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;=&aacute;&aring;=&oacute;W= ∆&oacute; =
=
=
838. &Ccedil;&oacute; = &oacute;′&Ccedil;&ntilde; =
=
839. &Ntilde; (&ntilde; + ∆&ntilde; ) = &Ntilde; (&ntilde; ) + &Ntilde; ′(&ntilde; )∆&ntilde; =
=
=
=
Figure 178.
221
CHAPTER 8. DIFFERENTIAL CALCULUS
840. p&atilde;~&auml;&auml;=`&Uuml;~&aring;&Ouml;&Eacute;=&aacute;&aring;=&oacute;=
∆&oacute; = &Ntilde; (&ntilde; + ∆&ntilde; ) − &Ntilde; (&ntilde; ) =
=
841. &Ccedil;(&igrave; + &icirc; ) = &Ccedil;&igrave; + &Ccedil;&icirc; =
=
842. &Ccedil;(&igrave; − &icirc; ) = &Ccedil;&igrave; − &Ccedil;&icirc; =
=
843. &Ccedil;(`&igrave; ) = `&Ccedil;&igrave; =
=
844. &Ccedil;(&igrave;&icirc; ) = &icirc;&Ccedil;&igrave; + &igrave;&Ccedil;&icirc; =
=
 &igrave;  &icirc;&Ccedil;&igrave; − &igrave;&Ccedil;&icirc;
845. &Ccedil;  =
=
&icirc;O
&icirc;
=
=
=
8.8 Multivariable Functions
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&iuml;&ccedil;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W= &ograve; (&ntilde; I &oacute; ) I= &Ntilde; (&ntilde; I &oacute; ) I= &Ouml; (&ntilde; I &oacute; ) I= &Uuml;(&ntilde; I &oacute; ) ==
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;&euml;W=&ntilde;I=&oacute;I=&iacute;=
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;=&aacute;&aring;=&ntilde;I=&oacute;I=&ograve;I=&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;W= ∆&ntilde; I= ∆&oacute; I= ∆&ograve; K=
=
=
846. c&aacute;&ecirc;&euml;&iacute;=l&ecirc;&Ccedil;&Eacute;&ecirc;=m~&ecirc;&iacute;&aacute;~&auml;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;=
q&Uuml;&Eacute;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&iuml;&aacute;&iacute;&Uuml;=&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;=&iacute;&ccedil;=&ntilde;=
∂&Ntilde;
∂&ograve;
= &Ntilde; &ntilde; =E~&auml;&euml;&ccedil;= = &ograve; &ntilde; FI=
∂&ntilde;
∂&ntilde;
q&Uuml;&Eacute;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&iuml;&aacute;&iacute;&Uuml;=&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;=&iacute;&ccedil;=&oacute;=
∂&Ntilde;
∂&ograve;
= &Ntilde; &oacute; =E~&auml;&euml;&ccedil;= = &ograve; &oacute; FK=
∂&oacute;
∂&oacute;
=
=
222
CHAPTER 8. DIFFERENTIAL CALCULUS
847. p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=l&ecirc;&Ccedil;&Eacute;&ecirc;=m~&ecirc;&iacute;&aacute;~&auml;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;=
∂  ∂&Ntilde;  ∂ O&Ntilde;
= &Ntilde; &ntilde;&ntilde; I==
 =
∂&ntilde;  ∂&ntilde;  ∂&ntilde; O
∂  ∂&Ntilde;  ∂ O&Ntilde;
 =
= &Ntilde; &oacute;&oacute; I==
∂&oacute;  ∂&oacute;  ∂&oacute; O
∂  ∂&Ntilde;  ∂ O&Ntilde;
= &Ntilde; &ntilde;&oacute; I==
 =
∂&oacute;  ∂&ntilde;  ∂&oacute;∂&ntilde;
∂  ∂&Ntilde;  ∂ O&Ntilde;
 =
= &Ntilde; &oacute;&ntilde; K==
∂&ntilde;  ∂&oacute;  ∂&ntilde;∂&oacute;
f&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;=~&ecirc;&Eacute;=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I=&iacute;&Uuml;&Eacute;&aring;==
∂ O&Ntilde;
∂ O&Ntilde;
=
K==
∂&oacute;∂&ntilde; ∂&ntilde;∂&oacute;
=
848. `&Uuml;~&aacute;&aring;=o&igrave;&auml;&Eacute;&euml;==
f&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) = &Ouml;(&Uuml;(&ntilde; I &oacute; )) =E&Ouml;=&aacute;&euml;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&ccedil;&aring;&Eacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;=&Uuml;FI=&iacute;&Uuml;&Eacute;&aring;==
∂&Ntilde;
∂&Uuml; ∂&Ntilde;
∂&Uuml;
= &Ouml;′(&Uuml;(&ntilde; I &oacute; )) I= = &Ouml;′(&Uuml;(&ntilde; I &oacute; )) K==
∂&oacute;
∂&ntilde;
∂&ntilde; ∂&oacute;
=
∂&Ntilde; &Ccedil;&ntilde; ∂&Ntilde; &Ccedil;&oacute;
+
K==
f&Ntilde;= &Uuml;(&iacute; ) = &Ntilde; (&ntilde; (&iacute; )I &oacute; (&iacute; )) I=&iacute;&Uuml;&Eacute;&aring;= &Uuml;′(&iacute; ) =
∂&ntilde; &Ccedil;&iacute; ∂&oacute; &Ccedil;&iacute;
=
f&Ntilde;= &ograve; = &Ntilde; (&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )) I=&iacute;&Uuml;&Eacute;&aring;==
∂&ograve; ∂&Ntilde; ∂&ntilde; ∂&Ntilde; ∂&oacute; ∂&ograve; ∂&Ntilde; ∂&ntilde; ∂&Ntilde; ∂&oacute;
+
K==
+
=
I= =
∂&igrave; ∂&ntilde; ∂&igrave; ∂&oacute; ∂&igrave; ∂&icirc; ∂&ntilde; ∂&icirc; ∂&oacute; ∂&icirc;
=
849. p&atilde;~&auml;&auml;=`&Uuml;~&aring;&Ouml;&Eacute;&euml;=
∂&Ntilde;
∂&Ntilde;
∆&ograve; ≈ ∆&ntilde; + ∆&oacute; =
∂&ntilde;
∂&oacute;
=
=
223
CHAPTER 8. DIFFERENTIAL CALCULUS
850. i&ccedil;&Aring;~&auml;=j~&ntilde;&aacute;&atilde;~=~&aring;&Ccedil;=j&aacute;&aring;&aacute;&atilde;~=
&Ntilde; (&ntilde; I &oacute; ) =&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;=~&iacute;= (&ntilde; M I &oacute; M ) =&aacute;&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) ≤ &Ntilde; (&ntilde; M I &oacute; M ) =
&Ntilde;&ccedil;&ecirc;=~&auml;&auml;= (&ntilde; I &oacute; ) =&euml;&igrave;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&auml;&oacute;=&Aring;&auml;&ccedil;&euml;&Eacute;=&iacute;&ccedil;= (&ntilde; M I &oacute; M ) K==
=
&Ntilde; (&ntilde; I &oacute; ) =&Uuml;~&euml;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;=~&iacute;= (&ntilde; M I &oacute; M ) =&aacute;&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) ≥ &Ntilde; (&ntilde; M I &oacute; M ) =
&Ntilde;&ccedil;&ecirc;=~&auml;&auml;= (&ntilde; I &oacute; ) =&euml;&igrave;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&auml;&oacute;=&Aring;&auml;&ccedil;&euml;&Eacute;=&iacute;&ccedil;= (&ntilde; M I &oacute; M ) K=
=
851. p&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute;=m&ccedil;&aacute;&aring;&iacute;&euml;=
∂&Ntilde; ∂&Ntilde;
=
= M K=
∂&ntilde; ∂&oacute;
i&ccedil;&Aring;~&auml;=&atilde;~&ntilde;&aacute;&atilde;~=~&aring;&Ccedil;=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;~=&ccedil;&Aring;&Aring;&igrave;&ecirc;=~&iacute;=&euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;K=
==
852. p~&Ccedil;&Ccedil;&auml;&Eacute;=m&ccedil;&aacute;&aring;&iacute;=
^=&euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute;==&eacute;&ccedil;&aacute;&aring;&iacute;==&iuml;&Uuml;&aacute;&Aring;&Uuml;==&aacute;&euml;==&aring;&Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc;==~==&auml;&ccedil;&Aring;~&auml;==&atilde;~&ntilde;&aacute;&atilde;&igrave;&atilde;=
&aring;&ccedil;&ecirc;=~=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;&igrave;&atilde;=
=
853. p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=q&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=p&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute;=m&ccedil;&aacute;&aring;&iacute;&euml;=
∂&Ntilde; ∂&Ntilde;
=
= M FK==
i&Eacute;&iacute;= (&ntilde; M I &oacute; M ) =&Auml;&Eacute;=~=&euml;&iacute;~&iacute;&aacute;&ccedil;&aring;~&ecirc;&oacute;=&eacute;&ccedil;&aacute;&aring;&iacute;=E
∂&ntilde; ∂&oacute;
&Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &Ntilde; &ntilde;&oacute; (&ntilde; M I &oacute; M )
K==
a=
&Ntilde; &oacute;&ntilde; (&ntilde; M I &oacute; M ) &Ntilde; &oacute;&oacute; (&ntilde; M I &oacute; M )
=
f&Ntilde;= a &gt; M I= &Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &gt; M I== (&ntilde; M I &oacute; M ) ==&aacute;&euml;=~=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=&auml;&ccedil;&Aring;~&auml;=&atilde;&aacute;&aring;&aacute;&atilde;~K=
f&Ntilde;= a &gt; M I= &Ntilde; &ntilde;&ntilde; (&ntilde; M I &oacute; M ) &lt; M I== (&ntilde; M I &oacute; M ) ==&aacute;&euml;=~=&eacute;&ccedil;&aacute;&aring;&iacute;=&ccedil;&Ntilde;=&auml;&ccedil;&Aring;~&auml;=&atilde;~&ntilde;&aacute;&atilde;~K=
f&Ntilde;= a &lt; M I= (&ntilde; M I &oacute; M ) =&aacute;&euml;=~=&euml;~&Ccedil;&Ccedil;&auml;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;K=
f&Ntilde;= a = M I=&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&euml;&iacute;=&Ntilde;~&aacute;&auml;&euml;K=
=
854. q~&aring;&Ouml;&Eacute;&aring;&iacute;=m&auml;~&aring;&Eacute;=
q&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=&eacute;&auml;~&aring;&Eacute;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;= &ograve; = &Ntilde; (&ntilde; I &oacute; ) =
~&iacute;= (&ntilde; M I &oacute; M I &ograve; M ) =&aacute;&euml;==
&ograve; − &ograve; M = &Ntilde; &ntilde; (&ntilde; M I &oacute; M )(&ntilde; − &ntilde; M ) + &Ntilde; &oacute; (&ntilde; M I &oacute; M )(&oacute; − &oacute; M ) K=
=
224
CHAPTER 8. DIFFERENTIAL CALCULUS
855. k&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=
q&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;= &ograve; = &Ntilde; (&ntilde; I &oacute; ) =~&iacute;=
(&ntilde; M I &oacute; M I &ograve; M ) =&aacute;&euml;==
&ntilde; − &ntilde;M
&oacute; − &oacute;M
&ograve; − &ograve;M
=
=
K=
&Ntilde; &ntilde; ( &ntilde; M I &oacute; M ) &Ntilde; &oacute; (&ntilde; M I &oacute; M )
−N
=
=
=
8.9 Differential Operators
=
r r r
r&aring;&aacute;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;=~&auml;&ccedil;&aring;&Ouml;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=~&ntilde;&Eacute;&euml;W= &aacute; I= &agrave; I= &acirc; =
p&Aring;~&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=E&euml;&Aring;~&auml;~&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;&euml;FW= &Ntilde; (&ntilde; I &oacute; I &ograve; ) I= &igrave;(&ntilde; N I &ntilde; O I KI &ntilde; &aring; ) =
d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=&euml;&Aring;~&auml;~&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= &Ouml;&ecirc;~&Ccedil; &igrave; I= ∇&igrave; =
∂&Ntilde;
a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;W= =
∂&auml;
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=E&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;FW= c (mI nI o ) =
r
r
a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= &Ccedil;&aacute;&icirc; c I= ∇ ⋅ c =
r
r
`&igrave;&ecirc;&auml;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= &Aring;&igrave;&ecirc;&auml; c I= ∇ &times; c =
i~&eacute;&auml;~&Aring;&aacute;~&aring;=&ccedil;&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;W= ∇ O =
=
=
856. d&ecirc;~&Ccedil;&aacute;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=p&Aring;~&auml;~&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
 ∂&Ntilde; ∂&Ntilde; ∂&Ntilde; 
&Ouml;&ecirc;~&Ccedil; &Ntilde; = ∇&Ntilde; =  I I  I==
 ∂&ntilde; ∂&oacute; ∂&ograve; 
 ∂&igrave; ∂&igrave;
∂&igrave; 
 K=
&Ouml;&ecirc;~&Ccedil; &igrave; = ∇&igrave; = 
I
IKI
∂&ntilde; &aring; 
 ∂&ntilde; N ∂&ntilde; O
=
857. a&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;~&auml;=a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=
∂&Ntilde; ∂&Ntilde;
∂&Ntilde;
∂&Ntilde;
= &Aring;&ccedil;&euml; α + &Aring;&ccedil;&euml; β + &Aring;&ccedil;&euml; γ I==
∂&auml; ∂&ntilde;
∂&oacute;
∂&ograve;
225
CHAPTER 8. DIFFERENTIAL CALCULUS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&ecirc;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=
r
&auml; (&Aring;&ccedil;&euml; αI &Aring;&ccedil;&euml; βI &Aring;&ccedil;&euml; γ ) I= &Aring;&ccedil;&euml; O α + &Aring;&ccedil;&euml; O β + &Aring;&ccedil;&euml; O γ = N K==
=
858. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=~=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=c&aacute;&Eacute;&auml;&Ccedil;=
r
r ∂m ∂n ∂o
+
+
=
&Ccedil;&aacute;&icirc; c = ∇ ⋅ c =
∂&ntilde; ∂&oacute; ∂&ograve;
=
859. `&igrave;&ecirc;&auml;=&ccedil;&Ntilde;=~=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=c&aacute;&Eacute;&auml;&Ccedil;=
r
r
r
&aacute;
&agrave;
&acirc;
r
r ∂
∂
∂
=
&Aring;&igrave;&ecirc;&auml; c = ∇ &times; c =
∂&ntilde; ∂&ntilde; ∂&ntilde;
m n o
 ∂o ∂n  r  ∂m ∂o  r  ∂n ∂m  r
 &aacute; +  −
= 
−
− &acirc; =
 &agrave; + 
 ∂&oacute; ∂&ograve;   ∂&ograve; ∂&ntilde;   ∂&ntilde; ∂&oacute; 
=
860. i~&eacute;&auml;~&Aring;&aacute;~&aring;=l&eacute;&Eacute;&ecirc;~&iacute;&ccedil;&ecirc;=
∂ O&Ntilde; ∂ O&Ntilde; ∂ O&Ntilde;
∇ O&Ntilde; = O + O + O =
∂&ograve;
∂&oacute;
∂&ntilde;
=
r
r
861. &Ccedil;&aacute;&icirc; &Aring;&igrave;&ecirc;&auml; c = ∇ ⋅ ∇ &times; c ≡ M =
=
862. &Aring;&igrave;&ecirc;&auml;(&Ouml;&ecirc;~&Ccedil; &Ntilde; ) = ∇ &times; (∇&Ntilde; ) ≡ M =
=
863. &Ccedil;&aacute;&icirc; (&Ouml;&ecirc;~&Ccedil; &Ntilde; ) = ∇ ⋅ (∇&Ntilde; ) = ∇ O&Ntilde; =
=
r
r
r
r
r
864. &Aring;&igrave;&ecirc;&auml; &Aring;&igrave;&ecirc;&auml; c = &Ouml;&ecirc;~&Ccedil; &Ccedil;&aacute;&icirc; c − ∇ Oc = ∇ ∇ ⋅ c − ∇ Oc =
=
=
(
)
(
)
(
)
(
)
226
( )
Chapter 9
Integral Calculus
=
=
=
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W=&Ntilde;I=&Ouml;I=&igrave;I=&icirc;=
f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W=&ntilde;I=&iacute;I= ξ =
f&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; I= ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; I=&pound;=
a&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &oacute;′(&ntilde; ) I= &Ntilde; ′(&ntilde; ) I= c′(&ntilde; ) I=&pound;=
o&Eacute;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;W=`I=~I=&Auml;I=&Aring;I=&Ccedil;I=&acirc;=
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&atilde;I=&aring;I=&aacute;I=&agrave;=
=
=
9.1 Indefinite Integral
=
865.
=
866.
=
867.
=
868.
=
869.
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&ntilde; ) + ` =&aacute;&Ntilde;= c′(&ntilde; ) = &Ntilde; (&ntilde; ) K=
(∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ) ′= &Ntilde; (&ntilde; ) =
∫ &acirc;&Ntilde; (&ntilde; )&Ccedil;&ntilde; = &acirc; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
∫ [&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; =
∫ [&Ntilde; (&ntilde; ) − &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; − ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; =
=
870.
N
∫ &Ntilde; (~&ntilde; )&Ccedil;&ntilde; = ~ c(~&ntilde; ) + ` =
227
CHAPTER 9. INTEGRAL CALCULUS
N
∫ &Ntilde; (~&ntilde; + &Auml;)&Ccedil;&ntilde; = ~ c(~&ntilde; + &Auml;) + ` =
871.
=
N
∫ &Ntilde; (&ntilde; )&Ntilde; ′(&ntilde; )&Ccedil;&ntilde; = O &Ntilde; (&ntilde; ) + ` =
O
872.
=
&Ntilde; ′(&ntilde; )
∫ &Ntilde; (&ntilde; ) &Ccedil;&ntilde; = &auml;&aring; &Ntilde; (&ntilde; ) + ` =
873.
=
874. j&Eacute;&iacute;&Uuml;&ccedil;&Ccedil;=&ccedil;&Ntilde;=p&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;=
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&igrave;(&iacute; ))&igrave;′(&iacute; )&Ccedil;&iacute; =&aacute;&Ntilde;= &ntilde; = &igrave;(&iacute; ) K=
=
875. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&Auml;&oacute;=m~&ecirc;&iacute;&euml;=
∫ &igrave;&Ccedil;&icirc; = &igrave;&icirc; − ∫ &icirc;&Ccedil;&igrave; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &igrave;(&ntilde; ) I= &icirc; (&ntilde; ) =~&ecirc;&Eacute;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K==
=
=
=
9.2 Integrals of Rational Functions
=
∫ ~&Ccedil;&ntilde; = ~&ntilde; + ` =
876.
=
∫ &ntilde;&Ccedil;&ntilde; =
877.
&ntilde;O
+`=
O
=
878.
O
∫ &ntilde; &Ccedil;&ntilde; =
&ntilde;P
+`=
P
&eacute;
∫ &ntilde; &Ccedil;&ntilde; =
&ntilde; &eacute; +N
+ ` I= &eacute; ≠ −N K=
&eacute;+N
=
879.
=
228
CHAPTER 9. INTEGRAL CALCULUS
&aring; +N
(
~&ntilde; + &Auml;)
∫ (~&ntilde; + &Auml;) &Ccedil;&ntilde; = ~(&aring; + N) + ` I= &aring; ≠ −N K=
&aring;
880.
=
∫
881.
&Ccedil;&ntilde;
= &auml;&aring; &ntilde; + ` =
&ntilde;
=
&Ccedil;&ntilde;
N
∫ ~&ntilde; + &Auml; = ~ &auml;&aring; ~&ntilde; + &Auml; + ` =
882.
=
~&ntilde; + &Auml;
~
∫ &Aring;&ntilde; + &Ccedil; &Ccedil;&ntilde; = &Aring; &ntilde; +
883.
&Auml;&Aring; − ~&Ccedil;
&auml;&aring; &Aring;&ntilde; + &Ccedil; + ` =
&Aring;O
=
&Ccedil;&ntilde;
&ntilde;+&Auml;
N
∫ (&ntilde; + ~ )(&ntilde; + &Auml;) = ~ − &Auml; &auml;&aring; &ntilde; + ~ + ` I= ~ ≠ &Auml; K=
884.
=
N
&ntilde;&Ccedil;&ntilde;
∫ ~ + &Auml;&ntilde; = &Auml; (~ + &Auml;&ntilde; − ~ &auml;&aring; ~ + &Auml;&ntilde; ) + ` =
885.
O
=
&ntilde; O&Ccedil;&ntilde;
N N

O
O
∫ ~ + &Auml;&ntilde; = &Auml;P  O (~ + &Auml;&ntilde; ) − O~(~ + &Auml;&ntilde; ) + ~ &auml;&aring; ~ + &Auml;&ntilde;  + ` =
886.
=
&Ccedil;&ntilde;
N
∫ &ntilde; (~ + &Auml;&ntilde; ) = ~ &auml;&aring;
887.
~ + &Auml;&ntilde;
+`=
&ntilde;
=
888.
&Ccedil;&ntilde;
N
&Auml;
∫ &ntilde; (~ + &Auml;&ntilde; ) = − ~&ntilde; + ~
O
O
&auml;&aring;
~ + &Auml;&ntilde;
+`=
&ntilde;
=
889.
&ntilde;&Ccedil;&ntilde;
∫ (~ + &Auml;&ntilde; )
O
=
N
~ 
&auml;&aring; ~ + &Auml;&ntilde; +
+`=
O 
&Auml; 
~ + &Auml;&ntilde; 
=
229
CHAPTER 9. INTEGRAL CALCULUS
&ntilde; O&Ccedil;&ntilde;
N
~O 

+`=
=
+
−
+
−
~
&Auml;&ntilde;
O
~
&auml;&aring;
~
&Auml;&ntilde;
∫ (~ + &Auml;&ntilde; )O &Auml;P 
~ + &Auml;&ntilde; 
890.
=
&Ccedil;&ntilde;
∫ &ntilde; (~ + &Auml;&ntilde; )
891.
O
=
N
N ~ + &Auml;&ntilde;
+ O &auml;&aring;
+`=
~ (~ + &Auml;&ntilde; ) ~
&ntilde;
=
∫&ntilde;
892.
&Ccedil;&ntilde;
N &ntilde; −N
+`=
= &auml;&aring;
O
−N O &ntilde; +N
=
&Ccedil;&ntilde;
∫ N− &ntilde;
893.
O
N N+ &ntilde;
= &auml;&aring;
+`=
O N− &ntilde;
=
894.
&Ccedil;&ntilde;
N ~+&ntilde;
+`=
= &auml;&aring;
O
O~ ~ − &ntilde;
−&ntilde;
∫~
O
∫&ntilde;
O
=
895.
&ntilde;−~
&Ccedil;&ntilde;
N
+`=
= &auml;&aring;
O
O~ &ntilde; + ~
−~
=
&Ccedil;&ntilde;
∫ N+ &ntilde;
896.
O
= &iacute;~&aring; −N &ntilde; + ` =
=
897.
∫~
O
∫&ntilde;
O
=
898.
N
&ntilde;
&Ccedil;&ntilde;
= &iacute;~&aring; −N + ` =
O
~
~
+&ntilde;
N
&ntilde;&Ccedil;&ntilde;
= &auml;&aring;(&ntilde; O + ~ O ) + ` =
O
+~
O
=
&Ccedil;&ntilde;
∫ ~ + &Auml;&ntilde;
899.
O
=
 &Auml;
N
 + ` I= ~&Auml; &gt; M K=
~&ecirc;&Aring;&iacute;~&aring; &ntilde;

~
~&Auml;


=
230
CHAPTER 9. INTEGRAL CALCULUS
&ntilde;&Ccedil;&ntilde;
∫ ~ + &Auml;&ntilde;
900.
O
=
~
N
&auml;&aring; &ntilde; O + + ` =
&Auml;
O&Auml;
=
&Ccedil;&ntilde;
N
&ntilde;O
=
&auml;&aring;
∫ &ntilde; (~ + &Auml;&ntilde; O ) O~ ~ + &Auml;&ntilde; O + ` =
901.
=
∫~
902.
O
&Ccedil;&ntilde;
N
~ + &Auml;&ntilde;
=
&auml;&aring;
+`=
O O
−&Auml; &ntilde;
O~&Auml; ~ − &Auml;&ntilde;
=
&Ccedil;&ntilde;
N
O~&ntilde; + &Auml; − &Auml;O − Q~&Aring;
903. ∫ O
=
&auml;&aring;
+ ` I=
~&ntilde; + &Auml;&ntilde; + &Aring;
&Auml;O − Q~&Aring; O~&ntilde; + &Auml; + &Auml;O − Q~&Aring;
&Auml;O − Q~&Aring; &gt; M K=
=
&Ccedil;&ntilde;
O
O~&ntilde; + &Auml;
~&ecirc;&Aring;&iacute;~&aring;
+ ` I=
=
+ &Auml;&ntilde; + &Aring;
Q~&Aring; − &Auml;O
Q~&Aring; − &Auml;O
&Auml;O − Q~&Aring; &lt; M K=
∫ ~&ntilde;
904.
O
=
=
=
9.3 Integrals of Irrational Functions
=
905.
∫
&Ccedil;&ntilde;
O
~&ntilde; + &Auml; + ` =
=
~&ntilde; + &Auml; ~
∫
~&ntilde; + &Auml; &Ccedil;&ntilde; =
∫
&ntilde;&Ccedil;&ntilde;
O(~&ntilde; − O&Auml;)
=
~&ntilde; + &Auml; + ` =
P~ O
~&ntilde; + &Auml;
=
906.
=
907.
P
O
(~&ntilde; + &Auml;) O + ` =
P~
=
231
CHAPTER 9. INTEGRAL CALCULUS
∫&ntilde;
908.
~&ntilde; + &Auml; &Ccedil;&ntilde; =
P
O(P~&ntilde; − O&Auml;)
(~&ntilde; + &Auml;) O + ` =
O
NR~
=
~&ntilde; + &Auml; − &Auml; − ~&Aring;
&Ccedil;&ntilde;
N
&auml;&aring;
=
+ ` I==
~&ntilde; + &Auml; + &Auml; − ~&Aring;
~&ntilde; + &Auml;
&Auml; − ~&Aring;
&Auml; − ~&Aring; &gt; M K=
∫ (&ntilde; + &Aring; )
909.
=
~&ntilde; + &Auml;
&Ccedil;&ntilde;
N
=
~&ecirc;&Aring;&iacute;~&aring;
+ ` I==
~&Aring; − &Auml;
~&ntilde; + &Auml;
~&Aring; − &Auml;
&Auml; − ~&Aring; &lt; M K=
∫ (&ntilde; + &Aring; )
910.
=
911.
∫
N
~&ntilde; + &Auml;
(~&ntilde; + &Auml;)(&Aring;&ntilde; + &Ccedil; ) − =
&Ccedil;&ntilde; =
&Aring;&ntilde; + &Ccedil;
&Aring;
~&Ccedil; − &Auml;&Aring;
&auml;&aring; ~ (&Aring;&ntilde; + &Ccedil; ) + &Aring;(~&ntilde; + &Auml;) + ` I= ~ &gt; M K=
−
&Aring; ~&Aring;
∫
~&ntilde; + &Auml;
N
&Ccedil;&ntilde; =
&Aring;&ntilde; + &Ccedil;
&Aring;
=
912.
−
~ (&Aring;&ntilde; + &Ccedil; )
~&Ccedil; − &Auml;&Aring;
~&ecirc;&Aring;&iacute;~&aring;
+ ` I=E ~ &lt; M I= &Aring; &gt; M FK==
&Aring;(~&ntilde; + &Auml;)
&Aring; ~&Aring;
=
O
∫ &ntilde; ~ + &Auml;&ntilde; &Ccedil;&ntilde; =
913.
=
∫
914.
(~&ntilde; + &Auml;)(&Aring;&ntilde; + &Ccedil; ) − =
O(U~ O − NO~&Auml;&ntilde; + NR&Auml;O &ntilde; O )
NMR&Auml;P
(~ + &Auml;&ntilde; )P + ` =
&ntilde; O&Ccedil;&ntilde;
O(U~ O − Q~&Auml;&ntilde; + P&Auml;O &ntilde; O )
=
~ + &Auml;&ntilde; + ` =
NR&Auml;P
~ + &Auml;&ntilde;
=
∫&ntilde;
915.
&Ccedil;&ntilde;
N
~ + &Auml;&ntilde; − ~
=
+ ` I= ~ &gt; M K=
&auml;&aring;
~ + &Auml;&ntilde;
~
~ + &Auml;&ntilde; + ~
=
232
CHAPTER 9. INTEGRAL CALCULUS
∫&ntilde;
916.
&Ccedil;&ntilde;
O
~ + &Auml;&ntilde;
=
~&ecirc;&Aring;&iacute;~&aring;
+ ` I= ~ &lt; M K=
−~
~ + &Auml;&ntilde;
−~
=
917.
∫
~−&ntilde;
&Ccedil;&ntilde; =
&Auml;+ &ntilde;
∫
~+&ntilde;
&Auml;−&ntilde;
&Ccedil;&ntilde; = − (~ + &ntilde; )(&Auml; − &ntilde; ) − (~ + &Auml;)~&ecirc;&Aring;&euml;&aacute;&aring;
+`=
&Auml;−&ntilde;
~+&Auml;
∫
N+ &ntilde;
&Ccedil;&ntilde; = − N − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; + ` =
N− &ntilde;
(~ − &ntilde; )(&Auml; + &ntilde; ) + (~ + &Auml;)~&ecirc;&Aring;&euml;&aacute;&aring;
&ntilde;+&Auml;
+`=
~+&Auml;
=
918.
=
919.
=
&Ccedil;&ntilde;
∫ (&ntilde; − ~ )(&Auml; − ~ ) = O ~&ecirc;&Aring;&euml;&aacute;&aring;
920.
=
∫
921.
O&Aring;&ntilde; − &Auml;
~ + &Auml;&ntilde; − &Aring;&ntilde; O + =
Q&Aring;
&Auml;O − Q~&Aring;
O&Aring;&ntilde; − &Auml;
+
~&ecirc;&Aring;&euml;&aacute;&aring;
+`=
U &Aring;P
&Auml;O + Q~&Aring;
~ + &Auml;&ntilde; − &Aring;&ntilde; O &Ccedil;&ntilde; =
=
&Ccedil;&ntilde;
∫
922.
~&ntilde; O + &Auml;&ntilde; + &Aring;
~ &gt; M K=
=
N
&auml;&aring; O~&ntilde; + &Auml; + O ~ (~&ntilde; O + &Auml;&ntilde; + &Aring; ) + ` I==
~
=
∫
923.
&ntilde; −~
+` =
&Auml;−~
&Ccedil;&ntilde;
~&ntilde; + &Auml;&ntilde; + &Aring;
O
=−
N
O~&ntilde; + &Auml; O
~&ecirc;&Aring;&euml;&aacute;&aring;
&Auml; − Q~&Aring; + ` I= ~ &lt; M K=
Q~
~
=
∫
924.
&ntilde; O + ~ O &Ccedil;&ntilde; =
&ntilde; O O ~O
&ntilde; + ~ + &auml;&aring; &ntilde; + &ntilde; O + ~ O + ` =
O
O
=
233
CHAPTER 9. INTEGRAL CALCULUS
∫&ntilde;
925.
&ntilde; O + ~ O &Ccedil;&ntilde; =
N O O PO
(&ntilde; + ~ ) + ` =
P
=
∫&ntilde;
926.
&ntilde; O + ~ O &Ccedil;&ntilde; =
O
−
&ntilde;
(
O&ntilde; O + ~ O ) &ntilde; O + ~ O − =
U
~Q
&auml;&aring; &ntilde; + &ntilde; O + ~ O + ` =
U
=
∫
927.
&ntilde;O + ~O
&ntilde;O + ~O
&Ccedil;&ntilde;
=
−
+ &auml;&aring; &ntilde; + &ntilde; O + ~ O + ` =
&ntilde;O
&ntilde;
=
∫
928.
&Ccedil;&ntilde;
&ntilde; +~
O
O
= &auml;&aring; &ntilde; + &ntilde; O + ~ O + ` =
=
929.
∫
&ntilde; O + ~O
&ntilde;
&Ccedil;&ntilde; = &ntilde; O + ~ O + ~ &auml;&aring;
+`=
&ntilde;
~ + &ntilde; O + ~O
∫
&ntilde;&Ccedil;&ntilde;
=
930.
&ntilde; +~
O
O
= &ntilde; O + ~O + ` =
=
∫
931.
&ntilde; O&Ccedil;&ntilde;
&ntilde; O + ~O
=
&ntilde; O O ~O
&ntilde; + ~ − &auml;&aring; &ntilde; + &ntilde; O + ~ O + ` =
O
O
=
∫&ntilde;
932.
&Ccedil;&ntilde;
N
&ntilde;
= &auml;&aring;
+`=
&ntilde; O + ~O ~ ~ + &ntilde; O + ~O
=
∫
933.
&ntilde; O − ~ O &Ccedil;&ntilde; =
&ntilde; O O ~O
&ntilde; − ~ − &auml;&aring; &ntilde; + &ntilde; O − ~ O + ` =
O
O
=
934.
∫&ntilde;
&ntilde; O − ~ O &Ccedil;&ntilde; =
N O O PO
(&ntilde; − ~ ) + ` =
P
234
CHAPTER 9. INTEGRAL CALCULUS
935.
∫
&ntilde;O − ~O
~
&Ccedil;&ntilde; = &ntilde; O − ~ O + ~ ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
&ntilde;
&ntilde;
∫
&ntilde;O − ~O
&ntilde; O − ~O
&Ccedil;&ntilde;
=−
+ &auml;&aring; &ntilde; + &ntilde; O − ~ O + ` =
O
&ntilde;
&ntilde;
=
936.
=
∫
937.
&Ccedil;&ntilde;
&ntilde; −~
O
= &auml;&aring; &ntilde; + &ntilde; O − ~ O + ` =
O
=
∫
938.
&ntilde;&Ccedil;&ntilde;
&ntilde; −~
O
= &ntilde; O − ~O + ` =
O
=
∫
939.
&ntilde; O&Ccedil;&ntilde;
&ntilde; O O ~O
=
&ntilde; − ~ + &auml;&aring; &ntilde; + &ntilde; O − ~ O + ` =
O
O
O
O
&ntilde; −~
=
&Ccedil;&ntilde;
N
~
= − ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
~
&ntilde;
&ntilde; −~
∫&ntilde;
940.
O
O
=
&Ccedil;&ntilde;
∫ (&ntilde; + ~ )
941.
&ntilde; O − ~O
=
N &ntilde; −~
+`=
~ &ntilde;+~
=
&Ccedil;&ntilde;
∫ (&ntilde; − ~ )
942.
&ntilde; −~
O
O
=−
N &ntilde;+~
+`=
~ &ntilde; −~
=
∫&ntilde;
943.
&Ccedil;&ntilde;
O
&ntilde; O − ~O
=
&ntilde; O − ~O
+`=
~O&ntilde;
=
∫ (&ntilde;
944.
&Ccedil;&ntilde;
O
− ~O ) O
P
=−
&ntilde;
~O &ntilde; O − ~O
+`=
=
235
CHAPTER 9. INTEGRAL CALCULUS
∫ (&ntilde;
945.
O
− ~ O ) O &Ccedil;&ntilde; = −
P
&ntilde;
(
O&ntilde; O − R~ O ) &ntilde; O − ~ O + =
U
P~ Q
+
&auml;&aring; &ntilde; + &ntilde; O − ~ O + ` =
U
=
∫
946.
~ O − &ntilde; O &Ccedil;&ntilde; =
&ntilde;
~O
&ntilde; O
~ − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
~
O
O
=
O
O
∫ &ntilde; ~ − &ntilde; &Ccedil;&ntilde; = −
947.
P
N O
(
~ − &ntilde;O ) O + ` =
P
=
O
O
O
∫ &ntilde; ~ − &ntilde; &Ccedil;&ntilde; =
948.
&ntilde;
~Q
&ntilde;
(
O&ntilde; O − ~ O ) ~ O − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
U
U
~
=
949.
∫
~O − &ntilde; O
&ntilde;
&Ccedil;&ntilde; = ~ O − &ntilde; O + ~ &auml;&aring;
+`=
&ntilde;
~ + ~O − &ntilde; O
∫
&ntilde;
~O − &ntilde;O
~O − &ntilde;O
&Ccedil;&ntilde;
=
−
− ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
O
~
&ntilde;
&ntilde;
=
950.
=
∫
951.
&Ccedil;&ntilde;
= ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; + ` =
N− &ntilde;O
=
∫
952.
&Ccedil;&ntilde;
~O − &ntilde;O
= &euml;&aacute;&aring;
&ntilde;
+`=
~
=
∫
953.
&ntilde;&Ccedil;&ntilde;
~ −&ntilde;
O
O
= − ~O − &ntilde; O + ` =
=
954.
∫
&ntilde; O&Ccedil;&ntilde;
~O − &ntilde; O
=−
&ntilde; O
~O
&ntilde;
~ − &ntilde; O + ~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
O
O
~
236
CHAPTER 9. INTEGRAL CALCULUS
&Ccedil;&ntilde;
∫ (&ntilde; + ~ )
955.
~O − &ntilde; O
=−
N ~−&ntilde;
+`=
O ~+&ntilde;
=−
N ~+&ntilde;
+`=
O ~−&ntilde;
=
&Ccedil;&ntilde;
∫ (&ntilde; − ~ )
956.
~O − &ntilde; O
=
&Ccedil;&ntilde;
∫ (&ntilde; + &Auml; )
957.
~O − &ntilde; O
N
=
&Auml;O − ~ O
~&ecirc;&Aring;&euml;&aacute;&aring;
&Auml;&ntilde; + ~ O
+ ` I= &Auml; &gt; ~ K=
~ (&ntilde; + &Auml; )
=
&Ccedil;&ntilde;
∫ (&ntilde; + &Auml; )
958.
~ −&ntilde;
O
O
N
=
~ −&Auml;
O
O
&auml;&aring;
&ntilde;+&Auml;
~ −&Auml;
O
O
~ O − &ntilde; O + ~ O + &Auml;&ntilde;
&Auml; &lt; ~ K=
=
∫&ntilde;
959.
&Ccedil;&ntilde;
O
~O − &ntilde;O
=−
~O − &ntilde;O
+`=
~O&ntilde;
=
∫ (~
960.
O
− &ntilde; O ) O &Ccedil;&ntilde; =
P
&ntilde; O
P~ Q
&ntilde;
(
R~ − O&ntilde; O ) ~ O − &ntilde; O +
~&ecirc;&Aring;&euml;&aacute;&aring; + ` =
U
U
~
=
∫ (~
961.
&Ccedil;&ntilde;
O
−&ntilde;
O
)
P
=
O
&ntilde;
~O ~O − &ntilde; O
+`=
=
=
=
9.4 Integrals of Trigonometric Functions
=
∫ &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde; = − &Aring;&ccedil;&euml; &ntilde; + ` =
962.
=
963.
∫ &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde; = &euml;&aacute;&aring; &ntilde; + ` =
237
+ `I =
CHAPTER 9. INTEGRAL CALCULUS
∫ &euml;&aacute;&aring;
964.
O
&ntilde; &Ccedil;&ntilde; =
&ntilde; N
− &euml;&aacute;&aring; O&ntilde; + ` =
O Q
&ntilde; &Ccedil;&ntilde; =
&ntilde; N
+ &euml;&aacute;&aring; O&ntilde; + ` =
O Q
=
∫ &Aring;&ccedil;&euml;
965.
O
=
∫ &euml;&aacute;&aring;
966.
P
N
N
P
&ntilde; &Ccedil;&ntilde; = &Aring;&ccedil;&euml; P &ntilde; − &Aring;&ccedil;&euml; &ntilde; + ` = &Aring;&ccedil;&euml; P&ntilde; − &Aring;&ccedil;&euml; &ntilde; + ` =
P
NO
Q
=
∫ &Aring;&ccedil;&euml;
967.
P
N
N
P
&ntilde; &Ccedil;&ntilde; = &euml;&aacute;&aring; &ntilde; − &euml;&aacute;&aring; P &ntilde; + ` = &euml;&aacute;&aring; P&ntilde; + &euml;&aacute;&aring; &ntilde; + ` =
P
NO
Q
=
&Ccedil;&ntilde;
&ntilde;
&Ccedil;&ntilde;
&ntilde;
∫ &euml;&aacute;&aring; &ntilde; = ∫ &Aring;&euml;&Aring; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + ` =
968.
=
π
∫ &Aring;&ccedil;&euml; &ntilde; = ∫ &euml;&Eacute;&Aring; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + Q  + ` =
969.
=
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring;
970.
O
&ntilde;
= ∫ &Aring;&euml;&Aring; O &ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&iacute; &ntilde; + ` =
&ntilde;
= ∫ &euml;&Eacute;&Aring; O &ntilde; &Ccedil;&ntilde; = &iacute;~&aring; &ntilde; + ` =
&ntilde;
= ∫ &Aring;&euml;&Aring; P &ntilde; &Ccedil;&ntilde; = −
&ntilde;
= ∫ &euml;&Eacute;&Aring;P &ntilde; &Ccedil;&ntilde; =
=
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml;
971.
O
=
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring;
972.
P
&Aring;&ccedil;&euml; &ntilde;
N
&ntilde;
+ &auml;&aring; &iacute;~&aring; + ` =
O
O &euml;&aacute;&aring; &ntilde; O
O
=
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml;
973.
P
&euml;&aacute;&aring; &ntilde;
N
 &ntilde; π
+ &auml;&aring; &iacute;~&aring; +  + ` =
O
O &Aring;&ccedil;&euml; &ntilde; O
O Q
=
974.
N
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = − Q &Aring;&ccedil;&euml; O&ntilde; + ` =
238
CHAPTER 9. INTEGRAL CALCULUS
∫ &euml;&aacute;&aring;
975.
O
N
&ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &euml;&aacute;&aring; P &ntilde; + ` =
P
=
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml;
976.
O
N
&ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&euml;P &ntilde; + ` =
P
=
∫ &euml;&aacute;&aring;
977.
O
&ntilde; &Aring;&ccedil;&euml; O &ntilde; &Ccedil;&ntilde; =
&ntilde; N
− &euml;&aacute;&aring; Q &ntilde; + ` =
U PO
=
∫ &iacute;~&aring; &ntilde;&Ccedil;&ntilde; = − &auml;&aring; &Aring;&ccedil;&euml; &ntilde; + ` =
978.
=
&euml;&aacute;&aring; &ntilde;
N
&Ccedil;&ntilde; =
+ ` = &euml;&Eacute;&Aring; &ntilde; + ` =
O
&ntilde;
&Aring;&ccedil;&euml; &ntilde;
∫ &Aring;&ccedil;&euml;
979.
=
&euml;&aacute;&aring; O &ntilde;
 &ntilde; π
∫ &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; O + Q  − &euml;&aacute;&aring; &ntilde; + ` =
980.
=
∫ &iacute;~&aring;
981.
O
&ntilde; &Ccedil;&ntilde; = &iacute;~&aring; &ntilde; − &ntilde; + ` =
=
∫ &Aring;&ccedil;&iacute; &ntilde;&Ccedil;&ntilde; = &auml;&aring; &euml;&aacute;&aring; &ntilde; + ` =
982.
=
&Aring;&ccedil;&euml; &ntilde;
N
&Ccedil;&ntilde; = −
+ ` = − &Aring;&euml;&Aring; &ntilde; + ` =
O
&ntilde;
&euml;&aacute;&aring; &ntilde;
∫ &euml;&aacute;&aring;
983.
=
&Aring;&ccedil;&euml; O &ntilde;
&ntilde;
&Ccedil;&ntilde; = &auml;&aring; &iacute;~&aring; + &Aring;&ccedil;&euml; &ntilde; + ` =
984. ∫
&euml;&aacute;&aring; &ntilde;
O
=
985. ∫ &Aring;&ccedil;&iacute; O &ntilde; &Ccedil;&ntilde; = − &Aring;&ccedil;&iacute; &ntilde; − &ntilde; + ` =
=
986.
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml; &ntilde; &euml;&aacute;&aring; &ntilde; = &auml;&aring; &iacute;~&aring; &ntilde; + ` =
239
CHAPTER 9. INTEGRAL CALCULUS
∫ &euml;&aacute;&aring;
987.
O
&Ccedil;&ntilde;
N
 &ntilde; π
=−
+ &auml;&aring; &iacute;~&aring; +  + ` =
&ntilde; &Aring;&ccedil;&euml; &ntilde;
&euml;&aacute;&aring; &ntilde;
O Q
=
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml;
988.
O
&ntilde;
=
N
&ntilde;
+ &auml;&aring; &iacute;~&aring; + ` =
&Aring;&ccedil;&euml; &ntilde;
O
=
∫ &euml;&aacute;&aring;
989.
O
&Ccedil;&ntilde;
= &iacute;~&aring; &ntilde; − &Aring;&ccedil;&iacute; &ntilde; + ` =
&ntilde; &Aring;&ccedil;&euml; O &ntilde;
=
&euml;&aacute;&aring;(&atilde; + &aring; )&ntilde;
∫ &euml;&aacute;&aring; &atilde;&ntilde; &euml;&aacute;&aring; &aring;&ntilde; &Ccedil;&ntilde; = − O(&atilde; + &aring;)
990.
+
&euml;&aacute;&aring;(&atilde; − &aring; )&ntilde;
+ ` I=
O(&atilde; − &aring; )
−
&Aring;&ccedil;&euml;(&atilde; − &aring; )&ntilde;
+ ` I=
O(&atilde; − &aring; )
&atilde; O ≠ &aring; O K=
=
&Aring;&ccedil;&euml;(&atilde; + &aring; )&ntilde;
∫ &euml;&aacute;&aring; &atilde;&ntilde; &Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde; = − O(&atilde; + &aring;)
991.
&atilde; O ≠ &aring; O K=
=
&euml;&aacute;&aring;(&atilde; + &aring; )&ntilde;
∫ &Aring;&ccedil;&euml; &atilde;&ntilde; &Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde; = O(&atilde; + &aring;)
992.
&atilde; O ≠ &aring; O K=
=
∫ &euml;&Eacute;&Aring; &ntilde; &iacute;~&aring; &ntilde;&Ccedil;&ntilde; = &euml;&Eacute;&Aring; &ntilde; + ` =
993.
=
∫ &Aring;&euml;&Aring; &ntilde; &Aring;&ccedil;&iacute; &ntilde;&Ccedil;&ntilde; = − &Aring;&euml;&Aring; &ntilde; + ` =
994.
=
&aring;
∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = −
995.
&Aring;&ccedil;&euml; &aring;+N &ntilde;
+`=
&aring; +N
=
&euml;&aacute;&aring; &aring; +N &ntilde;
996. ∫ &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; =
+`=
&aring; +N
=
&aring;
240
+
&euml;&aacute;&aring;(&atilde; − &aring; )&ntilde;
+ ` I=
O(&atilde; − &aring; )
CHAPTER 9. INTEGRAL CALCULUS
997.
∫ ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; +
N− &ntilde;O + ` =
∫ ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; −
N− &ntilde;O + ` =
=
998.
=
∫ ~&ecirc;&Aring;&iacute;~&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring;&iacute;~&aring; &ntilde; − O &auml;&aring;(&ntilde;
N
999.
O
+ N) + ` =
=
N
1000. ∫ ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; &Ccedil;&ntilde; = &ntilde; ~&ecirc;&Aring; &Aring;&ccedil;&iacute; &ntilde; + &auml;&aring;(&ntilde; O + N) + ` =
O
=
=
=
9.5 Integrals of Hyperbolic Functions
=
1001. ∫ &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = &Aring;&ccedil;&euml;&Uuml; &ntilde; + ` =
=
1002. ∫ &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; = &euml;&aacute;&aring;&Uuml; &ntilde; + ` =
=
1003. ∫ &iacute;~&aring;&Uuml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &Aring;&ccedil;&euml;&Uuml; &ntilde; + ` =
=
1004. ∫ &Aring;&ccedil;&iacute;&Uuml; &ntilde; &Ccedil;&ntilde; = &auml;&aring; &euml;&aacute;&aring;&Uuml; &ntilde; + ` =
=
1005. ∫ &euml;&Eacute;&Aring;&Uuml; O &ntilde;&Ccedil;&ntilde; = &iacute;~&aring;&Uuml; &ntilde; + ` =
=
1006. ∫ &Aring;&euml;&Aring;&Uuml; O &ntilde;&Ccedil;&ntilde; = − &Aring;&ccedil;&iacute;&Uuml; &ntilde; + ` =
=
1007. ∫ &euml;&Eacute;&Aring;&Uuml;&ntilde; &iacute;~&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = −&euml;&Eacute;&Aring;&Uuml;&ntilde; + ` =
=
241
CHAPTER 9. INTEGRAL CALCULUS
1008. ∫ &Aring;&euml;&Aring;&Uuml;&ntilde; &Aring;&ccedil;&iacute;&Uuml; &ntilde;&Ccedil;&ntilde; = −&Aring;&euml;&Aring;&Uuml;&ntilde; + ` =
=
=
=
9.6 Integrals of Exponential and Logarithmic
Functions
=
1009. ∫ &Eacute; &ntilde; &Ccedil;&ntilde; = &Eacute; &ntilde; + ` =
=
~&ntilde;
1010. ∫ ~ &Ccedil;&ntilde; =
+`=
&auml;&aring; ~
=
&Eacute; ~&ntilde;
1011. ∫ &Eacute; ~&ntilde; &Ccedil;&ntilde; =
+`=
~
=
&Eacute; ~&ntilde;
1012. ∫ &ntilde;&Eacute; ~&ntilde; &Ccedil;&ntilde; = O (~&ntilde; − N) + ` =
~
=
1013. ∫ &auml;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; &auml;&aring; &ntilde; − &ntilde; + ` =
&ntilde;
=
&Ccedil;&ntilde;
∫ &ntilde; &auml;&aring; &ntilde; = &auml;&aring; &auml;&aring; &ntilde; + ` =
1014.
=
 &auml;&aring; &ntilde;
N 
1015. ∫ &ntilde; &aring; &auml;&aring; &ntilde; &Ccedil;&ntilde; = &ntilde; &aring; +N 
+`=
−
O
 &aring; + N (&aring; + N) 
=
~ &euml;&aacute;&aring; &Auml;&ntilde; − &Auml; &Aring;&ccedil;&euml; &Auml;&ntilde; ~&ntilde;
1016. ∫ &Eacute; ~&ntilde; &euml;&aacute;&aring; &Auml;&ntilde; &Ccedil;&ntilde; =
&Eacute; +`=
~ O + &Auml;O
=
242
CHAPTER 9. INTEGRAL CALCULUS
1017. ∫ &Eacute; ~&ntilde; &Aring;&ccedil;&euml; &Auml;&ntilde; &Ccedil;&ntilde; =
~ &Aring;&ccedil;&euml; &Auml;&ntilde; + &Auml; &euml;&aacute;&aring; &Auml;&ntilde; ~&ntilde;
&Eacute; +`=
~ O + &Auml;O
=
=
=
9.7 Reduction Formulas
=
1018. ∫ &ntilde; &aring; &Eacute; &atilde;&ntilde; &Ccedil;&ntilde; =
N &aring; &atilde;&ntilde; &aring;
&ntilde; &Eacute; − ∫ &ntilde; &aring; −N&Eacute; &atilde;&ntilde; &Ccedil;&ntilde; =
&atilde;
&atilde;
=
&Eacute; &atilde;&ntilde;
&Eacute; &atilde;&ntilde;
&atilde; &Eacute; &atilde;&ntilde;
&Ccedil;&ntilde;
=
−
+
&Ccedil;&ntilde; I= &aring; ≠ N K=
∫ &ntilde;&aring;
(&aring; − N)&ntilde; &aring;−N &aring; − N ∫ &ntilde; &aring;−N
1019.
=
1020. ∫ &euml;&aacute;&aring;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; =
=
&Ccedil;&ntilde;
∫ &euml;&aacute;&aring;&Uuml;
1021.
&aring;
&ntilde;
=−
=
1022. ∫ &Aring;&ccedil;&euml;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; =
=
&Ccedil;&ntilde;
∫ &Aring;&ccedil;&euml;&Uuml;
1023.
&aring;
&ntilde;
=−
N
&aring; −N
&euml;&aacute;&aring;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde; −
&euml;&aacute;&aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; =
∫
&aring;
&aring;
&Aring;&ccedil;&euml;&Uuml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
−
I= &aring; ≠ N K=
&aring; −N
∫
(&aring; − N) &euml;&aacute;&aring;&Uuml; &ntilde; &aring; − N &euml;&aacute;&aring;&Uuml; &aring;−O &ntilde;
N
&aring; −N
&euml;&aacute;&aring;&Uuml; &ntilde; &Aring;&ccedil;&euml;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde; +
&Aring;&ccedil;&euml;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; =
∫
&aring;
&aring;
&euml;&aacute;&aring;&Uuml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
+
I= &aring; ≠ N K=
&aring; −N
∫
(&aring; − N) &Aring;&ccedil;&euml;&Uuml; &ntilde; &aring; − N &Aring;&ccedil;&euml;&Uuml; &aring;−O &ntilde;
=
&euml;&aacute;&aring;&Uuml; &aring; +N &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; −N &ntilde;
1024. ∫ &euml;&aacute;&aring;&Uuml; &ntilde; &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; =
=
&aring;+&atilde;
&atilde; −N
+
&euml;&aacute;&aring;&Uuml; &aring; &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; −O &ntilde;&Ccedil;&ntilde; =
∫
&aring;+&atilde;
=
&euml;&aacute;&aring;&Uuml; &aring; −N &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; +N &ntilde;
1025. ∫ &euml;&aacute;&aring;&Uuml; &aring; &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; &ntilde;&Ccedil;&ntilde; =
=
&aring;+&atilde;
&aring;
&atilde;
243
CHAPTER 9. INTEGRAL CALCULUS
−
&aring; −N
&euml;&aacute;&aring;&Uuml; &aring; −O &ntilde; &Aring;&ccedil;&euml;&Uuml; &atilde; &ntilde;&Ccedil;&ntilde; =
∫
&aring;+&atilde;
=
1026. ∫ &iacute;~&aring;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&iacute;~&aring;&Uuml; &aring; −N &ntilde; + ∫ &iacute;~&aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
&aring; −N
=
1027. ∫ &Aring;&ccedil;&iacute;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&Aring;&ccedil;&iacute;&Uuml; &aring; −N &ntilde; + ∫ &Aring;&ccedil;&iacute;&Uuml; &aring; − O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
&aring; −N
=
1028. ∫ &euml;&Eacute;&Aring;&Uuml; &aring; &ntilde;&Ccedil;&ntilde; =
&euml;&Eacute;&Aring;&Uuml; &aring; −O &ntilde; &iacute;~&aring;&Uuml; &ntilde; &aring; − O
+
&euml;&Eacute;&Aring;&Uuml; &aring; −O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
∫
&aring; −N
&aring; −N
=
N
&aring; −N
1029. ∫ &euml;&aacute;&aring; &aring; &ntilde;&Ccedil;&ntilde; = − &euml;&aacute;&aring; &aring; −N &ntilde; &Aring;&ccedil;&euml; &ntilde; +
&euml;&aacute;&aring; &aring; −O &ntilde;&Ccedil;&ntilde; =
∫
&aring;
&aring;
=
&Ccedil;&ntilde;
&Aring;&ccedil;&euml; &ntilde;
&aring;−O
&Ccedil;&ntilde;
1030. ∫
=−
+
I= &aring; ≠ N K=
&aring;
&aring; −N
∫
(&aring; − N) &euml;&aacute;&aring; &ntilde; &aring; − N &euml;&aacute;&aring; &aring;−O &ntilde;
&euml;&aacute;&aring; &ntilde;
=
N
&aring; −N
1031. ∫ &Aring;&ccedil;&euml; &aring; &ntilde;&Ccedil;&ntilde; = &euml;&aacute;&aring; &ntilde; &Aring;&ccedil;&euml; &aring; −N &ntilde; +
&Aring;&ccedil;&euml; &aring; −O &ntilde;&Ccedil;&ntilde; =
∫
&aring;
&aring;
=
&Ccedil;&ntilde;
&euml;&aacute;&aring; &ntilde;
&aring;−O
&Ccedil;&ntilde;
1032. ∫
=
+
I= &aring; ≠ N K=
&aring;
&aring; −N
∫
&Aring;&ccedil;&euml; &ntilde; (&aring; − N) &Aring;&ccedil;&euml; &ntilde; &aring; − N &Aring;&ccedil;&euml; &aring; −O &ntilde;
=
&euml;&aacute;&aring; &aring;+N &ntilde; &Aring;&ccedil;&euml; &atilde; −N &ntilde;
1033. ∫ &euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde; =
=
&aring;+&atilde;
&atilde; −N
+
&euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; −O &ntilde;&Ccedil;&ntilde; =
&aring;+&atilde;∫
=
&euml;&aacute;&aring; &aring; −N &ntilde; &Aring;&ccedil;&euml; &atilde; +N &ntilde;
1034. ∫ &euml;&aacute;&aring; &aring; &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde; = −
=
&aring;+&atilde;
244
CHAPTER 9. INTEGRAL CALCULUS
+
&aring; −N
&euml;&aacute;&aring; &aring; −O &ntilde; &Aring;&ccedil;&euml; &atilde; &ntilde;&Ccedil;&ntilde; =
∫
&aring;+&atilde;
=
1035. ∫ &iacute;~&aring; &aring; &ntilde;&Ccedil;&ntilde; =
N
&iacute;~&aring; &aring; −N &ntilde; − ∫ &iacute;~&aring; &aring; −O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
&aring; −N
=
1036. ∫ &Aring;&ccedil;&iacute; &aring; &ntilde;&Ccedil;&ntilde; = −
N
&Aring;&ccedil;&iacute; &aring; −N &ntilde; − ∫ &Aring;&ccedil;&iacute; &aring;−O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
&aring; −N
=
1037. ∫ &euml;&Eacute;&Aring; &aring; &ntilde;&Ccedil;&ntilde; =
&euml;&Eacute;&Aring; &aring; −O &ntilde; &iacute;~&aring; &ntilde; &aring; − O
+
&euml;&Eacute;&Aring; &aring; −O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
∫
&aring; −N
&aring; −N
=
1038. ∫ &Aring;&euml;&Aring; &aring; &ntilde;&Ccedil;&ntilde; = −
&Aring;&euml;&Aring; &aring; −O &ntilde; &Aring;&ccedil;&iacute; &ntilde; &aring; − O
+
&Aring;&euml;&Aring; &aring; −O &ntilde;&Ccedil;&ntilde; I= &aring; ≠ N K=
∫
&aring; −N
&aring; −N
=
1039. ∫ &ntilde; &aring; &auml;&aring; &atilde; &ntilde;&Ccedil;&ntilde; =
&ntilde; &aring; +N &auml;&aring; &atilde; &ntilde; &atilde;
−
&ntilde; &aring; &auml;&aring; &atilde; −N &ntilde;&Ccedil;&ntilde; =
&aring; +N
&aring; +N ∫
=
&auml;&aring; &atilde; &ntilde;
&auml;&aring; &atilde; &ntilde;
&atilde; &auml;&aring; &atilde; −N &ntilde;
&Ccedil;&ntilde;
=
−
+
&Ccedil;&ntilde; I= &aring; ≠ N K=
∫ &ntilde;&aring;
(&aring; − N)&ntilde; &aring;−N &aring; − N ∫ &ntilde; &aring;
1040.
=
1041. ∫ &auml;&aring; &aring; &ntilde;&Ccedil;&ntilde; = &ntilde; &auml;&aring; &aring; &ntilde; − &aring; ∫ &auml;&aring; &aring; −N &ntilde;&Ccedil;&ntilde; =
=
1042. ∫ &ntilde; &aring; &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &Aring;&ccedil;&euml;&Uuml; &ntilde; − &aring; ∫ &ntilde; &aring; −N &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; =
=
1043. ∫ &ntilde; &aring; &Aring;&ccedil;&euml;&Uuml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &euml;&aacute;&aring;&Uuml; &ntilde; − &aring; ∫ &ntilde; &aring; −N &euml;&aacute;&aring;&Uuml; &ntilde;&Ccedil;&ntilde; =
=
1044. ∫ &ntilde; &aring; &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde; = − &ntilde; &aring; &Aring;&ccedil;&euml; &ntilde; + &aring; ∫ &ntilde; &aring; −N &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde; =
=
1045. ∫ &ntilde; &aring; &Aring;&ccedil;&euml; &ntilde;&Ccedil;&ntilde; = &ntilde; &aring; &euml;&aacute;&aring; &ntilde; − &aring; ∫ &ntilde; &aring; −N &euml;&aacute;&aring; &ntilde;&Ccedil;&ntilde; =
245
CHAPTER 9. INTEGRAL CALCULUS
1046. ∫ &ntilde; &aring; &euml;&aacute;&aring; −N &ntilde;&Ccedil;&ntilde; =
N
&ntilde; &aring; +N
&ntilde; &aring; +N
&Ccedil;&ntilde; =
&euml;&aacute;&aring; −N &ntilde; −
&aring; + N ∫ N− &ntilde;O
&aring; +N
=
1047. ∫ &ntilde; &aring; &Aring;&ccedil;&euml; −N &ntilde;&Ccedil;&ntilde; =
&ntilde; &aring; +N
N
&ntilde; &aring; +N
&Aring;&ccedil;&euml; −N &ntilde; +
&Ccedil;&ntilde; =
&aring; +N
&aring; + N ∫ N− &ntilde;O
=
1048. ∫ &ntilde; &aring; &iacute;~&aring; −N &ntilde;&Ccedil;&ntilde; =
&ntilde; &aring; +N
N
&ntilde; &aring; +N
&iacute;~&aring; −N &ntilde; −
&Ccedil;&ntilde; =
&aring; +N
&aring; + N ∫ N+ &ntilde;O
=
&ntilde; &aring; &Ccedil;&ntilde;
&ntilde; &Auml;
&Ccedil;&ntilde;
1049. ∫ &aring;
= − ∫ &aring;
=
~&ntilde; + &Auml; ~ ~ ~&ntilde; + &Auml;
=
&Ccedil;&ntilde;
− O~&ntilde; − &Auml;
1050. ∫
=
=
&aring;
O
O
(~&ntilde; + &Auml;&ntilde; + &Aring; ) (&aring; − N)(&Auml; − Q~&Aring; )(~&ntilde; O + &Auml;&ntilde; + &Aring; )&aring;−N
O(O&aring; − P)~
&Ccedil;&ntilde;
I= &aring; ≠ N K=
−
O
∫
O
(&aring; − N)(&Auml; − Q~&Aring; ) (~&ntilde; + &Auml;&ntilde; + &Aring; )&aring;−N
=
∫ (&ntilde;
1051.
&Ccedil;&ntilde;
+~
&aring; ≠ N K=
O
)
O &aring;
=
&ntilde;
O(&aring; − N)~ O (&ntilde; O + ~
)
O &aring; −N
+
O&aring; − P
O(&aring; − N)~ O
=
∫ (&ntilde;
1052.
&Ccedil;&ntilde;
O
=−
&ntilde;
=
&aring; −N
O(&aring; − N)~ O (&ntilde; O − ~ O )
O&aring; − P
&Ccedil;&ntilde;
−
I= &aring; ≠ N K=
O ∫
O(&aring; − N)~ (&ntilde; O − ~ O )&aring;−N
−~
)
O &aring;
=
=
=
=
=
246
∫ (&ntilde;
&Ccedil;&ntilde;
O
+ ~O )
&aring; −N
I
CHAPTER 9. INTEGRAL CALCULUS
9.8 Definite Integral
=
&Auml;
&Auml;
~
~
a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; I= ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; I=&pound;=
&aring;
o&aacute;&Eacute;&atilde;~&aring;&aring;=&euml;&igrave;&atilde;W= ∑ &Ntilde; (ξ &aacute; )∆&ntilde; &aacute; ==
&aacute; =N
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W= ∆&ntilde; &aacute; ==
^&aring;&iacute;&aacute;&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W= c(&ntilde; ) I= d(&ntilde; ) ==
i&aacute;&atilde;&aacute;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;W=~I=&Auml;I=&Aring;I=&Ccedil;=
=
=
&Auml;
1053. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
~
&aring;
&auml;&aacute;&atilde;
∑ &Ntilde; (ξ )∆&ntilde;
&aring;→∞
&atilde;~&ntilde; ∆&ntilde; &aacute; →M &aacute; =N
&aacute;
&aacute;
I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;== ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I== &ntilde; &aacute; −N ≤ ξ &aacute; ≤ &ntilde; &aacute; K==
=
=
=
Figure 179.
=
247
CHAPTER 9. INTEGRAL CALCULUS
&Auml;
1054. ∫ N&Ccedil;&ntilde; = &Auml; − ~ =
~
=
&Auml;
&Auml;
~
~
1055. ∫ &acirc;&Ntilde; (&ntilde; )&Ccedil;&ntilde; = &acirc; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
=
&Auml;
&Auml;
&Auml;
~
~
~
&Auml;
&Auml;
&Auml;
~
~
~
∫ [&Ntilde; (&ntilde; ) + &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; =
1056.
=
∫ [&Ntilde; (&ntilde; ) − &Ouml;(&ntilde; )]&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; − ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; =
1057.
=
~
1058. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = M =
~
=
&Auml;
~
~
&Auml;
1059. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = − ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
=
&Auml;
&Aring;
&Auml;
~
~
&Aring;
1060. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =&Ntilde;&ccedil;&ecirc;= ~ &lt; &Aring; &lt; &Auml; K=
=
&Auml;
1061. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ≥ M =&aacute;&Ntilde;= &Ntilde; (&ntilde; ) ≥ M =&ccedil;&aring;= [~ I &Auml;] K=
~
=
&Auml;
1062. ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ≤ M =&aacute;&Ntilde;= &Ntilde; (&ntilde; ) ≤ M =&ccedil;&aring;= [~ I &Auml;] K=
~
=
=
=
=
248
CHAPTER 9. INTEGRAL CALCULUS
1063. c&igrave;&aring;&Ccedil;~&atilde;&Eacute;&aring;&iacute;~&auml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=&ccedil;&Ntilde;=`~&auml;&Aring;&igrave;&auml;&igrave;&euml;=
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&ntilde; )
&Auml;
~
= c(&Auml;) − c(~ ) =&aacute;&Ntilde;= c′(&ntilde; ) = &Ntilde; (&ntilde; ) K=
~
=
1064. j&Eacute;&iacute;&Uuml;&ccedil;&Ccedil;=&ccedil;&Ntilde;=p&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;==
f&Ntilde;= &ntilde; = &Ouml; (&iacute; ) I=&iacute;&Uuml;&Eacute;&aring;==
&Auml;
&Ccedil;
~
&Aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&Ouml;(&iacute; ))&Ouml;′(&iacute; )&Ccedil;&iacute; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
&Aring; = &Ouml; −N (~ ) I= &Ccedil; = &Ouml; −N (&Auml;) K=
=
1065. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&Auml;&oacute;=m~&ecirc;&iacute;&euml;=
&Auml;
&Auml;
∫ &igrave;&Ccedil;&icirc; = (&igrave;&icirc; ) ~ − ∫ &icirc;&Ccedil;&igrave; =
~
&Auml;
~
=
1066. q&ecirc;~&eacute;&Eacute;&ograve;&ccedil;&aacute;&Ccedil;~&auml;=o&igrave;&auml;&Eacute;=
&Auml;
&aring; −N
&Auml;−~ 

(
)
(
)
&Ntilde;
&ntilde;
&Ccedil;&ntilde;
&Ntilde;
(
&ntilde;
)
&Ntilde;
&ntilde;
O
&Ntilde; (&ntilde; &aacute; ) =
=
+
+
∑
M
&aring;

∫~
O&aring; 
&aacute; =N

=
249
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 180.
=
1067. p&aacute;&atilde;&eacute;&euml;&ccedil;&aring;∞&euml;=o&igrave;&auml;&Eacute;==
&Auml;
&Auml;−~
∫~ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = P&aring; [&Ntilde; (&ntilde; M ) + Q&Ntilde; (&ntilde; N ) + O&Ntilde; (&ntilde; O ) + Q&Ntilde; (&ntilde; P ) + =
+ O&Ntilde; (&ntilde; Q ) + K + Q&Ntilde; (&ntilde; &aring; −N ) + &Ntilde; (&ntilde; &aring; )] I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&Auml;−~
&ntilde;&aacute; = ~ +
&aacute; I= &aacute; = MI NI OI KI &aring; K==
&aring;
=
250
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 181.
=
1068. ^&ecirc;&Eacute;~=r&aring;&Ccedil;&Eacute;&ecirc;=~=`&igrave;&ecirc;&icirc;&Eacute;=
&Auml;
p = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = c(&Auml;) − c(~ ) I==
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= c′(&ntilde; ) = &Ntilde; (&ntilde; ) K=
=
251
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 182.
=
1069. ^&ecirc;&Eacute;~=_&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=q&iuml;&ccedil;=`&igrave;&ecirc;&icirc;&Eacute;&euml;=
&Auml;
p = ∫ [&Ntilde; (&ntilde; ) − &Ouml; (&ntilde; )]&Ccedil;&ntilde; = c(&Auml;) − d(&Auml;) − c(~ ) + d(~ ) I==
~
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= c′(&ntilde; ) = &Ntilde; (&ntilde; ) I= d′(&ntilde; ) = &Ouml; (&ntilde; ) K=
=
252
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 183.
=
=
=
9.9 Improper Integral
=
&Auml;
1070. q&Uuml;&Eacute;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;== ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=~&aring;=&aacute;&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=
~
&aacute;&Ntilde;==
~=&ccedil;&ecirc;=&Auml;=&aacute;&euml;=&aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;I=
&Ntilde; (&ntilde; ) ==&Uuml;~&euml;==&ccedil;&aring;&Eacute;==&ccedil;&ecirc;==&atilde;&ccedil;&ecirc;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;==&Ccedil;&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&aacute;&iacute;&oacute;=
=====&aacute;&aring;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;= [~ I &Auml;] K=
=
1071. f&Ntilde;= &Ntilde; (&ntilde; ) =&aacute;&euml;=~=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&aring;= [~ I ∞ ) I=&iacute;&Uuml;&Eacute;&aring;==
•
•
∞
&aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K=
~
&aring; →∞
~
=
253
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 184.
=
1072. f&Ntilde;= &Ntilde; (&ntilde; ) =&aacute;&euml;=~=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&aring;= (− ∞I &Auml;] I=&iacute;&Uuml;&Eacute;&aring;==
&Auml;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K=
−∞
&aring;→ −∞
&aring;
=
=
=
Figure 185.
254
CHAPTER 9. INTEGRAL CALCULUS
k&ccedil;&iacute;&Eacute;=W=q&Uuml;&Eacute;=&aacute;&atilde;&eacute;&ecirc;&ccedil;&eacute;&Eacute;&ecirc;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&aacute;&aring;=NMTNI=NMTO=~&ecirc;&Eacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=
&aacute;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&atilde;&aacute;&iacute;&euml;=&Eacute;&ntilde;&aacute;&euml;&iacute;=~&aring;&Ccedil;=~&ecirc;&Eacute;=&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;X=&ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=~&ecirc;&Eacute;=
&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
∞
&Aring;
∞
−∞
−∞
&Aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
1073.
=
=
=
Figure 186.
=
f&Ntilde;=&Ntilde;&ccedil;&ecirc;=&euml;&ccedil;&atilde;&Eacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&Aring;I=&Auml;&ccedil;&iacute;&Uuml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ecirc;&aacute;&Ouml;&Uuml;&iacute;=
∞
&euml;&aacute;&Ccedil;&Eacute;= ~&ecirc;&Eacute;= &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I= &iacute;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &aacute;&euml;= ~&auml;&euml;&ccedil;=====
−∞
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;X=&ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;=&aacute;&iacute;=&aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
1074. `&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;&euml;=
i&Eacute;&iacute;== &Ntilde; (&ntilde; ) =~&aring;&Ccedil;== &Ouml;(&ntilde; ) ==&Auml;&Eacute;==&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;==&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;==&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=
&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;= [~ I ∞ ) K= p&igrave;&eacute;&eacute;&ccedil;&euml;&Eacute;= &iacute;&Uuml;~&iacute;= M ≤ &Ouml; (&ntilde; ) ≤ &Ntilde; (&ntilde; ) = &Ntilde;&ccedil;&ecirc;= ~&auml;&auml;= &ntilde;= &aacute;&aring;=
[~I ∞ ) K=
255
CHAPTER 9. INTEGRAL CALCULUS
•
∞
∞
~
~
f&Ntilde;= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=~&auml;&euml;&ccedil;=
=====&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I=
•
∞
∞
~
~
f&Ntilde;= ∫ &Ouml; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=~&auml;&euml;&ccedil;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
1075. ^&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=
=
∞
∞
~
~
f&Ntilde;= ∫ &Ntilde; (&ntilde; ) &Ccedil;&ntilde; =&aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =&aacute;&euml;=~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K===
=
1076. a&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil;=
i&Eacute;&iacute;= &Ntilde; (&ntilde; ) =&Auml;&Eacute;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&iuml;&Uuml;&aacute;&Aring;&Uuml;=&aacute;&euml;=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;==
[~I &Auml;) =&Auml;&igrave;&iacute;=&aacute;&euml;=&Ccedil;&aacute;&euml;&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=~&iacute;= &ntilde; = &Auml; K=q&Uuml;&Eacute;&aring;==
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde;
~
ε →M +
&Auml;−ε
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; =
~
=
=
=
Figure 187.
256
CHAPTER 9. INTEGRAL CALCULUS
1077. i&Eacute;&iacute;= &Ntilde; (&ntilde; ) =&Auml;&Eacute;=~=&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;==&ntilde;==&aacute;&aring;=
&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;== [~ I &Auml;] ==&Eacute;&ntilde;&Aring;&Eacute;&eacute;&iacute;==&Ntilde;&ccedil;&ecirc;==&euml;&ccedil;&atilde;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;==&Aring;==&aacute;&aring;= (~ I &Auml;) K=q&Uuml;&Eacute;&aring;=
&Aring; −ε
&Auml;
&Auml;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; + &auml;&aacute;&atilde; ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; K=
~
ε →M +
δ →M +
~
&Aring; +δ
=
=
=
Figure 188.
=
=
=
9.10 Double Integral
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&iuml;&ccedil;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W= &Ntilde; (&ntilde; I &oacute; ) I= &Ntilde; (&igrave;I &icirc; ) I=&pound;=
a&ccedil;&igrave;&Auml;&auml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;W= ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; I= ∫∫ &Ouml; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; I=&pound;=
o
&atilde;
&aring;
o
(
)
o&aacute;&Eacute;&atilde;~&aring;&aring;=&euml;&igrave;&atilde;W= ∑∑ &Ntilde; &igrave; &aacute; I &icirc; &agrave; ∆&ntilde; &aacute; ∆&oacute; &agrave; =
&aacute; =N &agrave;=N
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W= ∆&ntilde; &aacute; I= ∆&oacute; &agrave; =
o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W=oI=p==
m&ccedil;&auml;~&ecirc;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ecirc; I= θ =
257
CHAPTER 9. INTEGRAL CALCULUS
^&ecirc;&Eacute;~W=^=
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=p=
s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=&euml;&ccedil;&auml;&aacute;&Ccedil;W=s=
j~&euml;&euml;=&ccedil;&Ntilde;=~=&auml;~&atilde;&aacute;&aring;~W=&atilde;=
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= ρ(&ntilde; I &oacute; ) =
c&aacute;&ecirc;&euml;&iacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W= j &ntilde; I= j &oacute; =
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W= f &ntilde; I= f &oacute; I= fM =
`&Uuml;~&ecirc;&Ouml;&Eacute;=&ccedil;&Ntilde;=~=&eacute;&auml;~&iacute;&Eacute;W=n=
`&Uuml;~&ecirc;&Ouml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= σ(&ntilde; I &oacute; ) =
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&atilde;~&euml;&euml;W= &ntilde; I= &oacute; =
^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &micro; =
=
1078. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=a&ccedil;&igrave;&Auml;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=
q&Uuml;&Eacute;=&Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&icirc;&Eacute;&ecirc;=~=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;= [~I &Auml;]&times; [&Aring;I &Ccedil;] =&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=
&iacute;&ccedil;=&Auml;&Eacute;==
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = &auml;&aacute;&atilde;
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]
(
)
∑∑ &Ntilde; (&igrave; I &icirc; )∆&ntilde; ∆&oacute;
&atilde;
&aring;
&atilde;~&ntilde; ∆&ntilde; &aacute; →M
&atilde;~&ntilde; ∆&oacute; &agrave; → M &aacute; =N &agrave;=N
&aacute;
&agrave;
&aacute;
&agrave;
I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &igrave; &aacute; I &icirc; &agrave; =&aacute;&euml;=&euml;&ccedil;&atilde;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;=
(&ntilde; &aacute; −N I &ntilde; &aacute; )&times; &oacute; &agrave;−N I &oacute; &agrave; I=~&aring;&Ccedil;= ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I= ∆&oacute; &agrave; = &oacute; &agrave; − &oacute; &agrave;−N K=
=
===
(
)
Figure 189.
258
=
CHAPTER 9. INTEGRAL CALCULUS
q&Uuml;&Eacute;=&Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&icirc;&Eacute;&ecirc;=~=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=o=&aacute;&euml;==
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^ I==
o
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;= [~I &Auml;]&times; [&Aring;I &Ccedil;] =&Aring;&ccedil;&aring;&iacute;~&aacute;&aring;&euml;=oI==
&Ouml;(&ntilde; I &oacute; ) = &Ntilde; (&ntilde; I &oacute; ) =&aacute;&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) =&aacute;&euml;=&aacute;&aring;=o=~&aring;&Ccedil;= &Ouml;(&ntilde; I &oacute; ) = M =&ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;K=
=
=
=
Figure 190.
=
1079.
∫∫ [&Ntilde; (&ntilde; I &oacute; ) + &Ouml;(&ntilde; I &oacute; )]&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ + ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^ =
o
=
1080.
o
∫∫ [&Ntilde; (&ntilde; I &oacute; ) − &Ouml;(&ntilde; I &oacute; )]&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ − ∫∫ &Ouml;(&ntilde; I &oacute; )&Ccedil;^ =
o
=
1081.
o
o
o
∫∫ &acirc;&Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = &acirc; ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ I==
o
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&acirc;=&aacute;&euml;=~=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1082. f&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) ≤ &Ouml;(&ntilde; I &oacute; ) =&ccedil;&aring;=oI=&iacute;&Uuml;&Eacute;&aring;= ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ ≤ ∫∫ &Ouml; (&ntilde; I &oacute; )&Ccedil;^ K=
o
=
1083. f&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) ≥ M =&ccedil;&aring;=o=~&aring;&Ccedil;= p ⊂ o I=&iacute;&Uuml;&Eacute;&aring;=
259
o
CHAPTER 9. INTEGRAL CALCULUS
= ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ ≤ ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K=
p
o
=
=
=
Figure 191.
=
1084. f&Ntilde;= &Ntilde; (&ntilde; I &oacute; ) ≥ M =&ccedil;&aring;=o=~&aring;&Ccedil;=o=~&aring;&Ccedil;=p=~&ecirc;&Eacute;=&aring;&ccedil;&aring;-&ccedil;&icirc;&Eacute;&ecirc;&auml;~&eacute;&eacute;&aacute;&aring;&Ouml;=
&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;I=&iacute;&Uuml;&Eacute;&aring;=
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ + ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K==
o ∪p
o
p
e&Eacute;&ecirc;&Eacute;= o ∪ p =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;=o=~&aring;&Ccedil;=pK=
=
=
=
Figure 192.
=
=
260
CHAPTER 9. INTEGRAL CALCULUS
1085. f&iacute;&Eacute;&ecirc;~&iacute;&Eacute;&Ccedil;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=~&aring;&Ccedil;=c&igrave;&Auml;&aacute;&aring;&aacute;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
&Auml; &egrave;(&ntilde; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫( &Ntilde;) (&ntilde; I &oacute; )&Ccedil;&oacute;&Ccedil;&ntilde; ==
o
~&eacute; &ntilde;
&Ntilde;&ccedil;&ecirc;=~=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&oacute;&eacute;&Eacute;=fI==
o = {(&ntilde; I &oacute; ) &ouml; ~ ≤ &ntilde; ≤ &Auml;I &eacute;(&ntilde; ) ≤ &oacute; ≤ &egrave;(&ntilde; )} K=
=
=
=
Figure 193.
=
&Ccedil; &icirc; (&oacute; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; ==
o
&Aring; &igrave;(&oacute; )
&Ntilde;&ccedil;&ecirc;=~=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&oacute;&eacute;&Eacute;=ffI=
o = {(&ntilde; I &oacute; ) &ouml; &igrave;(&oacute; ) ≤ &ntilde; ≤ &icirc; (&oacute; )I &Aring; ≤ &oacute; ≤ &Ccedil;} K=
=
261
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 194.
=
1086. a&ccedil;&igrave;&Auml;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&icirc;&Eacute;&ecirc;=o&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;=
=
f&Ntilde;=o=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&igrave;&auml;~&ecirc;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;= [~I &Auml;]&times; [&Aring;I &Ccedil;] I=&iacute;&Uuml;&Eacute;&aring;==
&Auml; &Ccedil;
&Ccedil; &Auml;






 ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde; &Ccedil;&oacute; K==
(
)
(
)
=
=
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&oacute;
&Ccedil;&ntilde;
∫∫o
∫~  ∫&Aring;
∫



&Aring;~


=
f&aring;=&iacute;&Uuml;&Eacute;=&euml;&eacute;&Eacute;&Aring;&aacute;~&auml;=&Aring;~&euml;&Eacute;=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil;= &Ntilde; (&ntilde; I &oacute; ) =&Aring;~&aring;=&Auml;&Eacute;=&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring;=~&euml;= &Ouml; (&ntilde; )&Uuml;(&oacute; ) =&iuml;&Eacute;=&Uuml;~&icirc;&Eacute;==
&Auml;
&Ccedil;




 K==
(
)
(
)
(
)
(
)
(
)
=
=
&Ntilde;
&ntilde;
I
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ouml;
&ntilde;
&Uuml;
&oacute;
&Ccedil;&ntilde;&Ccedil;&oacute;
&Ouml;
&ntilde;
&Ccedil;&ntilde;
&Uuml;
&oacute;
&Ccedil;&oacute;
∫∫o
∫∫o
∫
∫

~
&Aring;

=
1087. `&Uuml;~&aring;&Ouml;&Eacute;=&ccedil;&Ntilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
∂ (&ntilde; I &oacute; )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫∫ &Ntilde; [&ntilde; (&igrave;I &icirc; )I &oacute;(&igrave;I &icirc; )] ∂(&igrave;I &icirc; ) &Ccedil;&igrave;&Ccedil;&icirc; I==
o
p
∂&ntilde; ∂&ntilde;
∂ (&ntilde; I &oacute; ) ∂&igrave; ∂&icirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
=
≠ M &aacute;&euml;= &iacute;&Uuml;&Eacute;= &agrave;~&Aring;&ccedil;&Auml;&aacute;~&aring;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &iacute;&ecirc;~&aring;&euml;∂ (&igrave;I &icirc; ) ∂&oacute; ∂&oacute;
∂&igrave; ∂&icirc;
&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring;&euml;= (&ntilde; I &oacute; ) → (&igrave;I &icirc; ) I=~&aring;&Ccedil;=p=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc;=&ccedil;&Ntilde;=o=&iuml;&Uuml;&aacute;&Aring;&Uuml;=
262
CHAPTER 9. INTEGRAL CALCULUS
&Aring;~&aring;=&Auml;&Eacute;=&Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil;=&Auml;&oacute;= &ntilde; = &ntilde; (&igrave;I &icirc; ) I= &oacute; = &oacute; (&igrave;I &icirc; ) =&aacute;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=oK==
=
1088. m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
&ntilde; = &ecirc; &Aring;&ccedil;&euml; θ I= &oacute; = &ecirc; &euml;&aacute;&aring; θ K==
=
=
=
Figure 195.
=
1089. a&ccedil;&igrave;&Auml;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
=
q&Uuml;&Eacute;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Ccedil;&ntilde;&Ccedil;&oacute;=&Ntilde;&ccedil;&ecirc;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&aacute;&euml;==
∂ (&ntilde; I &oacute; )
&Ccedil;&ntilde;&Ccedil;&oacute; =
&Ccedil;&ecirc;&Ccedil;θ = &ecirc;&Ccedil;&ecirc;&Ccedil;θ K==
∂ (&ecirc;I θ)
=
i&Eacute;&iacute;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=o=&aacute;&euml;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil;=~&euml;=&Ntilde;&ccedil;&auml;&auml;&ccedil;&iuml;&euml;W=
M ≤ &Ouml; (θ) ≤ &ecirc; ≤ &Uuml;(θ) I= α ≤ θ ≤ β I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= β − α ≤ Oπ K==
q&Uuml;&Eacute;&aring;==
β &Uuml; (θ )
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫ ∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K=
o
α &Ouml; (θ )
=
263
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 196.
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=o=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&auml;~&ecirc;=&ecirc;&Eacute;&Aring;&iacute;~&aring;&Ouml;&auml;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
M ≤ ~ ≤ &ecirc; ≤ &Auml; I= α ≤ θ ≤ β I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= β − α ≤ Oπ I===
&iacute;&Uuml;&Eacute;&aring;==
β &Auml;
∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; = ∫ ∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K==
o
α ~
=
=
=
Figure 197.
=
=
264
CHAPTER 9. INTEGRAL CALCULUS
1090. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=o&Eacute;&Ouml;&aacute;&ccedil;&aring;=
&Auml; &Ntilde; (&ntilde; )
^ = ∫ ∫ &Ccedil;&oacute;&Ccedil;&ntilde; =E&Ntilde;&ccedil;&ecirc;=~=&iacute;&oacute;&eacute;&Eacute;=f=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;FK=
~ &Ouml; (&ntilde; )
=
=
=
Figure 198.
=
&Ccedil; &egrave;(&oacute; )
^ = ∫ ∫ &Ccedil;&ntilde;&Ccedil;&oacute; =E&Ntilde;&ccedil;&ecirc;=~=&iacute;&oacute;&eacute;&Eacute;=ff=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;FK=
&Aring; &eacute;( &oacute; )
=
=
=
Figure 199.
=
=
265
CHAPTER 9. INTEGRAL CALCULUS
1091. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ K==
o
=
=
=
Figure 200.
=
f&Ntilde;=o=&aacute;&euml;=~=&iacute;&oacute;&eacute;&Eacute;=f=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=&Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;= &ntilde; = ~ I= &ntilde; = &Auml; I= &oacute; = &Uuml;(&ntilde; ) I=
&oacute; = &Ouml; (&ntilde; ) I=&iacute;&Uuml;&Eacute;&aring;==
&Auml; &Ouml;(&ntilde; )
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&oacute;&Ccedil;&ntilde; K==
o
~ &Uuml;(&ntilde; )
=
f&Ntilde;=o=&aacute;&euml;=~=&iacute;&oacute;&eacute;&Eacute;=ff=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=&Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;= &oacute; = &Aring; I= &oacute; = &Ccedil; I= &ntilde; = &egrave;(&oacute; ) I=
&ntilde; = &eacute;(&oacute; ) I=&iacute;&Uuml;&Eacute;&aring;=
&Ccedil; &egrave;(&oacute; )
s = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ = ∫ ∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;&ntilde;&Ccedil;&oacute; K==
o
&Aring; &eacute;( &oacute; )
=
266
CHAPTER 9. INTEGRAL CALCULUS
f&Ntilde;== &Ntilde; (&ntilde; I &oacute; ) ≥ &Ouml; (&ntilde; I &oacute; ) ==&ccedil;&icirc;&Eacute;&ecirc;==~==&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;==oI==&iacute;&Uuml;&Eacute;&aring;==&iacute;&Uuml;&Eacute;==&icirc;&ccedil;&auml;&igrave;&atilde;&Eacute;==&ccedil;&Ntilde;=
&iacute;&Uuml;&Eacute;= &euml;&ccedil;&auml;&aacute;&Ccedil;= &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;= &ograve;N = &Ntilde; (&ntilde; I &oacute; ) = ~&aring;&Ccedil;= &ograve; O = &Ouml;(&ntilde; I &oacute; ) = &ccedil;&icirc;&Eacute;&ecirc;= o= &aacute;&euml;=
&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=
s = ∫∫ [&Ntilde; (&ntilde; I &oacute; ) − &Ouml; (&ntilde; I &oacute; )]&Ccedil;^ K==
o
=
1092. ^&ecirc;&Eacute;~=~&aring;&Ccedil;=s&ccedil;&auml;&igrave;&atilde;&Eacute;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
f&Ntilde;=p=&aacute;&euml;=~=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=&Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;= θ = α I= θ = β I=
&ecirc; = &Uuml;(θ) I= &ecirc; = &Ouml; (θ) I==
&iacute;&Uuml;&Eacute;&aring;==
β &Ouml; (θ )
^ = ∫∫ &Ccedil;^ = ∫ ∫ &ecirc;&Ccedil;&ecirc;&Ccedil;θ I==
p
α &Uuml;(θ )
s = ∫∫ &Ntilde; (&ecirc;I θ)&ecirc;&Ccedil;&ecirc;&Ccedil;θ K==
p
=
=
=
Figure 201.
=
1093. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=^&ecirc;&Eacute;~=
p = ∫∫
o
O
 ∂&ograve;   ∂&ograve; 
N +   +   &Ccedil;&ntilde;&Ccedil;&oacute; =
 ∂&ntilde;   ∂&oacute; 
O
=
267
CHAPTER 9. INTEGRAL CALCULUS
1094. j~&euml;&euml;=&ccedil;&Ntilde;=~=i~&atilde;&aacute;&aring;~=
&atilde; = ∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^ I==
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==&iacute;&Uuml;&Eacute;==&auml;~&atilde;&aacute;&aring;~==&ccedil;&Aring;&Aring;&igrave;&eacute;&aacute;&Eacute;&euml;==~==&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;==o=~&aring;&Ccedil;==&aacute;&iacute;&euml;==&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;==~&iacute;=
~=&eacute;&ccedil;&aacute;&aring;&iacute;=E&ntilde;I&oacute;F=&aacute;&euml;= ρ(&ntilde; I &oacute; ) K===
=
1095. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=
q&Uuml;&Eacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;~&atilde;&aacute;&aring;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;==&ntilde;-~&ntilde;&aacute;&euml;==&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=
j &ntilde; = ∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^ K==
o
=
q&Uuml;&Eacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;~&atilde;&aacute;&aring;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&oacute;-~&ntilde;&aacute;&euml;=&aacute;&euml;=
j &oacute; = ∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^ K=
o
=
q&Uuml;&Eacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;=&aacute;&euml;=
f &ntilde; = ∫∫ &oacute; Oρ(&ntilde; I &oacute; )&Ccedil;^ K==
o
=
q&Uuml;&Eacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&oacute;-~&ntilde;&aacute;&euml;=&aacute;&euml;=
f &oacute; = ∫∫ &ntilde; Oρ(&ntilde; I &oacute; )&Ccedil;^ K==
o
=
q&Uuml;&Eacute;=&eacute;&ccedil;&auml;~&ecirc;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~=&aacute;&euml;=
fM = ∫∫ (&ntilde; O + &oacute; O )ρ(&ntilde; I &oacute; )&Ccedil;^ K==
o
=
1096. `&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=j~&euml;&euml;=
j&oacute;
N
&ntilde;=
= ∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^ =
&atilde; &atilde; o
∫∫ &ntilde;ρ(&ntilde; I &oacute; )&Ccedil;^
o
∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^
o
268
I==
CHAPTER 9. INTEGRAL CALCULUS
j
N
&oacute; = &ntilde; = ∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^ =
&atilde; &atilde; o
∫∫ &oacute;ρ(&ntilde; I &oacute; )&Ccedil;^
o
∫∫ ρ(&ntilde; I &oacute; )&Ccedil;^
K==
o
=
1097. `&Uuml;~&ecirc;&Ouml;&Eacute;=&ccedil;&Ntilde;=~=m&auml;~&iacute;&Eacute;=
n = ∫∫ σ(&ntilde; I &oacute; )&Ccedil;^ I==
o
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;~&auml;=&Aring;&Uuml;~&ecirc;&Ouml;&Eacute;=&aacute;&euml;=&Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil;=&ccedil;&icirc;&Eacute;&ecirc;=~=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;=o=~&aring;&Ccedil;=&aacute;&iacute;&euml;=
&Aring;&Uuml;~&ecirc;&Ouml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=~&iacute;=~=&eacute;&ccedil;&aacute;&aring;&iacute;=E&ntilde;I&oacute;F=&aacute;&euml;= σ(&ntilde; I &oacute; ) K===
=
1098. ^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute;=&ccedil;&Ntilde;=~=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
N
&micro; = ∫∫ &Ntilde; (&ntilde; I &oacute; )&Ccedil;^ I==
po
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= p = ∫∫ &Ccedil;^ K==
o
=
=
=
9.11 Triple Integral
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&ecirc;&Eacute;&Eacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W= &Ntilde; (&ntilde; I &oacute; I &ograve; ) I= &Ouml; (&ntilde; I &oacute; I &ograve; ) I=&pound;=
q&ecirc;&aacute;&eacute;&auml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;W= ∫∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s I= ∫∫∫ &Ouml; (&ntilde; I &oacute; I &ograve; )&Ccedil;s I=&pound;=
d
&atilde;
d
&aring;
&eacute;
(
)
o&aacute;&Eacute;&atilde;~&aring;&aring;=&euml;&igrave;&atilde;W= ∑∑∑ &Ntilde; &igrave; &aacute; I &icirc; &agrave; I &iuml; &acirc; ∆&ntilde; &aacute; ∆&oacute; &agrave; ∆&ograve; &acirc; =
&aacute; =N &agrave;=N &acirc; =N
p&atilde;~&auml;&auml;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;&euml;W= ∆&ntilde; &aacute; I= ∆&oacute; &agrave; I= ∆&ograve; &acirc; =
i&aacute;&atilde;&aacute;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W=~I=&Auml;I=&Aring;I=&Ccedil;I=&ecirc;I=&euml;=
o&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;W=dI=qI=p==
`&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ecirc; I= θ I=&ograve;=
p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ecirc; I= θ I= ϕ =
s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=&euml;&ccedil;&auml;&aacute;&Ccedil;W=s=
269
CHAPTER 9. INTEGRAL CALCULUS
j~&euml;&euml;=&ccedil;&Ntilde;=~=&euml;&ccedil;&auml;&aacute;&Ccedil;W=&atilde;==
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= &micro;(&ntilde; I &oacute; I &ograve; ) =
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&atilde;~&euml;&euml;W= &ntilde; I= &oacute; I= &ograve; =
c&aacute;&ecirc;&euml;&iacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W= j &ntilde;&oacute; I= j &oacute;&ograve; I= j &ntilde;&ograve; =
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W= f &ntilde;&oacute; I= f &oacute;&ograve; I= f &ntilde;&ograve; I= f &ntilde; I= f &oacute; I= f&ograve; I= fM =
=
=
1099. a&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=
q&Uuml;&Eacute;=&iacute;&ecirc;&aacute;&eacute;&auml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&icirc;&Eacute;&ecirc;=~=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;= [~I &Auml;]&times; [&Aring;I &Ccedil;]&times; [&ecirc;I &euml;] =
&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&iacute;&ccedil;=&Auml;&Eacute;==
∫∫∫
&Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s =
[~ I &Auml; ]&times;[&Aring; I &Ccedil; ]&times;[&ecirc; I &euml; ]
(
)
)
∑∑∑ &Ntilde; (&igrave; I &icirc; I &iuml; )∆&ntilde; ∆&oacute; ∆&ograve;
&atilde;
&auml;&aacute;&atilde;
&eacute;
&aring;
&atilde;~&ntilde; ∆&ntilde; &aacute; →M
&atilde;~&ntilde; ∆&oacute; &agrave; →M &aacute; =N &agrave;=N &acirc; =N
&atilde;~&ntilde; ∆&ograve; &acirc; →M
&aacute;
&agrave;
&acirc;
&aacute;
&agrave;
&acirc;
I
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &igrave; &aacute; I &icirc; &agrave; I &iuml; &acirc; =&aacute;&euml;=&euml;&ccedil;&atilde;&Eacute;=&eacute;&ccedil;&aacute;&aring;&iacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;=
(&ntilde; &aacute; −N I &ntilde; &aacute; )&times; &oacute; &agrave;−N I &oacute; &agrave; &times; (&ograve; &acirc; −N I &ograve; &acirc; ) I=~&aring;&Ccedil;= ∆&ntilde; &aacute; = &ntilde; &aacute; − &ntilde; &aacute; −N I=
∆&oacute; &agrave; = &oacute; &agrave; − &oacute; &agrave;−N I= ∆&ograve; &acirc; = &ograve; &acirc; − &ograve; &acirc; −N K=
=
1100. ∫∫∫ [&Ntilde; (&ntilde; I &oacute; I &ograve; ) + &Ouml; (&ntilde; I &oacute; I &ograve; )]&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s + ∫∫∫ &Ouml; (&ntilde; I &oacute; I &ograve; )&Ccedil;s
(
d
d
d
=
1101.
∫∫∫ [&Ntilde; (&ntilde; I &oacute; I &ograve; ) − &Ouml;(&ntilde; I &oacute;I &ograve; )]&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s − ∫∫∫ &Ouml;(&ntilde; I &oacute;I &ograve; )&Ccedil;s
d
d
d
=
1102.
∫∫∫ &acirc;&Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s = &acirc; ∫∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;s I==
d
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&acirc;=&aacute;&euml;=~=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1103. f&Ntilde;== &Ntilde; (&ntilde; I &oacute; I &ograve; ) ≥ M =~&aring;&Ccedil;=d=~&aring;&Ccedil;=q=~&ecirc;&Eacute;=&aring;&ccedil;&aring;&ccedil;&icirc;&Eacute;&ecirc;&auml;~&eacute;&eacute;&aacute;&aring;&Ouml;=&Auml;~&euml;&aacute;&Aring;=
&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;I=&iacute;&Uuml;&Eacute;&aring;==
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s = ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s + ∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;s K==
d∪q
d
q
e&Eacute;&ecirc;&Eacute;= d ∪ q =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;&euml;=d=~&aring;&Ccedil;=qK=
=
270
CHAPTER 9. INTEGRAL CALCULUS
1104. b&icirc;~&auml;&igrave;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&Auml;&oacute;=o&Eacute;&eacute;&Eacute;~&iacute;&Eacute;&Ccedil;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&ccedil;&auml;&aacute;&Ccedil;=d=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&iacute;=&ccedil;&Ntilde;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;= (&ntilde; I &oacute; I &ograve; ) =&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
= (&ntilde; I &oacute; )∈ oI χ N (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O (&ntilde; I &oacute; ) I=&iacute;&Uuml;&Eacute;&aring;==

χ O ( &ntilde; I &oacute; )
= ∫∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = ∫∫  ∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ograve;  &Ccedil;&ntilde;&Ccedil;&oacute; I==

d
o 
 χN ( &ntilde; I &oacute; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=o=&aacute;&euml;=&eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=d=&ccedil;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K=
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&ccedil;&auml;&aacute;&Ccedil;=d=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&iacute;=&ccedil;&Ntilde;=&eacute;&ccedil;&aacute;&aring;&iacute;&euml;= (&ntilde; I &oacute; I &ograve; ) =&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
~ ≤ &ntilde; ≤ &Auml;I ϕN (&ntilde; ) ≤ &oacute; ≤ ϕO (&ntilde; )I χN (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O (&ntilde; I &oacute; ) I=&iacute;&Uuml;&Eacute;&aring;==
∫∫∫
d
 ϕO ( &ntilde; )  χ O ( &ntilde; I &oacute; )
 
&Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = ∫  ∫  ∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;&ograve; &Ccedil;&oacute; &Ccedil;&ntilde; ==

 
~ 
 
 ϕN ( &ntilde; )  χN ( &ntilde; I &oacute; )
&Auml;
=
1105. q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&icirc;&Eacute;&ecirc;=m~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;=
f&Ntilde;=d=&aacute;&euml;=~=&eacute;~&ecirc;~&auml;&auml;&Eacute;&auml;&Eacute;&eacute;&aacute;&eacute;&Eacute;&Ccedil;= [~ I &Auml;]&times; [&Aring;I &Ccedil;]&times; [&ecirc;I &euml;] I=&iacute;&Uuml;&Eacute;&aring;=
&Auml; &Ccedil; &euml;
 
 

&Ccedil;&oacute; &Ccedil;&ntilde; K==
(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ograve;
=

∫∫∫
∫
∫
∫


d
~ &Aring;  &ecirc;
 
=
f&aring;==&iacute;&Uuml;&Eacute;=&euml;&eacute;&Eacute;&Aring;&aacute;~&auml;=&Aring;~&euml;&Eacute;==&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&aring;&Ccedil;== &Ntilde; (&ntilde; I &oacute; I &ograve; ) ==&Aring;~&aring;=&Auml;&Eacute;=
&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring;=~&euml;= &Ouml; (&ntilde; ) &Uuml;(&oacute; ) &acirc; (&ograve; ) =&iuml;&Eacute;=&Uuml;~&icirc;&Eacute;==
&Auml;
 &Ccedil;
 &euml;




 ∫ &acirc; (&ograve; )&Ccedil;&ograve;  K==
(
)
(
)
(
)
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;
&Ouml;
&ntilde;
&Ccedil;&ntilde;
&Uuml;
&oacute;
&Ccedil;&oacute;
=
∫∫∫

∫
 ∫

d

~
 &Aring;
 &ecirc;
=
1106. `&Uuml;~&aring;&Ouml;&Eacute;=&ccedil;&Ntilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = =
d
= ∫∫∫ &Ntilde; [&ntilde; (&igrave;I &icirc; I &iuml; )I &oacute; (&igrave;I &icirc; I &iuml; )I &ograve; (&igrave;I &icirc; I &iuml; )]
p
271
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;I =
∂ (&igrave;I &icirc; I &iuml; )
CHAPTER 9. INTEGRAL CALCULUS
∂&ntilde; ∂&ntilde; ∂&ntilde;
∂&igrave; ∂&icirc; ∂&iuml;
∂ (&ntilde; I &oacute; I &ograve; ) ∂&oacute; ∂&oacute; ∂&oacute;
=
≠ M ==&aacute;&euml;==&iacute;&Uuml;&Eacute;==&agrave;~&Aring;&ccedil;&Auml;&aacute;~&aring;==&ccedil;&Ntilde;=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
∂ (&igrave;I &icirc; I &iuml; ) ∂&igrave; ∂&icirc; ∂&iuml;
∂&ograve; ∂&ograve; ∂&ograve;
∂&igrave; ∂&icirc; ∂&iuml;
&iacute;&Uuml;&Eacute;= &iacute;&ecirc;~&aring;&euml;&Ntilde;&ccedil;&ecirc;&atilde;~&iacute;&aacute;&ccedil;&aring;&euml;= (&ntilde; I &oacute; I &ograve; ) → (&igrave;I &icirc; I &iuml; ) I= ~&aring;&Ccedil;= p= &aacute;&euml;= &iacute;&Uuml;&Eacute;= &eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc;= &ccedil;&Ntilde;= d= &iuml;&Uuml;&aacute;&Aring;&Uuml;= &Aring;~&aring;= &Auml;&Eacute;= &Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil;= &Auml;&oacute;= &ntilde; = &ntilde; (&igrave;I &icirc; I &iuml; ) I=
&oacute; = &oacute; (&igrave;I &icirc; I &iuml; ) =
&ograve; = &ograve; (&igrave;I &icirc; I &iuml; ) =&aacute;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=dK=
=
=
1107. q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&aacute;&aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
q&Uuml;&Eacute;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;=&Ntilde;&ccedil;&ecirc;=&Aring;&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&aacute;&euml;==
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; = &ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; K==
∂ (&ecirc;I θI &ograve; )
=
i&Eacute;&iacute;=&iacute;&Uuml;&Eacute;=&euml;&ccedil;&auml;&aacute;&Ccedil;=d=&aacute;&euml;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil;=~&euml;=&Ntilde;&ccedil;&auml;&auml;&ccedil;&iuml;&euml;W=
(&ntilde; I &oacute; )∈ oI χN (&ntilde; I &oacute; ) ≤ &ograve; ≤ χ O (&ntilde; I &oacute; ) I=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=o=&aacute;&euml;=&eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=d=&ccedil;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K=q&Uuml;&Eacute;&aring;==
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = ∫∫∫ &Ntilde; (&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θI &ograve; )&ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; =
d
p
χ O ( &ecirc; &Aring;&ccedil;&euml; θ I&ecirc; &euml;&aacute;&aring; θ )


(
)
&Ntilde;
&ecirc;
&Aring;&ccedil;&euml;
I
&ecirc;
&euml;&aacute;&aring;
I
&ograve;
&Ccedil;&ograve;
θ
θ
 &ecirc;&Ccedil;&ecirc;&Ccedil;θ K=

∫∫  χ (&ecirc; &Aring;&ccedil;&euml;∫θI&ecirc; &euml;&aacute;&aring; θ )
o ( &ecirc; Iθ ) 
N

e&Eacute;&ecirc;&Eacute;=p=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc;=&ccedil;&Ntilde;=d=&aacute;&aring;=&Aring;&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K=
=
1108. q&ecirc;&aacute;&eacute;&auml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&aacute;&aring;=p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
q&Uuml;&Eacute;=a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve;=&Ntilde;&ccedil;&ecirc;=p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&aacute;&euml;==
∂ (&ntilde; I &oacute; I &ograve; )
&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ = &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ ==
∂ (&ecirc;I θI ϕ)
=
∫∫∫ &Ntilde; (&ntilde; I &oacute;I &ograve; )&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; = =
=
d
272
CHAPTER 9. INTEGRAL CALCULUS
========= = ∫∫∫ &Ntilde; (&ecirc; &euml;&aacute;&aring; θ &Aring;&ccedil;&euml; ϕI &ecirc; &euml;&aacute;&aring; θ &euml;&aacute;&aring; ϕI &ecirc; &Aring;&ccedil;&euml; θ) &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ I==
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&euml;&ccedil;&auml;&aacute;&Ccedil;=p=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&igrave;&auml;&auml;&Auml;~&Aring;&acirc;=&ccedil;&Ntilde;=d=&aacute;&aring;=&euml;&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K=q&Uuml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;== θ ==&ecirc;~&aring;&Ouml;&Eacute;&euml;==&Ntilde;&ecirc;&ccedil;&atilde;==M=&iacute;&ccedil;== Oπ I==&iacute;&Uuml;&Eacute;==~&aring;&Ouml;&auml;&Eacute;== ϕ =
&ecirc;~&aring;&Ouml;&Eacute;&euml;=&Ntilde;&ecirc;&ccedil;&atilde;=M=&iacute;&ccedil;= π K==
=
=
=======
=
Figure 202.
=
1109. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=
s = ∫∫∫ &Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; =
d
=
1110. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&aacute;&aring;=`&oacute;&auml;&aacute;&aring;&Ccedil;&ecirc;&aacute;&Aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
s = ∫∫∫ &ecirc;&Ccedil;&ecirc;&Ccedil;θ&Ccedil;&ograve; =
p ( &ecirc; I θ I&ograve; )
=
1111. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&aacute;&aring;=p&eacute;&Uuml;&Eacute;&ecirc;&aacute;&Aring;~&auml;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
s = ∫∫∫ &ecirc; O &euml;&aacute;&aring; θ&Ccedil;&ecirc;&Ccedil;θ&Ccedil;ϕ =
p ( &ecirc; I θ Iϕ )
=
273
CHAPTER 9. INTEGRAL CALCULUS
1112. j~&euml;&euml;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=
&atilde; = ∫∫∫ &micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;s I==
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &iacute;&Uuml;&Eacute;= &euml;&ccedil;&auml;&aacute;&Ccedil;= &ccedil;&Aring;&Aring;&igrave;&eacute;&aacute;&Eacute;&euml;= ~= &ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;= d= ~&aring;&Ccedil;= &aacute;&iacute;&euml;= &Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;= ~&iacute;============
~=&eacute;&ccedil;&aacute;&aring;&iacute;= (&ntilde; I &oacute; I &ograve; ) =&aacute;&euml;= &micro;(&ntilde; I &oacute; I &ograve; ) K===
=
1113. `&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=j~&euml;&euml;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=
j &oacute;&ograve;
j &ntilde;&oacute;
j
&ntilde;=
I= &oacute; = &ntilde;&ograve; I= &ograve; =
I==
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
j &oacute;&ograve; = ∫∫∫ &ntilde;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
j &ntilde;&ograve; = ∫∫∫ &oacute;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
j &ntilde;&oacute; = ∫∫∫ &ograve;&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s =
d
~&ecirc;&Eacute;==&iacute;&Uuml;&Eacute;==&Ntilde;&aacute;&ecirc;&euml;&iacute;==&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;==~&Auml;&ccedil;&igrave;&iacute;==&iacute;&Uuml;&Eacute;==&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=&eacute;&auml;~&aring;&Eacute;&euml;= &ntilde; = M I=
&oacute; = M I= &ograve; = M I=&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;I== &micro;(&ntilde; I &oacute; I &ograve; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K==
=
1114. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;==&iacute;&Uuml;&Eacute;==&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=E&ccedil;&ecirc;= &ograve; = M FI=&oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;==
E &ntilde; = M FI=~&aring;&Ccedil;=&ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute;=E &oacute; = M F=
f &ntilde;&oacute; = ∫∫∫ &ograve; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
f &oacute;&ograve; = ∫∫∫ &ntilde; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
f &ntilde;&ograve; = ∫∫∫ &oacute; O&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s K==
d
=
1115. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;I=&oacute;-~&ntilde;&aacute;&euml;I=~&aring;&Ccedil;=&ograve;-~&ntilde;&aacute;&euml;=
f &ntilde; = f &ntilde;&oacute; + f &ntilde;&ograve; = ∫∫∫ (&ograve; O + &oacute; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
f &oacute; = f &ntilde;&oacute; + f &oacute;&ograve; = ∫∫∫ (&ograve; O + &ntilde; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s I==
d
274
CHAPTER 9. INTEGRAL CALCULUS
f&ograve; = f &ntilde;&ograve; + f &oacute;&ograve; = ∫∫∫ (&oacute; O + &ntilde; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s K==
d
=
1116. m&ccedil;&auml;~&ecirc;=j&ccedil;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=
fM = f &ntilde;&oacute; + f &oacute;&ograve; + f &ntilde;&ograve; = ∫∫∫ (&ntilde; O + &oacute; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; ) &Ccedil;s =
d
=
=
9.12 Line Integral
=
p&Aring;~&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W= c(&ntilde; I &oacute; I &ograve; ) I= c(&ntilde; I &oacute; ) I= &Ntilde; (&ntilde; ) =
p&Aring;~&auml;~&ecirc;=&eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;W= &igrave;(&ntilde; I &oacute; I &ograve; ) =
`&igrave;&ecirc;&icirc;&Eacute;&euml;W=`I= `N I= ` O =
i&aacute;&atilde;&aacute;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;W=~I=&Auml;I= α I= β =
m~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&euml;W=&iacute;I=&euml;=
m&ccedil;&auml;~&ecirc;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;W= &ecirc; I= θ =
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= c (mI nI o ) =
r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;W= &ecirc; (&euml; ) =
r r r r
r&aring;&aacute;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &aacute; I= &agrave; I= &acirc; I= τ =
^&ecirc;&Eacute;~=&ccedil;&Ntilde;=&ecirc;&Eacute;&Ouml;&aacute;&ccedil;&aring;W=p=
i&Eacute;&aring;&Ouml;&iacute;&Uuml;=&ccedil;&Ntilde;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;W=i=
j~&euml;&euml;=&ccedil;&Ntilde;=~=&iuml;&aacute;&ecirc;&Eacute;W=&atilde;=
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= ρ(&ntilde; I &oacute; I &ograve; ) I= ρ(&ntilde; I &oacute; ) =
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&atilde;~&euml;&euml;W= &ntilde; I= &oacute; I= &ograve; =
c&aacute;&ecirc;&euml;&iacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W= j &ntilde;&oacute; I= j &oacute;&ograve; I= j &ntilde;&ograve; =
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W= f &ntilde; I= f &oacute; I= f&ograve; =
s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=&euml;&ccedil;&auml;&aacute;&Ccedil;W=s=
t&ccedil;&ecirc;&acirc;W=t=
r
j~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= _ =
`&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;W=f=
b&auml;&Eacute;&Aring;&iacute;&ecirc;&ccedil;&atilde;&ccedil;&iacute;&aacute;&icirc;&Eacute;=&Ntilde;&ccedil;&ecirc;&Aring;&Eacute;W= ε =
j~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring;=&Ntilde;&auml;&igrave;&ntilde;W= ψ =
275
CHAPTER 9. INTEGRAL CALCULUS
1117. i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=p&Aring;~&auml;~&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
r r
i&Eacute;&iacute;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&Auml;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &ecirc; = &ecirc; (&euml; ) I=
M ≤ &euml; ≤ p I=~&aring;&Ccedil;=~=&euml;&Aring;~&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=c=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&ccedil;&icirc;&Eacute;&ecirc;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`K==
q&Uuml;&Eacute;&aring;==
p
r
∫ c(&ecirc; (&euml; ))&Ccedil;&euml; = ∫ c(&ntilde; I &oacute;I &ograve; )&Ccedil;&euml; = ∫ c&Ccedil;&euml; I==
M
`
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Ccedil;&euml;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&ecirc;&Aring;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;K==
=
1118. ∫ c &Ccedil;&euml; = ∫ c &Ccedil;&euml; + ∫ c &Ccedil;&euml; =
`N ∪ ` O
`N
`O
=
=
=
Figure 203.
=
r r
1119. f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil;=&Auml;&oacute;= &ecirc; = &ecirc; (&iacute; ) I=
α ≤ &iacute; ≤ β I=&iacute;&Uuml;&Eacute;&aring;==
β
∫ c(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; = ∫ c(&ntilde; (&iacute; )I &oacute;(&iacute; )I &ograve;(&iacute; )) (&ntilde; ′(&iacute; )) + (&oacute;′(&iacute; )) + (&ograve;′(&iacute; )) &Ccedil;&iacute; K=
O
O
O
α
`
=
1120. f&Ntilde;=`=&aacute;&euml;=~=&euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
&oacute; = &Ntilde; (&ntilde; ) I= ~ ≤ &ntilde; ≤ &Auml; I=&iacute;&Uuml;&Eacute;&aring;==
&Auml;
∫ c(&ntilde; I &oacute; )&Ccedil;&euml; = ∫ c(&ntilde; I &Ntilde; (&ntilde; )) N + (&Ntilde; ′(&ntilde; )) &Ccedil;&ntilde; K==
O
`
~
=
1121. i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=p&Aring;~&auml;~&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
276
CHAPTER 9. INTEGRAL CALCULUS
β
O
 &Ccedil;&ecirc; 
∫` c(&ntilde; I &oacute; )&Ccedil;&euml; = ∫α c(&ecirc; &Aring;&ccedil;&euml; θI &ecirc; &euml;&aacute;&aring; θ) &ecirc; +  &Ccedil;θ  &Ccedil;θ I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &ecirc;EθF K=
=
1122. i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=c&aacute;&Eacute;&auml;&Ccedil;=
r r
i&Eacute;&iacute;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&Auml;&Eacute;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &ecirc; = &ecirc; (&euml; ) I=
M ≤ &euml; ≤ p K=q&Uuml;&Eacute;&aring;==
r
&Ccedil;&ecirc; r
= τ = (&Aring;&ccedil;&euml; αI &Aring;&ccedil;&euml; βI &Aring;&ccedil;&euml; γ ) ==
&Ccedil;&euml;
&aacute;&euml;=&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=&auml;&aacute;&aring;&Eacute;=&iacute;&ccedil;=&iacute;&Uuml;&aacute;&euml;=&Aring;&igrave;&ecirc;&icirc;&Eacute;K==
=
O
=
=
Figure 204.
=
r
i&Eacute;&iacute;= ~= &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &Ntilde;&aacute;&Eacute;&auml;&Ccedil;= c (mI nI o ) = &aacute;&euml;= &Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;= &ccedil;&icirc;&Eacute;&ecirc;= &iacute;&Uuml;&Eacute;= &Aring;&igrave;&ecirc;&icirc;&Eacute;= `K=
r
q&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;= c =~&auml;&ccedil;&aring;&Ouml;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=
`=&aacute;&euml;==
p
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; = ∫ (m &Aring;&ccedil;&euml; α + n &Aring;&ccedil;&euml; β + o &Aring;&ccedil;&euml; γ )&Ccedil;&euml; K=
`
M
=
=
277
CHAPTER 9. INTEGRAL CALCULUS
1123. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=&ccedil;&Ntilde;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=c&aacute;&Eacute;&auml;&Ccedil;&euml;=
r r
r r
c
⋅
&Ccedil;
&ecirc;
=
−
c
∫
∫ ⋅ &Ccedil;&ecirc; I==
(
)
−`
(
)
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=-`=&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;==&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=&iuml;&aacute;&iacute;&Uuml;=&iacute;&Uuml;&Eacute;=&ccedil;&eacute;&eacute;&ccedil;&euml;&aacute;&iacute;&Eacute;=&ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;~&iacute;&aacute;&ccedil;&aring;K=
=
r r
r r
r r
r r
∫ c ⋅ &Ccedil;&ecirc; = ∫ c ⋅ &Ccedil;&ecirc; = ∫ c ⋅ &Ccedil;&ecirc; + ∫ c ⋅ &Ccedil;&ecirc; I==
(
)
(
) (
`N ∪ ` O
`
) (
`N
)
`O
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=`=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;&euml;= `N =~&aring;&Ccedil;= ` O K==
=
r
1124. f&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil;=&Auml;&oacute;= &ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ) I=
α ≤ &iacute; ≤ β I=&iacute;&Uuml;&Eacute;&aring;==
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; = =
`
β
&Ccedil;&oacute;
&Ccedil;&ntilde;
&Ccedil;&ograve; 

= ∫  m(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + n(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + o(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) &Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
=
1125. f&Ntilde;=`=&auml;&aacute;&Eacute;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=~&aring;&Ccedil;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;= &oacute; = &Ntilde; (&ntilde; ) I=
&iacute;&Uuml;&Eacute;&aring;==
&Auml;
&Ccedil;&Ntilde; 

m&Ccedil;&ntilde;
n&Ccedil;&oacute;
+
=
∫`
∫~  m(&ntilde; I &Ntilde; (&ntilde; )) + n(&ntilde; I &Ntilde; (&ntilde; )) &Ccedil;&ntilde; &Ccedil;&ntilde; K==
=
1126. d&ecirc;&Eacute;&Eacute;&aring;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
 ∂n ∂m 
∫∫o  ∂&ntilde; − ∂&oacute; &Ccedil;&ntilde;&Ccedil;&oacute; = ∫` m&Ccedil;&ntilde; + n&Ccedil;&oacute; I==
r
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= c = m(&ntilde; I &oacute; ) &aacute; + n(&ntilde; I &oacute; ) &agrave; = &aacute;&euml;= ~= &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;= &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &Ntilde;&igrave;&aring;&Aring;∂m ∂n
&iacute;&aacute;&ccedil;&aring;=&iuml;&aacute;&iacute;&Uuml;==&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;==&Ntilde;&aacute;&ecirc;&euml;&iacute;=&eacute;~&ecirc;&iacute;&aacute;~&auml;==&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;=
I=
=&aacute;&aring;=~=
∂&oacute; ∂&ntilde;
&euml;&ccedil;&atilde;&Eacute;= &Ccedil;&ccedil;&atilde;~&aacute;&aring;= oI= &iuml;&Uuml;&aacute;&Aring;&Uuml;= &aacute;&euml;= &Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;= &Auml;&oacute;= ~= &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;I= &eacute;&aacute;&Eacute;&Aring;&Eacute;&iuml;&aacute;&euml;&Eacute;=
&euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`K==
=
=
278
CHAPTER 9. INTEGRAL CALCULUS
1127. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=o&Eacute;&Ouml;&aacute;&ccedil;&aring;=o=_&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=`&igrave;&ecirc;&icirc;&Eacute;=`=
N
p = ∫∫ &Ccedil;&ntilde;&Ccedil;&oacute; = ∫ &ntilde;&Ccedil;&oacute; − &oacute;&Ccedil;&ntilde; =
O`
o
=
1128. m~&iacute;&Uuml;=f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=i&aacute;&aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;&euml;=
r
r
r
r
q&Uuml;&Eacute;= &auml;&aacute;&aring;&Eacute;= &aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;= &ccedil;&Ntilde;= ~= &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= c = m &aacute; + n &agrave; + o&acirc; = &aacute;&euml;=
&euml;~&aacute;&Ccedil;=&iacute;&ccedil;=&Auml;&Eacute;=&eacute;~&iacute;&Uuml;=&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;I=&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;=mI=nI=~&aring;&Ccedil;=o=~&ecirc;&Eacute;=
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;= &aacute;&aring;= ~= &Ccedil;&ccedil;&atilde;~&aacute;&aring;= aI= ~&aring;&Ccedil;= &aacute;&Ntilde;= &iacute;&Uuml;&Eacute;&ecirc;&Eacute;= &Eacute;&ntilde;&aacute;&euml;&iacute;&euml;= &euml;&ccedil;&atilde;&Eacute;= &euml;&Aring;~&auml;~&ecirc;=
&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;== &igrave; = &igrave;(&ntilde; I &oacute; I &ograve; ) ==E~==&euml;&Aring;~&auml;~&ecirc;==&eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;F==&aacute;&aring;==a==&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
r
∂&igrave;
∂&igrave;
∂&igrave;
= n I=
c = &Ouml;&ecirc;~&Ccedil; &igrave; I=&ccedil;&ecirc;=
= o K==
= m I=
∂&ograve;
∂&ntilde;
∂&oacute;
q&Uuml;&Eacute;&aring;==
rr r
∫ c(&ecirc; ) ⋅ &Ccedil;&ecirc; = ∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; = &igrave;(_) − &igrave;(^ ) K==
`
`
=
1129. q&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=~=`&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;=c&aacute;&Eacute;&auml;&Ccedil;=
r
^=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;= c = &Ouml;&ecirc;~&Ccedil; &igrave; =&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=~=&Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;=
r
r
r
r
&Ntilde;&aacute;&Eacute;&auml;&Ccedil;K=q&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= c = m &aacute; + n &agrave; + o&acirc; =
&aacute;&euml;=&eacute;~&iacute;&Uuml;=&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&aacute;&Ntilde;=~&aring;&Ccedil;=&ccedil;&aring;&auml;&oacute;=&aacute;&Ntilde;==
r
r
r
&aacute;
&agrave;
&acirc;
r ∂
∂
∂ r
&Aring;&igrave;&ecirc;&auml; c =
= M K==
∂&ntilde; ∂&oacute; ∂&ograve;
m n o
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&aacute;&euml;=&iacute;~&acirc;&Eacute;&aring;=&aacute;&aring;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=&euml;&ccedil;=&iacute;&Uuml;~&iacute;==
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; = &igrave;(_) − &igrave;(^ ) I==
`
&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&euml;&iacute;=&Ntilde;&ccedil;&ecirc;=&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&aacute;&aring;&Ouml;=&aacute;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;=&aacute;&euml;=&Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;=
&Aring;~&aring;=&Auml;&Eacute;=&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;==
∂m ∂n
=
K==
∂&oacute; ∂&ntilde;
=
279
CHAPTER 9. INTEGRAL CALCULUS
1130. i&Eacute;&aring;&Ouml;&iacute;&Uuml;=&ccedil;&Ntilde;=~=`&igrave;&ecirc;&icirc;&Eacute;=
r
O
O
O
β
β
&Ccedil;&oacute;
&Ccedil;&ecirc;
(&iacute; ) &Ccedil;&iacute; = ∫  &Ccedil;&ntilde;  +   +  &Ccedil;&ograve;  &Ccedil;&iacute; I=
i = ∫ &Ccedil;&euml; = ∫
&Ccedil;&iacute;
&Ccedil;&iacute;   &Ccedil;&iacute;   &Ccedil;&iacute; 
`
α
α 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=`=&aacute;~=~=&eacute;&aacute;&Eacute;&Aring;&Eacute;&iuml;&aacute;&euml;&Eacute;=&euml;&atilde;&ccedil;&ccedil;&iacute;&Uuml;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=&Ccedil;&Eacute;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;r
&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &ecirc; (&iacute; ) I= α ≤ &iacute; ≤ β K=
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&iacute;&iuml;&ccedil;-&Ccedil;&aacute;&atilde;&Eacute;&aring;&euml;&aacute;&ccedil;&aring;~&auml;I=&iacute;&Uuml;&Eacute;&aring;=
r
O
O
β
β
&Ccedil;&ecirc;
 &Ccedil;&ntilde;   &Ccedil;&oacute; 
(&iacute; ) &Ccedil;&iacute; = ∫   +   &Ccedil;&iacute; K==
i = ∫ &Ccedil;&euml; = ∫
&Ccedil;&iacute;
&Ccedil;&iacute;   &Ccedil;&iacute; 
`
α
α 
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ouml;&ecirc;~&eacute;&Uuml;=&ccedil;&Ntilde;=~=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &oacute; = &Ntilde; (&ntilde; ) =&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;&eacute;&auml;~&aring;&Eacute;= (~ ≤ &ntilde; ≤ &Auml;) I=&iacute;&Uuml;&Eacute;&aring;==
&Auml;
i=∫
~
O
 &Ccedil;&oacute; 
N +   &Ccedil;&ntilde; K==
 &Ccedil;&ntilde; 
=
1131. i&Eacute;&aring;&Ouml;&iacute;&Uuml;=&ccedil;&Ntilde;=~=`&igrave;&ecirc;&icirc;&Eacute;=&aacute;&aring;=m&ccedil;&auml;~&ecirc;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=
β
O
 &Ccedil;&ecirc; 
i = ∫   + &ecirc; O &Ccedil;θ I==
&Ccedil;θ 
α 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;= &ecirc; = &ecirc;(θ) I=
α ≤ θ ≤ β =&aacute;&aring;=&eacute;&ccedil;&auml;~&ecirc;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;K===
=
1132. j~&euml;&euml;=&ccedil;&Ntilde;=~=t&aacute;&ecirc;&Eacute;=
&atilde; = ∫ ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I==
`
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= ρ(&ntilde; I &oacute; I &ograve; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&atilde;~&euml;&euml;=&eacute;&Eacute;&ecirc;=&igrave;&aring;&aacute;&iacute;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&iuml;&aacute;&ecirc;&Eacute;K=
=
f&Ntilde;=`=&aacute;&euml;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;
r
&ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ) I==&iacute;&Uuml;&Eacute;&aring;==&iacute;&Uuml;&Eacute;==&atilde;~&euml;&euml;==&Aring;~&aring;==&Auml;&Eacute;==&Aring;&ccedil;&atilde;&eacute;&igrave;&iacute;&Eacute;&Ccedil;=&Auml;&oacute;=
&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=
280
CHAPTER 9. INTEGRAL CALCULUS
β
O
O
O
 &Ccedil;&ntilde;   &Ccedil;&oacute;   &Ccedil;&ograve; 
&atilde; = ∫ ρ(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ))   +   +   &Ccedil;&iacute; K==
 &Ccedil;&iacute;   &Ccedil;&iacute;   &Ccedil;&iacute; 
α
=
f&Ntilde;=`=&aacute;&euml;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=&aacute;&aring;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&atilde;~&euml;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&iuml;&aacute;&ecirc;&Eacute;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=
&Auml;&oacute;==
&atilde; = ∫ ρ(&ntilde; I &oacute; )&Ccedil;&euml; I=
`
&ccedil;&ecirc;=
β
&atilde; = ∫ ρ(&ntilde; (&iacute; )I &oacute; (&iacute; ))
α
O
O
 &Ccedil;&ntilde;   &Ccedil;&oacute; 
  +   &Ccedil;&iacute; =E&aacute;&aring;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&ccedil;&ecirc;&atilde;FK=
 &Ccedil;&iacute;   &Ccedil;&iacute; 
=
1133. `&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=j~&euml;&euml;=&ccedil;&Ntilde;=~=t&aacute;&ecirc;&Eacute;=
j &oacute;&ograve;
j &ntilde;&oacute;
j
&ntilde;=
I= &oacute; = &ntilde;&ograve; I= &ograve; =
I=
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
j &oacute;&ograve; = ∫ &ntilde;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I==
`
j &ntilde;&ograve; = ∫ &oacute;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I==
`
j &ntilde;&oacute; = ∫ &ograve;ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; K=
`
=
1134. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=
q&Uuml;&Eacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;I=&oacute;-~&ntilde;&aacute;&euml;I=~&aring;&Ccedil;=&ograve;-~&ntilde;&aacute;&euml;=
~&ecirc;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~&euml;=
f &ntilde; = ∫ (&oacute; O + &ograve; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I==
`
f &oacute; = ∫ (&ntilde; O + &ograve; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; I==
`
f &ograve; = ∫ (&ntilde; O + &oacute; O )ρ(&ntilde; I &oacute; I &ograve; )&Ccedil;&euml; K==
`
=
281
CHAPTER 9. INTEGRAL CALCULUS
1135. ^&ecirc;&Eacute;~=&ccedil;&Ntilde;=~=o&Eacute;&Ouml;&aacute;&ccedil;&aring;=_&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;=~=`&auml;&ccedil;&euml;&Eacute;&Ccedil;=`&igrave;&ecirc;&icirc;&Eacute;=
N
p = ∫ &ntilde;&Ccedil;&oacute; = − ∫ &oacute;&Ccedil;&ntilde; = ∫ &ntilde;&Ccedil;&oacute; − &oacute;&Ccedil;&ntilde; K==
O`
`
`
=
=
=
Figure 205.
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&aacute;&aring;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&ccedil;&ecirc;&atilde;=
r
&ecirc; (&iacute; ) = &ntilde; (&iacute; )I &oacute; (&iacute; ) I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=~&ecirc;&Eacute;~=&Aring;~&aring;=&Auml;&Eacute;=&Aring;~&auml;&Aring;&igrave;&auml;~&iacute;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=
β
β
β
&Ccedil;&oacute;
&Ccedil;&oacute;
&Ccedil;&ntilde;
N 
&Ccedil;&ntilde; 
p = ∫ &ntilde; (&iacute; ) &Ccedil;&iacute; = − ∫ &oacute; (&iacute; ) &Ccedil;&iacute; = ∫  &ntilde; (&iacute; ) − &oacute; (&iacute; ) &Ccedil;&iacute; K==
&Ccedil;&iacute;
&Ccedil;&iacute;
O α
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
α
=
1136. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=c&ccedil;&ecirc;&atilde;&Eacute;&Ccedil;=&Auml;&oacute;=o&ccedil;&iacute;~&iacute;&aacute;&aring;&Ouml;=~=`&auml;&ccedil;&euml;&Eacute;&Ccedil;=`&igrave;&ecirc;&icirc;&Eacute;=
~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;=
π
s = − π ∫ &oacute; O&Ccedil;&ntilde; = −Oπ ∫ &ntilde;&oacute;&Ccedil;&oacute; = − ∫ O&ntilde;&oacute;&Ccedil;&oacute; + &oacute; O&Ccedil;&ntilde; ==
O`
`
`
=
282
CHAPTER 9. INTEGRAL CALCULUS
=
=
Figure 206.
=
1137. t&ccedil;&ecirc;&acirc;=
r
t&ccedil;&ecirc;&acirc;=&Ccedil;&ccedil;&aring;&Eacute;=&Auml;&oacute;=~=&Ntilde;&ccedil;&ecirc;&Aring;&Eacute;= c =&ccedil;&aring;=~&aring;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;=&atilde;&ccedil;&icirc;&aacute;&aring;&Ouml;=~&auml;&ccedil;&aring;&Ouml;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=
`=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=
r r
t = ∫ c ⋅ &Ccedil; &ecirc; I=
`
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= c =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&ccedil;&ecirc;&Aring;&Eacute;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;=~&Aring;&iacute;&aacute;&aring;&Ouml;=&ccedil;&aring;=&iacute;&Uuml;&Eacute;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I= &Ccedil; &ecirc; =&aacute;&euml;=
&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&iacute;=&iacute;~&aring;&Ouml;&Eacute;&aring;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;K==
=
=
Figure 207.
283
CHAPTER 9. INTEGRAL CALCULUS
f&Ntilde;=&iacute;&Uuml;&Eacute;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;=&aacute;&euml;=&atilde;&ccedil;&icirc;&Eacute;&Ccedil;=~&auml;&ccedil;&aring;&Ouml;=~=&Aring;&igrave;&ecirc;&icirc;&Eacute;=`=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;I=&iacute;&Uuml;&Eacute;&aring;=
r r
t = ∫ c ⋅ &Ccedil; &ecirc; = ∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; I==
`
`
=
f&Ntilde;= ~= &eacute;~&iacute;&Uuml;= `= &aacute;&euml;= &euml;&eacute;&Eacute;&Aring;&aacute;&Ntilde;&aacute;&Eacute;&Ccedil;= &Auml;&oacute;= ~= &eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;= &iacute;= E&iacute;= &ccedil;&Ntilde;&iacute;&Eacute;&aring;= &atilde;&Eacute;~&aring;&euml;=
&iacute;&aacute;&atilde;&Eacute;FI=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=&Ntilde;&ccedil;&ecirc;=&Aring;~&auml;&Aring;&igrave;&auml;~&iacute;&aacute;&aring;&Ouml;=&iuml;&ccedil;&ecirc;&acirc;=&Auml;&Eacute;&Aring;&ccedil;&atilde;&Eacute;&euml;=
β
&Ccedil;&ntilde;
&Ccedil;&oacute;
&Ccedil;&ograve; 

t = ∫ m(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + n(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; )) + o(&ntilde; (&iacute; )I &oacute; (&iacute; )I &ograve; (&iacute; ))  &Ccedil;&iacute;I
&Ccedil;&iacute;
&Ccedil;&iacute;
&Ccedil;&iacute; 
α
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;=&Ouml;&ccedil;&Eacute;&euml;=&Ntilde;&ecirc;&ccedil;&atilde;= α =&iacute;&ccedil;= β K==
=
r
f&Ntilde;= ~= &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &Ntilde;&aacute;&Eacute;&auml;&Ccedil;= c = &aacute;&euml;= &Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;= ~&aring;&Ccedil;= &igrave;(&ntilde; I &oacute; I &ograve; ) = &aacute;&euml;= ~= &euml;&Aring;~&auml;~&ecirc;=
&eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &Ntilde;&aacute;&Eacute;&auml;&Ccedil;I= &iacute;&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= &iuml;&ccedil;&ecirc;&acirc;= &ccedil;&aring;= ~&aring;= &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;= &atilde;&ccedil;&icirc;&aacute;&aring;&Ouml;=
&Ntilde;&ecirc;&ccedil;&atilde;=^=&iacute;&ccedil;=_=&Aring;~&aring;=&Auml;&Eacute;=&Ntilde;&ccedil;&igrave;&aring;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=
t = &igrave;(_ ) − &igrave;(^ ) K==
=
1138. ^&atilde;&eacute;&Eacute;&ecirc;&Eacute;∞&euml;=i~&iuml;=
r r
_
∫ ⋅ &Ccedil;&ecirc; = &micro;Mf K==
`
r
q&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=&atilde;~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;= _ =~&ecirc;&ccedil;&igrave;&aring;&Ccedil;=~=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=&eacute;~&iacute;&Uuml;=
`= &aacute;&euml;= &Eacute;&egrave;&igrave;~&auml;= &iacute;&ccedil;= &iacute;&Uuml;&Eacute;= &iacute;&ccedil;&iacute;~&auml;= &Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;= f= &Ntilde;&auml;&ccedil;&iuml;&aacute;&aring;&Ouml;= &iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;= &iacute;&Uuml;&Eacute;= ~&ecirc;&Eacute;~=
&Auml;&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&eacute;~&iacute;&Uuml;K==
=
Figure 208.
284
=
CHAPTER 9. INTEGRAL CALCULUS
1139. c~&ecirc;~&Ccedil;~&oacute;∞&euml;=i~&iuml;=
r r
&Ccedil;ψ
=
ε = ∫ b ⋅ &Ccedil;&ecirc; = −
&Ccedil;&iacute;
`
=
q&Uuml;&Eacute;= &Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&ccedil;&atilde;&ccedil;&iacute;&aacute;&icirc;&Eacute;= &Ntilde;&ccedil;&ecirc;&Aring;&Eacute;= E&Eacute;&atilde;&Ntilde;F= ε = &aacute;&aring;&Ccedil;&igrave;&Aring;&Eacute;&Ccedil;= ~&ecirc;&ccedil;&igrave;&aring;&Ccedil;= ~= &Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=
&auml;&ccedil;&ccedil;&eacute;=`=&aacute;&euml;=&Eacute;&egrave;&igrave;~&auml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ecirc;~&iacute;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&Uuml;~&aring;&Ouml;&Eacute;=&ccedil;&Ntilde;=&atilde;~&Ouml;&aring;&Eacute;&iacute;&aacute;&Aring;=&Ntilde;&auml;&igrave;&ntilde;= ψ =
&eacute;~&euml;&euml;&aacute;&aring;&Ouml;=&iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;=&iacute;&Uuml;&Eacute;=&auml;&ccedil;&ccedil;&eacute;K===
=
=
=
Figure 209.
=
=
=
9.13 Surface Integral
=
p&Aring;~&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W= &Ntilde; (&ntilde; I &oacute; I &ograve; ) I= &ograve; (&ntilde; I &oacute; ) =
r
r
m&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &ecirc; (&igrave;I &icirc; ) I= &ecirc; (&ntilde; I &oacute; I &ograve; ) =
r r r
r&aring;&aacute;&iacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;W= &aacute; I= &agrave; I= &acirc; =
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W=p=
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= c (mI nI o ) =
r
r
a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= &Ccedil;&aacute;&icirc; c = ∇ ⋅ c =
285
CHAPTER 9. INTEGRAL CALCULUS
r
r
`&igrave;&ecirc;&auml;=&ccedil;&Ntilde;=~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= &Aring;&igrave;&ecirc;&auml; c = ∇ &times; c ==
r
s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=~=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W= &Ccedil;p =
r
k&ccedil;&ecirc;&atilde;~&auml;=&iacute;&ccedil;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W= &aring; =
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~W=^=
j~&euml;&euml;=&ccedil;&Ntilde;=~=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;W=&atilde;=
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= &micro;(&ntilde; I &oacute; I &ograve; ) =
`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;&euml;=&ccedil;&Ntilde;=&Aring;&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=&atilde;~&euml;&euml;W= &ntilde; I= &oacute; I= &ograve; =
c&aacute;&ecirc;&euml;&iacute;=&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;W= j &ntilde;&oacute; I= j &oacute;&ograve; I= j &ntilde;&ograve; =
j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=&aacute;&aring;&Eacute;&ecirc;&iacute;&aacute;~W= f &ntilde;&oacute; I= f &oacute;&ograve; I= f &ntilde;&ograve; I= f &ntilde; I= f &oacute; I= f&ograve; =
s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=&euml;&ccedil;&auml;&aacute;&Ccedil;W=s=
r
c&ccedil;&ecirc;&Aring;&Eacute;W= c =
d&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;W=d=
rr
c&auml;&igrave;&aacute;&Ccedil;=&icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;W= &icirc; (&ecirc; ) =
c&auml;&igrave;&aacute;&Ccedil;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= ρ =
r
m&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute;W= &eacute;(&ecirc; ) =
j~&euml;&euml;=&Ntilde;&auml;&igrave;&ntilde;I=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&auml;&igrave;&ntilde;W= Φ =
p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=&Aring;&Uuml;~&ecirc;&Ouml;&Eacute;W=n=
`&Uuml;~&ecirc;&Ouml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;W= σ(&ntilde; I &oacute; ) =
r
j~&Ouml;&aring;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;W= b =
=
=
1140. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=~=p&Aring;~&auml;~&ecirc;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
i&Eacute;&iacute;=~=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=&Auml;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=
r
r
r
r
&ecirc; (&igrave;I &icirc; ) = &ntilde; (&igrave;I &icirc; )&aacute; + &oacute; (&igrave;I &icirc; ) &agrave; + &ograve; (&igrave;I &icirc; )&acirc; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= (&igrave;I &icirc; ) = &ecirc;~&aring;&Ouml;&Eacute;&euml;= &ccedil;&icirc;&Eacute;&ecirc;= &euml;&ccedil;&atilde;&Eacute;= &Ccedil;&ccedil;&atilde;~&aacute;&aring;= a(&igrave;I &icirc; ) = &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &igrave;&icirc;&eacute;&auml;~&aring;&Eacute;K=
q&Uuml;&Eacute;==&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;==&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;==&ccedil;&Ntilde;==~==&euml;&Aring;~&auml;~&ecirc;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;== &Ntilde; (&ntilde; I &oacute; I &ograve; ) =&ccedil;&icirc;&Eacute;&ecirc;=
&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=&aacute;&euml;=&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;=~&euml;==
r
r
∂&ecirc; ∂&ecirc;
(
)
(
(
)
(
)
(
)
)
&Ntilde;
&ntilde;
I
&oacute;
I
&ograve;
&Ccedil;p
&Ntilde;
&ntilde;
&igrave;
I
&icirc;
I
&oacute;
&igrave;
I
&icirc;
I
&ograve;
&igrave;
I
&icirc;
&Ccedil;&igrave;&Ccedil;&icirc; I==
=
&times;
∫∫p
∫∫
&igrave;
&icirc;
∂
∂
a(&igrave; I &icirc; )
r
r
∂&ecirc;
∂&ecirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;= =~&aring;&Ccedil;= =~&ecirc;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
∂&igrave;
∂&icirc;
286
CHAPTER 9. INTEGRAL CALCULUS
r
r
r ∂&oacute;
r ∂&ograve;
∂&ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) &aacute; + (&igrave;I &icirc; ) &agrave; + (&igrave;I &icirc; )&acirc; I==
∂&igrave; ∂&igrave;
∂&igrave;
∂&igrave;
r
r
r ∂&oacute;
r ∂&ograve;
∂ &ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) &aacute; + (&igrave;I &icirc; ) &agrave; + (&igrave;I &icirc; )&acirc; =
∂&icirc; ∂&icirc;
∂&icirc;
∂&icirc;
r
r
∂&ecirc; ∂&ecirc;
~&aring;&Ccedil;= &times; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Aring;&ecirc;&ccedil;&euml;&euml;=&eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;K==
∂&igrave; ∂&icirc;
=
1141. f&Ntilde;==&iacute;&Uuml;&Eacute;==&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;==p==&aacute;&euml;==&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;= &ograve; = &ograve;(&ntilde; I &oacute; ) =&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
&ograve; (&ntilde; I &oacute; ) ==&aacute;&euml;==~==&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute;==&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==&aacute;&aring;=&iacute;&Uuml;&Eacute;=&Ccedil;&ccedil;&atilde;~&aacute;&aring;= a(&ntilde; I &oacute; ) I=
&iacute;&Uuml;&Eacute;&aring;==
∫∫ &Ntilde; (&ntilde; I &oacute; I &ograve; )&Ccedil;p = (∫∫ ) &Ntilde; (&ntilde; I &oacute; I &ograve;(&ntilde; I &oacute; ))
p
a &ntilde; I&oacute;
O
 ∂&ograve;   ∂&ograve; 
N +   +   &Ccedil;&ntilde;&Ccedil;&oacute; K==
 ∂&ntilde;   ∂&oacute; 
O
=
r
1142. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=s&Eacute;&Aring;&iacute;&ccedil;&ecirc;=c&aacute;&Eacute;&auml;&Ccedil;= c =&ccedil;&icirc;&Eacute;&ecirc;=&iacute;&Uuml;&Eacute;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=
• f&Ntilde;=p=&aacute;&euml;=&ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil;=&ccedil;&igrave;&iacute;&iuml;~&ecirc;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;==
r
r
r
r
===== ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &aring;&Ccedil;p =
p
======
p
r
r
r
 ∂&ecirc; ∂&ecirc; 
= ∫∫ c(&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) ⋅  &times;  &Ccedil;&igrave;&Ccedil;&icirc; K==
 ∂&igrave; ∂&icirc; 
a( &igrave; I &icirc; )
=
f&Ntilde;=p=&aacute;&euml;=&ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil;=&aacute;&aring;&iuml;~&ecirc;&Ccedil;I=&iacute;&Uuml;&Eacute;&aring;==
r
r
r
r
===== ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &aring;&Ccedil;p =
•
p
======
=
p
r
r
r
 ∂&ecirc; ∂&ecirc; 
= ∫∫ c(&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) ⋅  &times;  &Ccedil;&igrave;&Ccedil;&icirc; K==
 ∂&icirc; ∂&igrave; 
a( &igrave; I &icirc; )
r r
&Ccedil;p = &aring;&Ccedil;p ==&aacute;&euml;==&Aring;~&auml;&auml;&Eacute;&Ccedil;==&iacute;&Uuml;&Eacute;==&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;K==a&ccedil;&iacute;=
&atilde;&Eacute;~&aring;&euml;= = &iacute;&Uuml;&Eacute;= = &euml;&Aring;~&auml;~&ecirc;= = &eacute;&ecirc;&ccedil;&Ccedil;&igrave;&Aring;&iacute;= = &ccedil;&Ntilde;= = &iacute;&Uuml;&Eacute;= = ~&eacute;&eacute;&ecirc;&ccedil;&eacute;&ecirc;&aacute;~&iacute;&Eacute;= = &icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;&euml;K=
r
r
∂&ecirc;
∂&ecirc;
q&Uuml;&Eacute;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;= =~&aring;&Ccedil;== =~&ecirc;&Eacute;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
∂&igrave;
∂&icirc;
287
CHAPTER 9. INTEGRAL CALCULUS
r
r
r ∂&oacute;
r ∂&ograve;
∂&ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) ⋅ &aacute; + (&igrave;I &icirc; ) ⋅ &agrave; + (&igrave;I &icirc; ) ⋅ &acirc; I==
∂&igrave; ∂&igrave;
∂&igrave;
∂&igrave;
r
r
r ∂&oacute;
r ∂&ograve;
∂ &ecirc; ∂&ntilde;
= (&igrave;I &icirc; ) ⋅ &aacute; + (&igrave;I &icirc; ) ⋅ &agrave; + (&igrave;I &icirc; ) ⋅ &acirc; K==
∂&icirc; ∂&icirc;
∂&icirc;
∂&icirc;
=
1143. f&Ntilde;==&iacute;&Uuml;&Eacute;==&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;==p==&aacute;&euml;==&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;= &ograve; = &ograve; (&ntilde; I &oacute; ) I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
&ograve; (&ntilde; I &oacute; ) ==&aacute;&euml;==~==&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute;==&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==&aacute;&aring;==&iacute;&Uuml;&Eacute;=&Ccedil;&ccedil;&atilde;~&aacute;&aring;= a(&ntilde; I &oacute; ) I=
&iacute;&Uuml;&Eacute;&aring;==
• f&Ntilde;= p= &aacute;&euml;= &ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil;= &igrave;&eacute;&iuml;~&ecirc;&Ccedil;I= &aacute;K&Eacute;K= &iacute;&Uuml;&Eacute;= &acirc;-&iacute;&Uuml;= &Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;=
&aring;&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&aacute;&euml;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;I=&iacute;&Uuml;&Eacute;&aring;===
r
r
r
r
===== ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &aring;&Ccedil;p =
p
p
=
======
r
 ∂&ograve; r ∂&ograve; r r 
 −
(
)
c
&ntilde;
I
&oacute;
I
&ograve;
⋅
&aacute;−
&agrave; + &acirc; &Ccedil;&ntilde;&Ccedil;&oacute; I==
∫∫
∂&oacute;
 ∂&ntilde;

a( &ntilde; I &oacute; )
=
f&Ntilde;=p=&aacute;&euml;=&ccedil;&ecirc;&aacute;&Eacute;&aring;&iacute;&Eacute;&Ccedil;=&Ccedil;&ccedil;&iuml;&aring;&iuml;~&ecirc;&Ccedil;I=&aacute;K&Eacute;K=&iacute;&Uuml;&Eacute;=&acirc;-&iacute;&Uuml;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=
&aring;&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&aacute;&euml;=&aring;&Eacute;&Ouml;~&iacute;&aacute;&icirc;&Eacute;I=&iacute;&Uuml;&Eacute;&aring;===
r
r
r
r
===== ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &Ccedil;p = ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅ &aring;&Ccedil;p
•
p
p
r
 ∂&ograve; r ∂&ograve; r r 
= ∫∫ c(&ntilde; I &oacute; I &ograve; ) ⋅  &aacute; +
&agrave; − &acirc; &Ccedil;&ntilde;&Ccedil;&oacute; K==
∂&oacute;
 ∂&ntilde;

a( &ntilde; I &oacute; )
======
=
1144.
∫∫ (c ⋅ &aring;)&Ccedil;p = ∫∫ m&Ccedil;&oacute;&Ccedil;&ograve; + n&Ccedil;&ograve;&Ccedil;&ntilde; + o&Ccedil;&ntilde;&Ccedil;&oacute; =
r r
p
p
= ∫∫ (m &Aring;&ccedil;&euml; α + n &Aring;&ccedil;&euml; β + o &Aring;&ccedil;&euml; γ )&Ccedil;p I==
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= m(&ntilde; I &oacute; I &ograve; ) I= n(&ntilde; I &oacute; I &ograve; ) I= o(&ntilde; I &oacute; I &ograve; ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=
r
&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &Ntilde;&aacute;&Eacute;&auml;&Ccedil;= c K==
&Aring;&ccedil;&euml; α I= &Aring;&ccedil;&euml; β I= &Aring;&ccedil;&euml; γ ==~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;= ~&aring;&Ouml;&auml;&Eacute;&euml;= &Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&ccedil;&igrave;&iacute;&Eacute;&ecirc;=&igrave;&aring;&aacute;&iacute;=
r
&aring;&ccedil;&ecirc;&atilde;~&auml;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &aring; =~&aring;&Ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;I=&oacute;-~&ntilde;&aacute;&euml;I=~&aring;&Ccedil;=&ograve;-~&ntilde;&aacute;&euml;I=&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;K=
=
288
CHAPTER 9. INTEGRAL CALCULUS
1145. f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&aacute;&aring;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&ccedil;&ecirc;&atilde;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=
r
&ecirc; (&ntilde; (&igrave;I &icirc; )I &oacute; (&igrave;I &icirc; )I &ograve; (&igrave;I &icirc; )) I==&iacute;&Uuml;&Eacute;&aring;==&iacute;&Uuml;&Eacute;==&auml;~&iacute;&iacute;&Eacute;&ecirc;=&Ntilde;&ccedil;&ecirc;&atilde;&igrave;&auml;~=&Aring;~&aring;=&Auml;&Eacute;=
&iuml;&ecirc;&aacute;&iacute;&iacute;&Eacute;&aring;=~&euml;==
m n o
∂&ntilde; ∂&oacute; ∂&ograve;
&Ccedil;&igrave;&Ccedil;&icirc; I
∫∫p
∫∫
∂
∂
∂
&igrave;
&igrave;
&igrave;
p
a(&igrave; I&icirc; )
∂&ntilde; ∂&oacute; ∂&ograve;
∂&icirc; ∂&icirc; ∂&icirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= (&igrave;I &icirc; ) = &ecirc;~&aring;&Ouml;&Eacute;&euml;= &ccedil;&icirc;&Eacute;&ecirc;= &euml;&ccedil;&atilde;&Eacute;= &Ccedil;&ccedil;&atilde;~&aacute;&aring;= a(&igrave;I &icirc; ) = &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &igrave;&icirc;&eacute;&auml;~&aring;&Eacute;K=
=
1146. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
r r
r
c
⋅
&Ccedil;
p
=
∇
⋅
c
&Ccedil;s I==
∫∫
∫∫∫
( )
r r
c ⋅ &aring; &Ccedil;p = ∫∫ m&Ccedil;&oacute;&Ccedil;&ograve; + n&Ccedil;&ograve;&Ccedil;&ntilde; + o&Ccedil;&ntilde;&Ccedil;&oacute; =
(
p
)
d
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
r
c(&ntilde; I &oacute; I &ograve; ) = m(&ntilde; I &oacute; I &ograve; )I n(&ntilde; I &oacute; I &ograve; )I o(&ntilde; I &oacute; I &ograve; ) ===
&aacute;&euml;==~==&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;==&Ntilde;&aacute;&Eacute;&auml;&Ccedil;==&iuml;&Uuml;&ccedil;&euml;&Eacute;==&Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml;==mI==nI==~&aring;&Ccedil;==o==&Uuml;~&icirc;&Eacute;==
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;I==
r ∂m ∂n ∂o
∇⋅c =
+
+
==
∂&ntilde; ∂&oacute; ∂&ograve;
r
r
&aacute;&euml;==&iacute;&Uuml;&Eacute;==&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;==&ccedil;&Ntilde;== c I==~&auml;&euml;&ccedil;==&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;== &Ccedil;&aacute;&icirc;c K==q&Uuml;&Eacute;==&euml;&oacute;&atilde;&Auml;&ccedil;&auml;=
∫∫ =&aacute;&aring;&Ccedil;&aacute;&Aring;~&iacute;&Eacute;&euml;==&iacute;&Uuml;~&iacute;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&aacute;&euml;=&iacute;~&acirc;&Eacute;&aring;=&ccedil;&icirc;&Eacute;&ecirc;=~=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=
&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;K==
=
1147. a&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=&aacute;&aring;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=c&ccedil;&ecirc;&atilde;=
 ∂m ∂n ∂o 

&Ccedil;&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; K==
+
+
=
+
+
m&Ccedil;&oacute;&Ccedil;&ograve;
n&Ccedil;&ntilde;&Ccedil;&ograve;
o&Ccedil;&ntilde;&Ccedil;&oacute;
∫∫p
∫∫∫
∂
∂
∂
&ntilde;
&oacute;
&ograve;

d 
=
1148. p&iacute;&ccedil;&acirc;&Eacute;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
r r
r r
c
⋅
&Ccedil;
&ecirc;
=
∇
&times;
c
⋅ &Ccedil;p I==
∫
∫∫
(
`
)
p
289
CHAPTER 9. INTEGRAL CALCULUS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
r
c(&ntilde; I &oacute; I &ograve; ) = m(&ntilde; I &oacute; I &ograve; )I n(&ntilde; I &oacute; I &ograve; )I o(&ntilde; I &oacute; I &ograve; ) ===
&aacute;&euml;==~=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;==&Ntilde;&aacute;&Eacute;&auml;&Ccedil;==&iuml;&Uuml;&ccedil;&euml;&Eacute;==&Aring;&ccedil;&atilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&euml;==mI==nI==~&aring;&Ccedil;=o==&Uuml;~&icirc;&Eacute;=
&Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&eacute;~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;I==
r
r
r
&aacute;
&agrave;
&acirc;
r ∂
∂
∂  ∂o ∂n  r  ∂m ∂o  r  ∂n ∂m  r
&aacute; + 
∇&times;c=
=
−
−
− &acirc;
&agrave; +
∂&ntilde; ∂&ntilde; ∂&ntilde;  ∂&oacute; ∂&ograve;   ∂&ograve; ∂&ntilde;   ∂&ntilde; ∂&oacute; 
m n o
r
r
&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&auml;=&ccedil;&Ntilde;= c I=~&auml;&euml;&ccedil;=&Ccedil;&Eacute;&aring;&ccedil;&iacute;&Eacute;&Ccedil;= &Aring;&igrave;&ecirc;&auml; c K==
q&Uuml;&Eacute;=&euml;&oacute;&atilde;&Auml;&ccedil;&auml;== ∫ =&aacute;&aring;&Ccedil;&aacute;&Aring;~&iacute;&Eacute;&euml;=&iacute;&Uuml;~&iacute;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=&aacute;&euml;=&iacute;~&acirc;&Eacute;&aring;=&ccedil;&icirc;&Eacute;&ecirc;=
~=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=&Aring;&igrave;&ecirc;&icirc;&Eacute;K==
=
1149. p&iacute;&ccedil;&acirc;&Eacute;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=&aacute;&aring;=`&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=c&ccedil;&ecirc;&atilde;=
∫ m&Ccedil;&ntilde; + n&Ccedil;&oacute; + o&Ccedil;&ograve; =
`
 ∂o ∂n 
 ∂n ∂m 
∂m ∂o 
&Ccedil;&oacute;&Ccedil;&ograve; +  −
= ∫∫ 
−
− &Ccedil;&ntilde;&Ccedil;&oacute;
&Ccedil;&ograve;&Ccedil;&ntilde; + 
∂&oacute; ∂&ograve; 
 ∂&ograve; ∂&ntilde; 
 ∂&ntilde; ∂&oacute; 
p 
==
1150. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=^&ecirc;&Eacute;~=
^ = ∫∫ &Ccedil;p =
p
=
1151. f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=&aacute;&euml;=&eacute;~&ecirc;~&atilde;&Eacute;&iacute;&Eacute;&ecirc;&aacute;&ograve;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;==
r
r
r
r
&ecirc; (&igrave;I &icirc; ) = &ntilde; (&igrave;I &icirc; )&aacute; + &oacute; (&igrave;I &icirc; ) &agrave; + &ograve; (&igrave;I &icirc; )&acirc; I==
&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~=&aacute;&euml;==
r
r
∂&ecirc; ∂&ecirc;
^ = ∫∫
&times; &Ccedil;&igrave;&Ccedil;&icirc; I==
∂&igrave; ∂&icirc;
a( &igrave; I &icirc; )
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;== a(&igrave;I &icirc; ) ==&aacute;&euml;==&iacute;&Uuml;&Eacute;==&Ccedil;&ccedil;&atilde;~&aacute;&aring;==&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;= &ecirc; (&igrave;I &icirc; ) =&aacute;&euml;=
&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&Eacute;&Ccedil;K==
=
290
CHAPTER 9. INTEGRAL CALCULUS
1152. f&Ntilde;=p=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Eacute;&ntilde;&eacute;&auml;&aacute;&Aring;&aacute;&iacute;&auml;&oacute;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &ograve; (&ntilde; I &oacute; ) I==&iacute;&Uuml;&Eacute;&aring;==&iacute;&Uuml;&Eacute;==&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=~&ecirc;&Eacute;~=&aacute;&euml;==
O
 ∂&ograve;   ∂&ograve; 
^ = ∫∫ N +   +   &Ccedil;&ntilde;&Ccedil;&oacute; I==
 ∂&ntilde;   ∂&oacute; 
a( &ntilde; I &oacute; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= a(&ntilde; I &oacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&agrave;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=p=&ccedil;&aring;&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;&eacute;&auml;~&aring;&Eacute;K==
=
1153. j~&euml;&euml;=&ccedil;&Ntilde;=~=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=
&atilde; = ∫∫ &micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
O
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &micro;(&ntilde; I &oacute; I &ograve; ) = &aacute;&euml;= &iacute;&Uuml;&Eacute;= &atilde;~&euml;&euml;= &eacute;&Eacute;&ecirc;= &igrave;&aring;&aacute;&iacute;= ~&ecirc;&Eacute;~= = E&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;= &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;FK=
=
1154. `&Eacute;&aring;&iacute;&Eacute;&ecirc;=&ccedil;&Ntilde;=j~&euml;&euml;=&ccedil;&Ntilde;=~=p&Uuml;&Eacute;&auml;&auml;=
j &oacute;&ograve;
j &ntilde;&oacute;
j
&ntilde;=
I= &oacute; = &ntilde;&ograve; I= &ograve; =
I==
&atilde;
&atilde;
&atilde;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
j &oacute;&ograve; = ∫∫ &ntilde;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
j &ntilde;&ograve; = ∫∫ &oacute;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
j &ntilde;&oacute; = ∫∫ &ograve;&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p =
p
~&ecirc;&Eacute;==&iacute;&Uuml;&Eacute;==&Ntilde;&aacute;&ecirc;&euml;&iacute;==&atilde;&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;==~&Auml;&ccedil;&igrave;&iacute;===&iacute;&Uuml;&Eacute;=&Aring;&ccedil;&ccedil;&ecirc;&Ccedil;&aacute;&aring;~&iacute;&Eacute;=&eacute;&auml;~&aring;&Eacute;&euml;= &ntilde; = M I=
&oacute; = M I= &ograve; = M I==&ecirc;&Eacute;&euml;&eacute;&Eacute;&Aring;&iacute;&aacute;&icirc;&Eacute;&auml;&oacute;K== &micro;(&ntilde; I &oacute; I &ograve; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K=
=
1155. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;=E&ccedil;&ecirc;= &ograve; = M FI==&oacute;&ograve;-&eacute;&auml;~&aring;&Eacute;==
E &ntilde; = M FI=~&aring;&Ccedil;=&ntilde;&ograve;-&eacute;&auml;~&aring;&Eacute;=E &oacute; = M F=
f &ntilde;&oacute; = ∫∫ &ograve; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
f &oacute;&ograve; = ∫∫ &ntilde; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
291
CHAPTER 9. INTEGRAL CALCULUS
f &ntilde;&ograve; = ∫∫ &oacute; O&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p K=
p
=
1156. j&ccedil;&atilde;&Eacute;&aring;&iacute;&euml;=&ccedil;&Ntilde;=f&aring;&Eacute;&ecirc;&iacute;&aacute;~=~&Auml;&ccedil;&igrave;&iacute;=&iacute;&Uuml;&Eacute;=&ntilde;-~&ntilde;&aacute;&euml;I=&oacute;-~&ntilde;&aacute;&euml;I=~&aring;&Ccedil;=&ograve;-~&ntilde;&aacute;&euml;=
f &ntilde; = ∫∫ (&oacute; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
f &oacute; = ∫∫ (&ntilde; O + &ograve; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p I==
p
f&ograve; = ∫∫ (&ntilde; O + &oacute; O )&micro;(&ntilde; I &oacute; I &ograve; )&Ccedil;p K==
p
=
1157. s&ccedil;&auml;&igrave;&atilde;&Eacute;=&ccedil;&Ntilde;=~=p&ccedil;&auml;&aacute;&Ccedil;=_&ccedil;&igrave;&aring;&Ccedil;&Eacute;&Ccedil;=&Auml;&oacute;=~=`&auml;&ccedil;&euml;&Eacute;&Ccedil;=p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=
s=
N
&ntilde;&Ccedil;&oacute;&Ccedil;&ograve; + &oacute;&Ccedil;&ntilde;&Ccedil;&ograve; + &ograve;&Ccedil;&ntilde;&Ccedil;&oacute; ==
P ∫∫
p
=
1158. d&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml;=c&ccedil;&ecirc;&Aring;&Eacute;=
r
r
&ecirc;
c = d&atilde; ∫∫ &micro;(&ntilde; I &oacute; I &ograve; ) P &Ccedil;p I=
&ecirc;
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&atilde;=&aacute;&euml;=~=&atilde;~&euml;&euml;=~&iacute;=~=&eacute;&ccedil;&aacute;&aring;&iacute;= &ntilde; M I &oacute; M I &ograve; M =&ccedil;&igrave;&iacute;&euml;&aacute;&Ccedil;&Eacute;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;I==
r
&ecirc; = &ntilde; − &ntilde; M I &oacute; − &oacute; M I &ograve; − &ograve; M I==
&micro;(&ntilde; I &oacute; I &ograve; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I==
~&aring;&Ccedil;=d=&aacute;&euml;=&Ouml;&ecirc;~&icirc;&aacute;&iacute;~&iacute;&aacute;&ccedil;&aring;~&auml;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1159. m&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute;=c&ccedil;&ecirc;&Aring;&Eacute;=
r
r r
c = ∫∫ &eacute;(&ecirc; )&Ccedil;p I==
p
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&Eacute;&euml;&euml;&igrave;&ecirc;&Eacute;== &eacute;(&ecirc; ) ==~&Aring;&iacute;&euml;==&ccedil;&aring;==&iacute;&Uuml;&Eacute;==&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;==p==&Ouml;&aacute;&icirc;&Eacute;&aring;==&Auml;&oacute;=
r
&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&ccedil;&aring;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;= &ecirc; K=
=
1160. c&auml;&igrave;&aacute;&Ccedil;=c&auml;&igrave;&ntilde;=E~&Aring;&ecirc;&ccedil;&euml;&euml;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=pF=
r r r
Φ = ∫∫ &icirc; (&ecirc; ) ⋅ &Ccedil;p I==
p
292
CHAPTER 9. INTEGRAL CALCULUS
r r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &icirc; (&ecirc; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&auml;&igrave;&aacute;&Ccedil;=&icirc;&Eacute;&auml;&ccedil;&Aring;&aacute;&iacute;&oacute;K=
==
1161. j~&euml;&euml;=c&auml;&igrave;&ntilde;=E~&Aring;&ecirc;&ccedil;&euml;&euml;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=pF=
r r r
Φ = ∫∫ ρ&icirc; (&ecirc; ) ⋅ &Ccedil;p I==
p
r
r
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= c = ρ&icirc; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;&Eacute;&Aring;&iacute;&ccedil;&ecirc;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;I= ρ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&auml;&igrave;&aacute;&Ccedil;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;K=
=
1162. p&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=`&Uuml;~&ecirc;&Ouml;&Eacute;=
n = ∫∫ σ(&ntilde; I &oacute; )&Ccedil;p I==
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= σ(&ntilde; I &oacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=&Aring;&Uuml;~&ecirc;&Ouml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;K=
=
1163. d~&igrave;&euml;&euml;∞=i~&iuml;=
q&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&auml;&igrave;&ntilde;=&iacute;&Uuml;&ecirc;&ccedil;&igrave;&Ouml;&Uuml;=~&aring;&oacute;=&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=&aacute;&euml;=&eacute;&ecirc;&ccedil;&eacute;&ccedil;&ecirc;&iacute;&aacute;&ccedil;&aring;~&auml;=
&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Aring;&Uuml;~&ecirc;&Ouml;&Eacute;=n=&Eacute;&aring;&Aring;&auml;&ccedil;&euml;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&ecirc;&Ntilde;~&Aring;&Eacute;=
r r n
Φ = ∫∫ b ⋅ &Ccedil;p = I==
εM
p
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
Φ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&auml;&igrave;&ntilde;I==
r
b =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&atilde;~&Ouml;&aring;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Eacute;&auml;&Eacute;&Aring;&iacute;&ecirc;&aacute;&Aring;=&Ntilde;&aacute;&Eacute;&auml;&Ccedil;=&euml;&iacute;&ecirc;&Eacute;&aring;&Ouml;&iacute;&Uuml;I=
c
ε M = UIUR &times; NM −NO =&aacute;&euml;=&eacute;&Eacute;&ecirc;&atilde;&aacute;&iacute;&iacute;&aacute;&icirc;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&Ntilde;&ecirc;&Eacute;&Eacute;=&euml;&eacute;~&Aring;&Eacute;K==
&atilde;
=
=
293
Chapter 10
Differential Equations
=
=
=
=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&ccedil;&aring;&Eacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;W=&oacute;I=&eacute;I=&egrave;I=&igrave;I=&Ouml;I=&Uuml;I=dI=eI=&ecirc;I=&ograve;==
^&ecirc;&Ouml;&igrave;&atilde;&Eacute;&aring;&iacute;&euml;=E&aacute;&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;FW=&ntilde;I=&oacute;=
c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&iuml;&ccedil;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W= &Ntilde; (&ntilde; I &oacute; ) I= j(&ntilde; I &oacute; ) I= k(&ntilde; I &oacute; ) =
&Ccedil;&oacute;
c&aacute;&ecirc;&euml;&iacute;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;W= &oacute; ′ I= &igrave;′ I= &oacute;&amp; I= I=&pound;=
&Ccedil;&iacute;
&Ccedil; Of
p&Eacute;&Aring;&ccedil;&aring;&Ccedil;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W= &oacute;′′ I= &amp;&oacute;&amp; I= O I=&pound;=
&Ccedil;&iacute;
O
∂&igrave; ∂ &igrave;
m~&ecirc;&iacute;&aacute;~&auml;=&Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;&euml;W= I= O I=&pound;=
∂&iacute; ∂&ntilde;
k~&iacute;&igrave;&ecirc;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
m~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;&euml;W= &oacute; N I= &oacute; &eacute; =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&acirc;I=&iacute;I=`I= `N I= ` O I=&eacute;I=&egrave;I= α I= β =
o&ccedil;&ccedil;&iacute;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;W= λN I= λ O =
q&aacute;&atilde;&Eacute;W=&iacute;=
q&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;W=qI=p=
m&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= m(&iacute; ) ==
j~&euml;&euml;=&ccedil;&Ntilde;=~&aring;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;W=&atilde;=
p&iacute;&aacute;&Ntilde;&Ntilde;&aring;&Eacute;&euml;&euml;=&ccedil;&Ntilde;=~=&euml;&eacute;&ecirc;&aacute;&aring;&Ouml;W=&acirc;=
a&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&atilde;~&euml;&euml;=&Ntilde;&ecirc;&ccedil;&atilde;=&Eacute;&egrave;&igrave;&aacute;&auml;&aacute;&Auml;&ecirc;&aacute;&igrave;&atilde;W=&oacute;=
^&atilde;&eacute;&auml;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W=^=
c&ecirc;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&oacute;W= ω =
a~&atilde;&eacute;&aacute;&aring;&Ouml;=&Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;W= γ =
m&Uuml;~&euml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W= δ =
^&aring;&Ouml;&igrave;&auml;~&ecirc;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;W= θ =
m&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde;=&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;W=i=
294
CHAPTER 10. DIFFERENTIAL EQUATIONS
^&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&Ouml;&ecirc;~&icirc;&aacute;&iacute;&oacute;W=&Ouml;=
`&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;W=f=
o&Eacute;&euml;&aacute;&euml;&iacute;~&aring;&Aring;&Eacute;W=o=
f&aring;&Ccedil;&igrave;&Aring;&iacute;~&aring;&Aring;&Eacute;W=i=
`~&eacute;~&Aring;&aacute;&iacute;~&aring;&Aring;&Eacute;W=`=
=
=
10.1 First Order Ordinary Differential
Equations
=
1164. i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=
&Ccedil;&oacute;
+ &eacute;(&ntilde; )&oacute; = &egrave;(&ntilde; ) K==
&Ccedil;&ntilde;
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
∫ &igrave;(&ntilde; )&egrave;(&ntilde; )&Ccedil;&ntilde; + ` I==
&oacute;=
&igrave; (&ntilde; )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&igrave;(&ntilde; ) = &Eacute;&ntilde;&eacute; ∫ &eacute;(&ntilde; )&Ccedil;&ntilde; K=
(
)
=
1165. p&Eacute;&eacute;~&ecirc;~&Auml;&auml;&Eacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=
&Ccedil;&oacute;
= &Ntilde; (&ntilde; I &oacute; ) = &Ouml; (&ntilde; )&Uuml;(&oacute; ) =
&Ccedil;&ntilde;
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;=
&Ccedil;&oacute;
∫ &Uuml;(&oacute; ) = ∫ &Ouml;(&ntilde; )&Ccedil;&ntilde; + ` I==
&ccedil;&ecirc;=
e(&oacute; ) = d(&ntilde; ) + ` K=
=
=
=
295
CHAPTER 10. DIFFERENTIAL EQUATIONS
1166. e&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=
&Ccedil;&oacute;
= &Ntilde; (&ntilde; I &oacute; ) = &aacute;&euml;= &Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;I= &aacute;&Ntilde;=
&Ccedil;&ntilde;
&iacute;&Uuml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &Ntilde; (&ntilde; I &oacute; ) =&aacute;&euml;=&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;I=&iacute;&Uuml;~&iacute;=&aacute;&euml;==
&Ntilde; (&iacute;&ntilde; I &iacute;&oacute; ) = &Ntilde; (&ntilde; I &oacute; ) K==
=
&oacute;
q&Uuml;&Eacute;=&euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;= &ograve; = =E&iacute;&Uuml;&Eacute;&aring;= &oacute; = &ograve;&ntilde; F=&auml;&Eacute;~&Ccedil;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&eacute;~&ecirc;~&Auml;&auml;&Eacute;=
&ntilde;
&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
&Ccedil;&ograve;
&ntilde;
+ &ograve; = &Ntilde; (NI &ograve; ) K==
&Ccedil;&ntilde;
=
1167. _&Eacute;&ecirc;&aring;&ccedil;&igrave;&auml;&auml;&aacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
&Ccedil;&oacute;
+ &eacute;(&ntilde; )&oacute; = &egrave;(&ntilde; )&oacute; &aring; K==
&Ccedil;&ntilde;
=
q&Uuml;&Eacute;=&euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;= &ograve; = &oacute; N−&aring; =&auml;&Eacute;~&Ccedil;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&auml;&aacute;&aring;&Eacute;~&ecirc;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;==
&Ccedil;&ograve;
+ (N − &aring; )&eacute;(&ntilde; ) &ograve; = (N − &aring; )&egrave;(&ntilde; ) K==
&Ccedil;&ntilde;
=
1168. o&aacute;&Aring;&Aring;~&iacute;&aacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
&Ccedil;&oacute;
= &eacute;(&ntilde; ) + &egrave;(&ntilde; ) &oacute; + &ecirc;(&ntilde; ) &oacute; O =
&Ccedil;&ntilde;
=
f&Ntilde;=~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;= &oacute; N =&aacute;&euml;=&acirc;&aring;&ccedil;&iuml;&aring;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&Aring;~&aring;=&Auml;&Eacute;=&ccedil;&Auml;&iacute;~&aacute;&aring;&Eacute;&Ccedil;=&iuml;&aacute;&iacute;&Uuml;=&iacute;&Uuml;&Eacute;=&Uuml;&Eacute;&auml;&eacute;=&ccedil;&Ntilde;=&euml;&igrave;&Auml;&euml;&iacute;&aacute;&iacute;&igrave;&iacute;&aacute;&ccedil;&aring;=
N
&ograve;=
I=&iuml;&Uuml;&aacute;&Aring;&Uuml;=&auml;&Eacute;~&Ccedil;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=&auml;&aacute;&aring;&Eacute;~&ecirc;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;==
&oacute; − &oacute;N
&Ccedil;&ograve;
= −[&egrave;(&ntilde; ) + O&oacute; N&ecirc;(&ntilde; )] &ograve; − &ecirc;(&ntilde; ) K==
&Ccedil;&ntilde;
=
=
=
q&Uuml;&Eacute;= &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;= &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
296
CHAPTER 10. DIFFERENTIAL EQUATIONS
1169. b&ntilde;~&Aring;&iacute;=~&aring;&Ccedil;=k&ccedil;&aring;&Eacute;&ntilde;~&Aring;&iacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=
q&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;==
j(&ntilde; I &oacute; )&Ccedil;&ntilde; + k(&ntilde; I &oacute; )&Ccedil;&oacute; = M ==
&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=&Eacute;&ntilde;~&Aring;&iacute;=&aacute;&Ntilde;==
∂j ∂k
=
I==
∂&oacute; ∂&ntilde;
~&aring;&Ccedil;=&aring;&ccedil;&aring;&Eacute;&ntilde;~&Aring;&iacute;=&ccedil;&iacute;&Uuml;&Eacute;&ecirc;&iuml;&aacute;&euml;&Eacute;K=
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
∫ j(&ntilde; I &oacute; )&Ccedil;&ntilde; + ∫ k(&ntilde; I &oacute; )&Ccedil;&oacute; = ` K==
=
1170. o~&Ccedil;&aacute;&ccedil;~&Aring;&iacute;&aacute;&icirc;&Eacute;=a&Eacute;&Aring;~&oacute;=
&Ccedil;&oacute;
= − &acirc;&oacute; I==
&Ccedil;&iacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &oacute; (&iacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&atilde;&ccedil;&igrave;&aring;&iacute;=&ccedil;&Ntilde;=&ecirc;~&Ccedil;&aacute;&ccedil;~&Aring;&iacute;&aacute;&icirc;&Eacute;=&Eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=~&iacute;=&iacute;&aacute;&atilde;&Eacute;=&iacute;I=&acirc;=
&aacute;&euml;=&iacute;&Uuml;&Eacute;=&ecirc;~&iacute;&Eacute;=&ccedil;&Ntilde;=&Ccedil;&Eacute;&Aring;~&oacute;K==
=
q&Uuml;&Eacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
&oacute; (&iacute; ) = &oacute; M &Eacute; − &acirc;&iacute; I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &oacute; M = &oacute; (M) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&aacute;&iacute;&aacute;~&auml;=~&atilde;&ccedil;&igrave;&aring;&iacute;K=
=
1171. k&Eacute;&iuml;&iacute;&ccedil;&aring;∞&euml;=i~&iuml;=&ccedil;&Ntilde;=`&ccedil;&ccedil;&auml;&aacute;&aring;&Ouml;=
&Ccedil;q
= − &acirc; (q − p ) I==
&Ccedil;&iacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= q(&iacute; ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;=&ccedil;&Ntilde;=~&aring;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;=~&iacute;=&iacute;&aacute;&atilde;&Eacute;=&iacute;I=p=&aacute;&euml;=&iacute;&Uuml;&Eacute;=
&iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &euml;&igrave;&ecirc;&ecirc;&ccedil;&igrave;&aring;&Ccedil;&aacute;&aring;&Ouml;= &Eacute;&aring;&icirc;&aacute;&ecirc;&ccedil;&aring;&atilde;&Eacute;&aring;&iacute;I= &acirc;= &aacute;&euml;= ~= &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K==
=
q&Uuml;&Eacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;=
q(&iacute; ) = p + (qM − p) &Eacute; −&acirc;&iacute; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= qM = q(M) = &aacute;&euml;= &iacute;&Uuml;&Eacute;= &aacute;&aring;&aacute;&iacute;&aacute;~&auml;= &iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;= ~&iacute;=
&iacute;&aacute;&atilde;&Eacute;= &iacute; = M K==
=
=
297
CHAPTER 10. DIFFERENTIAL EQUATIONS
1172. m&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring;=a&oacute;&aring;~&atilde;&aacute;&Aring;&euml;=Ei&ccedil;&Ouml;&aacute;&euml;&iacute;&aacute;&Aring;=j&ccedil;&Ccedil;&Eacute;&auml;F=
&Ccedil;m
m

= &acirc;m N −  I==
&Ccedil;&iacute;
 j
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= m(&iacute; ) =&aacute;&euml;=&eacute;&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring;=~&iacute;=&iacute;&aacute;&atilde;&Eacute;=&iacute;I=&acirc;=&aacute;&euml;=~=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;I=
j=&aacute;&euml;=~=&auml;&aacute;&atilde;&aacute;&iacute;&aacute;&aring;&Ouml;=&euml;&aacute;&ograve;&Eacute;=&Ntilde;&ccedil;&ecirc;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&eacute;&igrave;&auml;~&iacute;&aacute;&ccedil;&aring;K==
=
q&Uuml;&Eacute;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
jmM
m(&iacute; ) =
I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= mM = m(M) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&aacute;&iacute;&aacute;~&auml;=&eacute;&ccedil;&eacute;&igrave;mM + (j − mM )&Eacute; − &acirc;&iacute;
&auml;~&iacute;&aacute;&ccedil;&aring;=~&iacute;=&iacute;&aacute;&atilde;&Eacute;= &iacute; = M K===
=
=
=
10.2 Second Order Ordinary Differential
Equations
=
1173. e&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;=i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&iuml;&aacute;&iacute;&Uuml;=`&ccedil;&aring;&euml;&iacute;~&aring;&iacute;=`&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;==
&oacute;′′ + &eacute;&oacute;′ + &egrave;&oacute; = M K==
q&Uuml;&Eacute;=&Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
λO + &eacute;λ + &egrave; = M K==
=
f&Ntilde;= λN = ~&aring;&Ccedil;= λ O = ~&ecirc;&Eacute;= &Ccedil;&aacute;&euml;&iacute;&aacute;&aring;&Aring;&iacute;= &ecirc;&Eacute;~&auml;= &ecirc;&ccedil;&ccedil;&iacute;&euml;= &ccedil;&Ntilde;= &iacute;&Uuml;&Eacute;= &Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;=
&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
&oacute; = `N&Eacute; λN&ntilde; + ` O&Eacute; λ O &ntilde; I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
`N =~&aring;&Ccedil;= ` O =~&ecirc;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;K==
==
&eacute;
f&Ntilde;= λN = λ O = − I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
O
&eacute;
− &ntilde;
&oacute; = (`N + ` O &ntilde; )&Eacute; O K==
=
f&Ntilde;= λN =~&aring;&Ccedil;= λ O =~&ecirc;&Eacute;=&Aring;&ccedil;&atilde;&eacute;&auml;&Eacute;&ntilde;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=
298
CHAPTER 10. DIFFERENTIAL EQUATIONS
λN = α + β &aacute; I= λ O = α − β &aacute; I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
Q&egrave; − &eacute; O
&eacute;
α = − I= β =
I==
O
O
&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
&oacute; = &Eacute; α&ntilde; (`N &Aring;&ccedil;&euml; β &ntilde; + ` O &euml;&aacute;&aring; β &ntilde; ) K==
=
1174. f&aring;&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;=i&aacute;&aring;&Eacute;~&ecirc;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&iuml;&aacute;&iacute;&Uuml;=`&ccedil;&aring;&euml;&iacute;~&aring;&iacute;=======================
`&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;==
&oacute;′′ + &eacute;&oacute;′ + &egrave;&oacute; = &Ntilde; (&ntilde; ) K==
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
&oacute; = &oacute; &eacute; + &oacute; &Uuml; I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&oacute; &eacute; =&aacute;&euml;==~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;==&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;==&iacute;&Uuml;&Eacute;=&aacute;&aring;&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
~&aring;&Ccedil;= &oacute; &Uuml; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=~&euml;&euml;&ccedil;&Aring;&aacute;~&iacute;&Eacute;&Ccedil;=&Uuml;&ccedil;&atilde;&ccedil;&Ouml;&Eacute;&aring;&Eacute;&ccedil;&igrave;&euml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=E&euml;&Eacute;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&Eacute;&icirc;&aacute;&ccedil;&igrave;&euml;=&iacute;&ccedil;&eacute;&aacute;&Aring;=NNTPFK=
=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&ecirc;&aacute;&Ouml;&Uuml;&iacute;=&euml;&aacute;&Ccedil;&Eacute;=&Uuml;~&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&ccedil;&ecirc;&atilde;==
&Ntilde; (&ntilde; ) = &Eacute; α&ntilde; (mN (&ntilde; )&Aring;&ccedil;&euml; β &ntilde; + mN (&ntilde; )&euml;&aacute;&aring; β&ntilde; ) I==
&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;==&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;= &oacute; &eacute; =&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
&oacute; &eacute; = &ntilde; &acirc; &Eacute; α&ntilde; (oN (&ntilde; )&Aring;&ccedil;&euml; β&ntilde; + o O (&ntilde; )&euml;&aacute;&aring; β &ntilde; ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &iacute;&Uuml;&Eacute;= &eacute;&ccedil;&auml;&oacute;&aring;&ccedil;&atilde;&aacute;~&auml;&euml;= oN (&ntilde; ) = ~&aring;&Ccedil;= o O (&ntilde; ) = &Uuml;~&icirc;&Eacute;= &iacute;&ccedil;= &Auml;&Eacute;= &Ntilde;&ccedil;&igrave;&aring;&Ccedil;=
&Auml;&oacute;=&igrave;&euml;&aacute;&aring;&Ouml;=&iacute;&Uuml;&Eacute;=&atilde;&Eacute;&iacute;&Uuml;&ccedil;&Ccedil;=&ccedil;&Ntilde;=&igrave;&aring;&Ccedil;&Eacute;&iacute;&Eacute;&ecirc;&atilde;&aacute;&aring;&Eacute;&Ccedil;=&Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;K==
• f&Ntilde;= α + β &aacute; =&aacute;&euml;=&aring;&ccedil;&iacute;=~=&ecirc;&ccedil;&ccedil;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Aring;&Uuml;~&ecirc;~&Aring;&iacute;&Eacute;&ecirc;&aacute;&euml;&iacute;&aacute;&Aring;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;I=&iacute;&Uuml;&Eacute;&aring;=
&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&iuml;&Eacute;&ecirc;= &acirc; = M I=
• f&Ntilde;= α + β &aacute; =&aacute;&euml;=~=&euml;&aacute;&atilde;&eacute;&auml;&Eacute;=&ecirc;&ccedil;&ccedil;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= &acirc; = N I=
• f&Ntilde;= α + β &aacute; =&aacute;&euml;=~=&Ccedil;&ccedil;&igrave;&Auml;&auml;&Eacute;=&ecirc;&ccedil;&ccedil;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= &acirc; = O K==
=
1175. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&iuml;&aacute;&iacute;&Uuml;=&oacute;=j&aacute;&euml;&euml;&aacute;&aring;&Ouml;=
&oacute;′′ = &Ntilde; (&ntilde; I &oacute;′) K==
p&Eacute;&iacute;= &igrave; = &oacute;′ K=q&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&aring;&Eacute;&iuml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=&euml;~&iacute;&aacute;&euml;&Ntilde;&aacute;&Eacute;&Ccedil;=&Auml;&oacute;=&icirc;=&aacute;&euml;==
&igrave;′ = &Ntilde; (&ntilde; I &igrave; ) I==
&iuml;&Uuml;&aacute;&Aring;&Uuml;=&aacute;&euml;=~=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K=
299
CHAPTER 10. DIFFERENTIAL EQUATIONS
1176. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&iuml;&aacute;&iacute;&Uuml;=&ntilde;=j&aacute;&euml;&euml;&aacute;&aring;&Ouml;=
&oacute;′′ = &Ntilde; (&oacute; I &oacute;′) K==
p&Eacute;&iacute;= &igrave; = &oacute;′ K=p&aacute;&aring;&Aring;&Eacute;==
&Ccedil;&igrave; &Ccedil;&igrave; &Ccedil;&oacute;
&Ccedil;&igrave;
&oacute;′′ =
=
= &igrave; I==
&Ccedil;&ntilde; &Ccedil;&oacute; &Ccedil;&ntilde;
&Ccedil;&oacute;
&iuml;&Eacute;=&Uuml;~&icirc;&Eacute;=
&Ccedil;&igrave;
&igrave;
= &Ntilde; (&oacute; I &igrave; ) I==
&Ccedil;&oacute;
&iuml;&Uuml;&aacute;&Aring;&Uuml;=&aacute;&euml;=~=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&ccedil;&ecirc;&Ccedil;&Eacute;&ecirc;=&Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;K==
=
1177. c&ecirc;&Eacute;&Eacute;=r&aring;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;=s&aacute;&Auml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;=
q&Uuml;&Eacute;=&atilde;&ccedil;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=~=j~&euml;&euml;=&ccedil;&aring;=~=p&eacute;&ecirc;&aacute;&aring;&Ouml;=&aacute;&euml;=&Ccedil;&Eacute;&euml;&Aring;&ecirc;&aacute;&Auml;&Eacute;&Ccedil;=&Auml;&oacute;=&iacute;&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;==
&atilde;&amp;&oacute;&amp; + &acirc;&oacute; = M I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&atilde;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&atilde;~&euml;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&ccedil;&Auml;&agrave;&Eacute;&Aring;&iacute;I=
&acirc;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&iacute;&aacute;&Ntilde;&Ntilde;&aring;&Eacute;&euml;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&eacute;&ecirc;&aacute;&aring;&Ouml;I=
&oacute;=&aacute;&euml;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&atilde;~&euml;&euml;=&Ntilde;&ecirc;&ccedil;&atilde;=&Eacute;&egrave;&igrave;&aacute;&auml;&aacute;&Auml;&ecirc;&aacute;&igrave;&atilde;K=
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
&oacute; = ^ &Aring;&ccedil;&euml;(ωM &iacute; − δ ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
^=&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&atilde;&eacute;&auml;&aacute;&iacute;&igrave;&Ccedil;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;I=
Oπ
I=
ωM =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&igrave;&aring;&Ccedil;~&atilde;&Eacute;&aring;&iacute;~&auml;=&Ntilde;&ecirc;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&oacute;I=&iacute;&Uuml;&Eacute;=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;=&aacute;&euml;= q =
ωM
δ =&aacute;&euml;=&eacute;&Uuml;~&euml;&Eacute;=~&aring;&Ouml;&auml;&Eacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;K=
q&Uuml;&aacute;&euml;=&aacute;&euml;=~&aring;=&Eacute;&ntilde;~&atilde;&eacute;&auml;&Eacute;=&ccedil;&Ntilde;=&euml;&aacute;&atilde;&eacute;&auml;&Eacute;=&Uuml;~&ecirc;&atilde;&ccedil;&aring;&aacute;&Aring;=&atilde;&ccedil;&iacute;&aacute;&ccedil;&aring;K==
=
1178. c&ecirc;&Eacute;&Eacute;=a~&atilde;&eacute;&Eacute;&Ccedil;=s&aacute;&Auml;&ecirc;~&iacute;&aacute;&ccedil;&aring;&euml;=
&atilde;&amp;&oacute;&amp; + γ&oacute;&amp; + &acirc;&oacute; = M I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
γ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ccedil;~&atilde;&eacute;&aacute;&aring;&Ouml;=&Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;K==
q&Uuml;&Eacute;&ecirc;&Eacute;=~&ecirc;&Eacute;=P=&Aring;~&euml;&Eacute;&euml;=&Ntilde;&ccedil;&ecirc;=&iacute;&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;W=
=
300
CHAPTER 10. DIFFERENTIAL EQUATIONS
`~&euml;&Eacute;=NK γ O &gt; Q&acirc;&atilde; =E&ccedil;&icirc;&Eacute;&ecirc;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F=
&oacute; (&iacute; ) = ^&Eacute; λN&iacute; + _&Eacute; λ O&iacute; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
λN =
− γ − γ O − Q &acirc;&atilde;
− γ + γ O − Q &acirc;&atilde;
I= λ O =
K==
O&atilde;
O&atilde;
=
`~&euml;&Eacute;=OK= γ O = Q&acirc;&atilde; E&Aring;&ecirc;&aacute;&iacute;&aacute;&Aring;~&auml;&auml;&oacute;=&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F=
&oacute; (&iacute; ) = (^ + _&iacute; )&Eacute; λ&iacute; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
γ
λ=−
K=
O&atilde;
=
`~&euml;&Eacute;=PK= γ O &lt; Q&acirc;&atilde; =E&igrave;&aring;&Ccedil;&Eacute;&ecirc;&Ccedil;~&atilde;&eacute;&Eacute;&Ccedil;F==
&oacute; (&iacute; ) = &Eacute;
−
γ
&iacute;
O&atilde;
^ &Aring;&ccedil;&euml;(ω&iacute; − δ ) I=&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
ω = Q&acirc;&atilde; − γ O K==
=
1179. p&aacute;&atilde;&eacute;&auml;&Eacute;=m&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde;=
&Ccedil; Oθ &Ouml;
+ θ = M I=
&Ccedil;&iacute; O i
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= θ = &aacute;&euml;= &iacute;&Uuml;&Eacute;= ~&aring;&Ouml;&igrave;&auml;~&ecirc;= &Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;I= i= &aacute;&euml;= &iacute;&Uuml;&Eacute;= &eacute;&Eacute;&aring;&Ccedil;&igrave;&auml;&igrave;&atilde;=
&auml;&Eacute;&aring;&Ouml;&iacute;&Uuml;I=&Ouml;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=~&Aring;&Aring;&Eacute;&auml;&Eacute;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&Ouml;&ecirc;~&icirc;&aacute;&iacute;&oacute;K=
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&Ntilde;&ccedil;&ecirc;=&euml;&atilde;~&auml;&auml;=~&aring;&Ouml;&auml;&Eacute;&euml;= θ =&aacute;&euml;==
&Ouml;
i
θ(&iacute; ) = θ&atilde;~&ntilde; &euml;&aacute;&aring;
&iacute; I=&iacute;&Uuml;&Eacute;=&eacute;&Eacute;&ecirc;&aacute;&ccedil;&Ccedil;=&aacute;&euml;= q = Oπ
K==
i
&Ouml;
=
1180. oi`=`&aacute;&ecirc;&Aring;&igrave;&aacute;&iacute;=
&Ccedil; Of
&Ccedil;f N
i O + o + f = s′(&iacute; ) = ωb M &Aring;&ccedil;&euml;(ω&iacute; ) I=
&Ccedil;&iacute;
&Ccedil;&iacute; `
301
CHAPTER 10. DIFFERENTIAL EQUATIONS
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=f=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&iacute;=&aacute;&aring;=~&aring;=oi`=&Aring;&aacute;&ecirc;&Aring;&igrave;&aacute;&iacute;=&iuml;&aacute;&iacute;&Uuml;=~&aring;=~&Aring;=&icirc;&ccedil;&auml;&iacute;~&Ouml;&Eacute;=
&euml;&ccedil;&igrave;&ecirc;&Aring;&Eacute;= s(&iacute; ) = b M &euml;&aacute;&aring;(ω&iacute; ) K===
=
q&Uuml;&Eacute;=&Ouml;&Eacute;&aring;&Eacute;&ecirc;~&auml;=&euml;&ccedil;&auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&aacute;&euml;==
f(&iacute; ) = `N&Eacute; &ecirc;N&iacute; + ` O&Eacute; &ecirc;O&iacute; + ^ &euml;&aacute;&aring;(ω&iacute; − ϕ) I=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
Qi
− o &plusmn; oO −
` I==
&ecirc; NI O =
Oi
ωb M
^=
I==
O
N
 O

O O
 iω −  + o ω
`


N 
 iω
ϕ = ~&ecirc;&Aring;&iacute;~&aring;
−
 I==
 o o`ω 
`N I= ` O =~&ecirc;&Eacute;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;&euml;=&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&aacute;&aring;&Ouml;=&ccedil;&aring;=&aacute;&aring;&aacute;&iacute;&aacute;~&auml;=&Aring;&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;&euml;K=
=
=
=
10.3. Some Partial Differential Equations
=
1181. q&Uuml;&Eacute;=i~&eacute;&auml;~&Aring;&Eacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
∂ O&igrave; ∂ O&igrave;
+
= M=
∂&ntilde; O ∂&oacute; O
~&eacute;&eacute;&auml;&aacute;&Eacute;&euml;=&iacute;&ccedil;=&eacute;&ccedil;&iacute;&Eacute;&aring;&iacute;&aacute;~&auml;=&Eacute;&aring;&Eacute;&ecirc;&Ouml;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &igrave;(&ntilde; I &oacute; ) ==&Ntilde;&ccedil;&ecirc;==~==&Aring;&ccedil;&aring;&euml;&Eacute;&ecirc;&icirc;~&iacute;&aacute;&icirc;&Eacute;= &Ntilde;&ccedil;&ecirc;&Aring;&Eacute;= &Ntilde;&aacute;&Eacute;&auml;&Ccedil;= &aacute;&aring;= &iacute;&Uuml;&Eacute;= &ntilde;&oacute;-&eacute;&auml;~&aring;&Eacute;K= m~&ecirc;&iacute;&aacute;~&auml;= &Ccedil;&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&auml;= &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&aacute;&euml;=&iacute;&oacute;&eacute;&Eacute;=~&ecirc;&Eacute;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=&Eacute;&auml;&auml;&aacute;&eacute;&iacute;&aacute;&Aring;K==
=
1182. q&Uuml;&Eacute;=e&Eacute;~&iacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
∂ O &igrave; ∂ O &igrave; ∂&igrave;
+
=
=
∂&ntilde; O ∂&oacute; O ∂&iacute;
302
CHAPTER 10. DIFFERENTIAL EQUATIONS
~&eacute;&eacute;&auml;&aacute;&Eacute;&euml;= &iacute;&ccedil;= &iacute;&Uuml;&Eacute;= &iacute;&Eacute;&atilde;&eacute;&Eacute;&ecirc;~&iacute;&igrave;&ecirc;&Eacute;= &Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;= &igrave;(&ntilde; I &oacute; ) = &aacute;&aring;= &iacute;&Uuml;&Eacute;= &ntilde;&oacute;&eacute;&auml;~&aring;&Eacute;=&iuml;&Uuml;&Eacute;&aring;=&Uuml;&Eacute;~&iacute;=&aacute;&euml;=~&auml;&auml;&ccedil;&iuml;&Eacute;&Ccedil;=&iacute;&ccedil;=&Ntilde;&auml;&ccedil;&iuml;=&Ntilde;&ecirc;&ccedil;&atilde;=&iuml;~&ecirc;&atilde;=~&ecirc;&Eacute;~&euml;=&iacute;&ccedil;=&Aring;&ccedil;&ccedil;&auml;=
&ccedil;&aring;&Eacute;&euml;K=q&Uuml;&Eacute;=&Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;=&ccedil;&Ntilde;=&iacute;&Uuml;&aacute;&euml;=&iacute;&oacute;&eacute;&Eacute;=~&ecirc;&Eacute;=&Aring;~&auml;&auml;&Eacute;&Ccedil;=&eacute;~&ecirc;~&Auml;&ccedil;&auml;&aacute;&Aring;K==
=
1183. q&Uuml;&Eacute;=t~&icirc;&Eacute;=b&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;=
∂ O&igrave; ∂ O&igrave; ∂ O&igrave;
+
=
=
∂&ntilde; O ∂&oacute; O ∂&iacute; O
~&eacute;&eacute;&auml;&aacute;&Eacute;&euml;=&iacute;&ccedil;=&iacute;&Uuml;&Eacute;=&Ccedil;&aacute;&euml;&eacute;&auml;~&Aring;&Eacute;&atilde;&Eacute;&aring;&iacute;= &igrave;(&ntilde; I &oacute; ) =&ccedil;&Ntilde;=&icirc;&aacute;&Auml;&ecirc;~&iacute;&aacute;&aring;&Ouml;=&atilde;&Eacute;&atilde;&Auml;&ecirc;~&aring;&Eacute;&euml;=
~&aring;&Ccedil;= &ccedil;&iacute;&Uuml;&Eacute;&ecirc;= &iuml;~&icirc;&Eacute;= &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;K= q&Uuml;&Eacute;= &Eacute;&egrave;&igrave;~&iacute;&aacute;&ccedil;&aring;&euml;= &ccedil;&Ntilde;= &iacute;&Uuml;&aacute;&euml;= &iacute;&oacute;&eacute;&Eacute;= ~&ecirc;&Eacute;=
&Aring;~&auml;&auml;&Eacute;&Ccedil;=&Uuml;&oacute;&eacute;&Eacute;&ecirc;&Auml;&ccedil;&auml;&aacute;&Aring;K==
=
=
303
Chapter 11
Series
=
=
=
=
11.1 Arithmetic Series
=
f&aring;&aacute;&iacute;&aacute;~&auml;=&iacute;&Eacute;&ecirc;&atilde;W= ~N =
k&iacute;&Uuml;=&iacute;&Eacute;&ecirc;&atilde;W= ~ &aring; =
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&Aring;&Eacute;=&Auml;&Eacute;&iacute;&iuml;&Eacute;&Eacute;&aring;=&euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&aacute;&icirc;&Eacute;=&iacute;&Eacute;&ecirc;&atilde;&euml;W=&Ccedil;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Eacute;&ecirc;&atilde;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W=&aring;=
p&igrave;&atilde;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&aring;=&iacute;&Eacute;&ecirc;&atilde;&euml;W= p&aring; =
=
=
1184. ~ &aring; = ~ &aring; −N + &Ccedil; = ~ &aring; −O + O&Ccedil; = K = ~N + (&aring; − N)&Ccedil; =
=
1185. ~N + ~ &aring; = ~ O + ~ &aring;−N = K = ~ &aacute; + ~ &aring; +N−&aacute; =
=
~ +~
1186. ~ &aacute; = &aacute; −N &aacute; +N =
O
=
~ +~
O~ + (&aring; − N)&Ccedil;
1187. p&aring; = N &aring; ⋅ &aring; = N
⋅&aring;=
O
O
=
=
=
=
=
304
CHAPTER 11. SERIES
11.2 Geometric Series
=
f&aring;&aacute;&iacute;&aacute;~&auml;=&iacute;&Eacute;&ecirc;&atilde;W= ~N =
k&iacute;&Uuml;=&iacute;&Eacute;&ecirc;&atilde;W= ~ &aring; =
`&ccedil;&atilde;&atilde;&ccedil;&aring;=&ecirc;~&iacute;&aacute;&ccedil;W=&egrave;=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Eacute;&ecirc;&atilde;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W=&aring;=
p&igrave;&atilde;=&ccedil;&Ntilde;=&iacute;&Uuml;&Eacute;=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&aring;=&iacute;&Eacute;&ecirc;&atilde;&euml;W= p&aring; =
p&igrave;&atilde;=&iacute;&ccedil;=&aacute;&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&oacute;W=p=
=
=
1188. ~ &aring; = &egrave;~ &aring; −N = ~N&egrave; &aring; −N =
=
1189. ~N ⋅ ~ &aring; = ~ O ⋅ ~ &aring; −N = K = ~ &aacute; ⋅ ~ &aring; +N−&aacute; =
=
1190. ~ &aacute; = ~ &aacute; −N ⋅ ~ &aacute; +N =
=
~ &aring; &egrave; − ~N ~N (&egrave; &aring; − N)
1191. p&aring; =
=
=
&egrave; −N
&egrave; −N
=
~
1192. p = &auml;&aacute;&atilde; p&aring; = N =
&aring; →∞
N− &egrave;
c&ccedil;&ecirc;= &egrave; &lt; N I=&iacute;&Uuml;&Eacute;=&euml;&igrave;&atilde;=p=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;=~&euml;= &aring; → ∞ K=
=
=
=
11.3 Some Finite Series
=
k&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&Eacute;&ecirc;&atilde;&euml;=&aacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W=&aring;=
=
=
305
CHAPTER 11. SERIES
1193. N + O + P + K + &aring; =
&aring;(&aring; + N)
=
O
=
1194. O + Q + S + K + O&aring; = &aring;(&aring; + N) =
=
1195. N + P + R + K + (O&aring; − N) = &aring; O =
=
1196. &acirc; + (&acirc; + N) + (&acirc; + O) + K + (&acirc; + &aring; − N) =
=
1197. NO + OO + PO + K + &aring; O =
&aring;(O&acirc; + &aring; − N)
=
O
&aring;(&aring; + N)(O&aring; + N)
=
S
=
 &aring;(&aring; + N) 
1198. NP + OP + PP + K + &aring;P = 
=
 O 
=
&aring;(Q&aring; O − N)
O
O
O
O
1199. N + P + R + K + (O&aring; − N) =
=
P
=
P
1200. NP + PP + RP + K + (O&aring; − N) = &aring; O (O&aring; O − N) =
=
N N N
N
1201. N + + + + K + &aring; + K = O =
O Q U
O
=
N
N
N
N
1202.
+
+
+K+
+K = N=
N⋅ O O ⋅ P P ⋅ Q
&aring;(&aring; + N)
=
N N N
N
1203. N + + + + K +
+K = &Eacute; =
(&aring; − N)&gt;
N&gt; O&gt; P&gt;
=
=
=
O
306
CHAPTER 11. SERIES
11.4 Infinite Series
=
p&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute;W= {~ &aring; }=
c&aacute;&ecirc;&euml;&iacute;=&iacute;&Eacute;&ecirc;&atilde;W= ~N =
k&iacute;&Uuml;=&iacute;&Eacute;&ecirc;&atilde;W= ~ &aring; =
=
=
1204. f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
∑~
&aring; =N
&aring;
= ~N + ~ O + K + ~ &aring; + K =
=
1205. k&iacute;&Uuml;=m~&ecirc;&iacute;&aacute;~&auml;=p&igrave;&atilde;=
&aring;
p&aring; = ∑ ~ &aring; = ~N + ~ O + K + ~ &aring; =
&aring; =N
=
1206. `&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=f&aring;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
∑~
&aring; =N
&aring;
= i I=&aacute;&Ntilde;= &auml;&aacute;&atilde; p&aring; = i =
&aring; →∞
=
1207. k&iacute;&Uuml;=q&Eacute;&ecirc;&atilde;=q&Eacute;&euml;&iacute;=
∞
•
f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;= ∑ ~ &aring; &aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;I=&iacute;&Uuml;&Eacute;&aring;= &auml;&aacute;&atilde; ~ &aring; = M K==
&aring;→∞
&aring; =N
•
f&Ntilde;= &auml;&aacute;&atilde; ~ &aring; ≠ M I=&iacute;&Uuml;&Eacute;&aring;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=&aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
&aring;→∞
=
=
=
11.5 Properties of Convergent Series
=
∞
∞
&aring; =N
&aring; =N
`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;W= ∑ ~ &aring; = ^ I= ∑ &Auml;&aring; = _ =
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&Aring;=
307
CHAPTER 11. SERIES
∞
∞
∞
&aring; =N
&aring; =N
&aring; =N
1208. ∑ (~ &aring; + &Auml; &aring; ) = ∑ ~ &aring; + ∑ &Auml; &aring; = ^ + _ =
=
∞
∑ &Aring;~
1209.
&aring; =N
∞
&aring;
= &Aring;∑ ~ &aring; = &Aring;^ K==
&aring; =N
=
=
=
11.6 Convergence Tests
=
1210. q&Uuml;&Eacute;=`&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring;=q&Eacute;&euml;&iacute;=
∞
∞
&aring; =N
∞
&aring; =N
i&Eacute;&iacute;= ∑ ~ &aring; =~&aring;&Ccedil;= ∑ &Auml;&aring; =&Auml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;= M &lt; ~ &aring; ≤ &Auml;&aring; =&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&aring;K==
•
•
∞
f&Ntilde;= ∑ &Auml;&aring; &aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; &aacute;&euml;=~&auml;&euml;&ccedil;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K==
&aring; =N
∞
∞
&aring; =N
&aring; =N
&aring; =N
f&Ntilde;= ∑ ~ &aring; &aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&iacute;&Uuml;&Eacute;&aring;= ∑ &Auml;&aring; &aacute;&euml;=~&auml;&euml;&ccedil;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
1211. q&Uuml;&Eacute;=i&aacute;&atilde;&aacute;&iacute;=`&ccedil;&atilde;&eacute;~&ecirc;&aacute;&euml;&ccedil;&aring;=q&Eacute;&euml;&iacute;=
∞
∞
&aring; =N
&aring; =N
i&Eacute;&iacute;= ∑ ~ &aring; =~&aring;&Ccedil;= ∑ &Auml;&aring; =&Auml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;= ~ &aring; =~&aring;&Ccedil;= &Auml;&aring; =~&ecirc;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&aring;K==
∞
∞
~&aring;
• f&Ntilde;= M &lt; &auml;&aacute;&atilde;
&lt; ∞ = &iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; = ~&aring;&Ccedil;= ∑ &Auml;&aring; ~&ecirc;&Eacute;= &Eacute;&aacute;&iacute;&Uuml;&Eacute;&ecirc;= &Auml;&ccedil;&iacute;&Uuml;=
&aring; →∞ &Auml;
&aring; =N
&aring; =N
&aring;
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&ccedil;&ecirc;=&Auml;&ccedil;&iacute;&Uuml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
∞
∞
~&aring;
• f&Ntilde;= &auml;&aacute;&atilde;
= M =&iacute;&Uuml;&Eacute;&aring;= ∑ &Auml;&aring; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&aacute;&atilde;&eacute;&auml;&aacute;&Eacute;&euml;=&iacute;&Uuml;~&iacute;= ∑ ~ &aring; =&aacute;&euml;=
&aring; →∞ &Auml;
&aring; =N
&aring; =N
&aring;
~&auml;&euml;&ccedil;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
308
CHAPTER 11. SERIES
∞
∞
~&aring;
= ∞ =&iacute;&Uuml;&Eacute;&aring;= ∑ &Auml;&aring; &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&aacute;&atilde;&eacute;&auml;&aacute;&Eacute;&euml;=&iacute;&Uuml;~&iacute;= ∑ ~ &aring; =&aacute;&euml;=
&aring; →∞ &Auml;
&aring; =N
&aring; =N
&aring;
~&auml;&euml;&ccedil;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
f&Ntilde;= &auml;&aacute;&atilde;
•
=
1212. &eacute;-&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
N
=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;=&Ntilde;&ccedil;&ecirc;= &eacute; &gt; N =~&aring;&Ccedil;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;=&Ntilde;&ccedil;&ecirc;=
&eacute;
&aring; =N &aring;
M &lt; &eacute; ≤ N K=
=
1213. q&Uuml;&Eacute;=f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;=q&Eacute;&euml;&iacute;=
i&Eacute;&iacute;= &Ntilde; (&ntilde; ) = &Auml;&Eacute;= ~= &Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;= &iuml;&Uuml;&aacute;&Aring;&Uuml;= &aacute;&euml;= &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I= &eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;I= ~&aring;&Ccedil;=
&Ccedil;&Eacute;&Aring;&ecirc;&Eacute;~&euml;&aacute;&aring;&Ouml;=&Ntilde;&ccedil;&ecirc;=~&auml;&auml;= &ntilde; ≥ N K=q&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;==
&eacute;-&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;= ∑
∞
∑ &Ntilde; (&aring;) = &Ntilde; (N) + &Ntilde; (O) + &Ntilde; (P) + K + &Ntilde; (&aring;) + K =
&aring; =N
∞
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;=&aacute;&Ntilde;= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;I=~&aring;&Ccedil;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&euml;=&aacute;&Ntilde;=
N
&aring;
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; → ∞ =~&euml;= &aring; → ∞ K=
N
=
1214. q&Uuml;&Eacute;=o~&iacute;&aacute;&ccedil;=q&Eacute;&euml;&iacute;=
∞
i&Eacute;&iacute;= ∑ ~ &aring; =&Auml;&Eacute;=~=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=&iuml;&aacute;&iacute;&Uuml;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&iacute;&Eacute;&ecirc;&atilde;&euml;K=
&aring; =N
∞
~ &aring; +N
&lt; N =&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; &aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
&aring; →∞ ~
&aring; =N
&aring;
•
f&Ntilde;= &auml;&aacute;&atilde;
•
f&Ntilde;= &auml;&aacute;&atilde;
•
∞
~ &aring; +N
&gt; N =&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; =&aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
&aring; →∞ ~
&aring; =N
&aring;
∞
~
f&Ntilde;= &auml;&aacute;&atilde; &aring; +N = N = &iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; = &atilde;~&oacute;= &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;= &ccedil;&ecirc;= &Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;= ~&aring;&Ccedil;=
&aring; →∞ ~
&aring; =N
&aring;
&iacute;&Uuml;&Eacute;= &ecirc;~&iacute;&aacute;&ccedil;= &iacute;&Eacute;&euml;&iacute;= &aacute;&euml;= &aacute;&aring;&Aring;&ccedil;&aring;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute;X= &euml;&ccedil;&atilde;&Eacute;= &ccedil;&iacute;&Uuml;&Eacute;&ecirc;= &iacute;&Eacute;&euml;&iacute;&euml;= &atilde;&igrave;&euml;&iacute;= &Auml;&Eacute;=
&igrave;&euml;&Eacute;&Ccedil;K==
=
309
CHAPTER 11. SERIES
1215. q&Uuml;&Eacute;=o&ccedil;&ccedil;&iacute;=q&Eacute;&euml;&iacute;=
∞
i&Eacute;&iacute;= ∑ ~ &aring; =&Auml;&Eacute;=~=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=&iuml;&aacute;&iacute;&Uuml;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&iacute;&Eacute;&ecirc;&atilde;&euml;K=
&aring; =N
∞
•
•
•
f&Ntilde;= &auml;&aacute;&atilde; &aring; ~ &aring; &lt; N =&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; =&aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
&aring; →∞
&aring; =N
∞
f&Ntilde;= &auml;&aacute;&atilde; &aring; ~ &aring; &gt; N =&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; =&aacute;&euml;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
&aring; →∞
&aring; =N
∞
f&Ntilde;= &auml;&aacute;&atilde; &aring; ~ &aring; = N =&iacute;&Uuml;&Eacute;&aring;= ∑ ~ &aring; =&atilde;~&oacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;=&ccedil;&ecirc;=&Ccedil;&aacute;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;I=&Auml;&igrave;&iacute;=
&aring; →∞
&aring; =N
&aring;&ccedil;=&Aring;&ccedil;&aring;&Aring;&auml;&igrave;&euml;&aacute;&ccedil;&aring;=&Aring;~&aring;=&Auml;&Eacute;=&Ccedil;&ecirc;~&iuml;&aring;=&Ntilde;&ecirc;&ccedil;&atilde;=&iacute;&Uuml;&aacute;&euml;=&iacute;&Eacute;&euml;&iacute;K=
=
=
=
11.7 Alternating Series
=
1216. q&Uuml;&Eacute;=^&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=q&Eacute;&euml;&iacute;=Ei&Eacute;&aacute;&Auml;&aring;&aacute;&ograve;∞&euml;=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;F=
=
i&Eacute;&iacute;= {~ &aring; }=&Auml;&Eacute;=~=&euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=&euml;&igrave;&Aring;&Uuml;=&iacute;&Uuml;~&iacute;=
~ &aring;+N &lt; ~ &aring; =&Ntilde;&ccedil;&ecirc;=~&auml;&auml;=&aring;K=
&auml;&aacute;&atilde; ~ &aring; = M K==
&aring;→∞
∞
q&Uuml;&Eacute;&aring;= &iacute;&Uuml;&Eacute;= ~&auml;&iacute;&Eacute;&ecirc;&aring;~&iacute;&aacute;&aring;&Ouml;= &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∑ (− N) ~
&aring; =N
∞
&aring;
&aring;
= ~&aring;&Ccedil;=
∑ (− N)
&aring; =N
&aring; −N
~&aring; =
&Auml;&ccedil;&iacute;&Uuml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;K====
=
1217. ^&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=
∞
•
^= &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∑~
&aring; =N
&aring;
= &aacute;&euml;= ~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute;= &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;= &aacute;&Ntilde;= &iacute;&Uuml;&Eacute;= &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
∑~
&aring; =N
&aring;
=&aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K==
310
CHAPTER 11. SERIES
∞
f&Ntilde;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;= ∑ ~ &aring; &aacute;&euml;=~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&iacute;&Uuml;&Eacute;&aring;=&aacute;&iacute;=&aacute;&euml;=&Aring;&ccedil;&aring;-
•
&aring; =N
&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
1218. `&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=
∞
^= &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∑~
&aring; =N
&aring;
&aacute;&euml;= &Aring;&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml;&auml;&oacute;= &Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;= &aacute;&Ntilde;= &iacute;&Uuml;&Eacute;= &euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;= &aacute;&euml;=
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&Auml;&igrave;&iacute;=&aacute;&euml;=&aring;&ccedil;&iacute;=~&Auml;&euml;&ccedil;&auml;&igrave;&iacute;&Eacute;&auml;&oacute;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;K=
=
=
=
11.8 Power Series
=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&ntilde;I= &ntilde; M =
∞
∞
&aring; =M
&aring; =M
m&ccedil;&iuml;&Eacute;&ecirc;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W= ∑ ~ &aring; &ntilde; &aring; I= ∑ ~ &aring; (&ntilde; − &ntilde; M ) =
&aring;
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;W=o=
=
1219. m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=&aacute;&aring;=&ntilde;=
∞
∑~
&aring; =M
&aring;
&ntilde; &aring; = ~ M + ~N&ntilde; + ~ O &ntilde; O + K + ~ &aring; &ntilde; &aring; + K =
=
1220. m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=&aacute;&aring;= (&ntilde; − &ntilde; M ) =
∞
∑ ~ (&ntilde; − &ntilde; )
&aring; =M
&aring;
&aring;
M
= ~ M + ~ N (&ntilde; − &ntilde; M ) + ~ O ( &ntilde; − &ntilde; M ) + K + ~ &aring; (&ntilde; − &ntilde; M ) + K
O
=
1221. f&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&ccedil;&Ntilde;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;===
q&Uuml;&Eacute;=&euml;&Eacute;&iacute;=&ccedil;&Ntilde;=&iacute;&Uuml;&ccedil;&euml;&Eacute;=&icirc;~&auml;&igrave;&Eacute;&euml;=&ccedil;&Ntilde;=&ntilde;=&Ntilde;&ccedil;&ecirc;=&iuml;&Uuml;&aacute;&Aring;&Uuml;=&iacute;&Uuml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
∞
&Ntilde; (&ntilde; ) = ∑ ~ &aring; (&ntilde; − &ntilde; M ) =&aacute;&euml;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&iacute;=&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;==&iacute;&Uuml;&Eacute;==&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&ccedil;&Ntilde;=
&aring;
&aring; =M
&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;K=
311
&aring;
CHAPTER 11. SERIES
1222. o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=
f&Ntilde;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&ccedil;&Ntilde;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;=&aacute;&euml;== (&ntilde; M − oI &ntilde; M + o ) ==&Ntilde;&ccedil;&ecirc;==&euml;&ccedil;&atilde;&Eacute;=
o ≥ M I=&iacute;&Uuml;&Eacute;=o=&aacute;&euml;=&Aring;~&auml;&auml;&Eacute;&Ccedil;==&iacute;&Uuml;&Eacute;=&ecirc;~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=&Aring;&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;K==f&iacute;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=
~&euml;=
~
N
=&ccedil;&ecirc;= o = &auml;&aacute;&atilde; &aring; K==
o = &auml;&aacute;&atilde;
&aring; →∞ &aring; ~
&aring; →∞ ~
&aring; +N
&aring;
=
=
=
11.9 Differentiation and Integration of Power
Series
=
`&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &Ntilde; (&ntilde; ) =
∞
m&ccedil;&iuml;&Eacute;&ecirc;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;W= ∑ ~ &aring; &ntilde; &aring; =
&aring; =M
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
o~&Ccedil;&aacute;&igrave;&euml;=&ccedil;&Ntilde;=`&ccedil;&aring;&icirc;&Eacute;&ecirc;&Ouml;&Eacute;&aring;&Aring;&Eacute;W=o=
=
=
1223. a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
i&Eacute;&iacute;= &Ntilde; (&ntilde; ) = ∑ ~ &aring; &ntilde; &aring; = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K =&Ntilde;&ccedil;&ecirc;= &ntilde; &lt; o K==
&aring; =M
q&Uuml;&Eacute;&aring;I= = &Ntilde;&ccedil;&ecirc;= &ntilde; &lt; o I= &Ntilde; (&ntilde; ) = &aacute;&euml;= &Aring;&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;I= &iacute;&Uuml;&Eacute;= &Ccedil;&Eacute;&ecirc;&aacute;&icirc;~&iacute;&aacute;&icirc;&Eacute;= &Ntilde; ′(&ntilde; ) =
&Eacute;&ntilde;&aacute;&euml;&iacute;&euml;=~&aring;&Ccedil;=
&Ccedil;
&Ccedil;
&Ccedil;
&Ntilde; ′(&ntilde; ) = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K =
&Ccedil;&ntilde;
&Ccedil;&ntilde;
&Ccedil;&ntilde;
∞
= ~N + O~ O &ntilde; + P~ P &ntilde; O + K = ∑ &aring;~ &aring; &ntilde; &aring; −N K=
&aring; =N
=
=
=
312
CHAPTER 11. SERIES
1224. f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=m&ccedil;&iuml;&Eacute;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
i&Eacute;&iacute;= &Ntilde; (&ntilde; ) = ∑ ~ &aring; &ntilde; &aring; = ~ M + ~ N&ntilde; + ~ O &ntilde; O + K =&Ntilde;&ccedil;&ecirc;= &ntilde; &lt; o K==
&aring; =M
q&Uuml;&Eacute;&aring;I==&Ntilde;&ccedil;&ecirc;= &ntilde; &lt; o I=&iacute;&Uuml;&Eacute;=&aacute;&aring;&Ccedil;&Eacute;&Ntilde;&aacute;&aring;&aacute;&iacute;&Eacute;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&auml;= ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; &Eacute;&ntilde;&aacute;&euml;&iacute;&euml;=~&aring;&Ccedil;==
∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; = ∫ ~ &Ccedil;&ntilde; + ∫ ~ &ntilde;&Ccedil;&ntilde; + ∫ ~ &ntilde; &Ccedil;&ntilde; + K =
O
M
= ~ M &ntilde; + ~N
N
O
∞
&ntilde;O
&ntilde;P
&ntilde; &aring; +N
+ ~O + K = ∑ ~&aring;
+ ` K=
O
P
&aring; +N
&aring; =M
=
=
=
11.10 Taylor and Maclaurin Series
=
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
a&aacute;&Ntilde;&Ntilde;&Eacute;&ecirc;&Eacute;&aring;&iacute;&aacute;~&Auml;&auml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &Ntilde; (&ntilde; ) =
o&Eacute;&atilde;~&aacute;&aring;&Ccedil;&Eacute;&ecirc;=&iacute;&Eacute;&ecirc;&atilde;W= o &aring; =
=
=
1225. q~&oacute;&auml;&ccedil;&ecirc;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
∞
(&ntilde; − ~ )&aring; = &Ntilde; (~ ) + &Ntilde; ′(~ )(&ntilde; − ~ ) + &Ntilde; ′′(~ )(&ntilde; − ~ )O + K
&Ntilde; (&ntilde; ) = ∑ &Ntilde; (&aring; ) (~ )
&aring;&gt;
O&gt;
&aring;=M
==
+
&Ntilde; (&aring; ) (~ )(&ntilde; − ~ )
+ o &aring; K==
&aring;&gt;
&aring;
=
1226. q&Uuml;&Eacute;=o&Eacute;&atilde;~&aacute;&aring;&Ccedil;&Eacute;&ecirc;=^&Ntilde;&iacute;&Eacute;&ecirc;=&aring;HN=q&Eacute;&ecirc;&atilde;&euml;=&aacute;&euml;=&Ouml;&aacute;&icirc;&Eacute;&aring;=&Auml;&oacute;==
&aring; +N
&Ntilde; (&aring; +N) (ξ )(&ntilde; − ~ )
I=== ~ &lt; ξ &lt; &ntilde; K=
o&aring; =
(&aring; + N)&gt;
=
1227. j~&Aring;&auml;~&igrave;&ecirc;&aacute;&aring;=p&Eacute;&ecirc;&aacute;&Eacute;&euml;=
313
CHAPTER 11. SERIES
∞
&Ntilde; (&ntilde; ) = ∑ &Ntilde; (&aring; ) (M )
&aring; =M
&ntilde;&aring;
&Ntilde; ′′(M)&ntilde; O
&Ntilde; (&aring; ) (M )&ntilde; &aring;
= &Ntilde; (M) + &Ntilde; ′(M)&ntilde; +
+K+
+ o&aring;
&aring;&gt;
O&gt;
&aring;&gt;
=
=
=
=
11.11 Power Series Expansions for Some
Functions
=
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&ntilde;=
=
=
&ntilde;O &ntilde;P
&ntilde;&aring;
1228. &Eacute; &ntilde; = N + &ntilde; +
+
+K+
+ K=
O&gt; P&gt;
&aring;&gt;
=
O
P
(&ntilde; &auml;&aring; ~ )&aring; + K =
&ntilde; &auml;&aring; ~ (&ntilde; &auml;&aring; ~ ) (&ntilde; &auml;&aring; ~ )
1229. ~ &ntilde; = N +
+
+
+K+
N&gt;
O&gt;
P&gt;
&aring;&gt;
=
&aring;
(
&ntilde; O &ntilde;P &ntilde; Q
− N) &ntilde; &aring;+N
1230. &auml;&aring;(N + &ntilde; ) = &ntilde; − + − + K +
&plusmn; K I= − N &lt; &ntilde; ≤ N K=
O P
Q
&aring; +N
=


&ntilde;P &ntilde;R &ntilde;T
N+ &ntilde;
= O &ntilde; + + + + K I= &ntilde; &lt; N K=
1231. &auml;&aring;
N− &ntilde;
P
R
T


=
 &ntilde; − N N  &ntilde; − N P N  &ntilde; − N  R 
1232. &auml;&aring; &ntilde; = O
+ 
 K I= &ntilde; &gt; M K=
 + 
 &ntilde; + N P  &ntilde; + N  R  &ntilde; + N  
=
(− N)&aring; &ntilde; O&aring; &plusmn; K =
&ntilde;O &ntilde;Q &ntilde;S
1233. &Aring;&ccedil;&euml; &ntilde; = N − + − + K +
(O&aring; )&gt;
O&gt; Q&gt; S&gt;
=
314
CHAPTER 11. SERIES
(− N) &ntilde; O&aring;+N &plusmn; K =
&ntilde;P &ntilde;R &ntilde;T
1234. &euml;&aacute;&aring; &ntilde; = &ntilde; − + − + K +
(O&aring; + N)&gt;
P&gt; R&gt; T&gt;
=
&ntilde; P O&ntilde; R NT &ntilde; T SO&ntilde; V
π
1235. &iacute;~&aring; &ntilde; = &ntilde; + +
+
+
+ K I= &ntilde; &lt; K=
O
P NR
PNR OUPR
=

N  &ntilde; &ntilde; P O&ntilde; R O&ntilde; T
1236. &Aring;&ccedil;&iacute; &ntilde; = −  + +
+
+ K I= &ntilde; &lt; π K=
&ntilde;  P QR VQR QTOR

=
&ntilde; P N ⋅ P&ntilde; R
N ⋅ P ⋅ RK (O&aring; − N)&ntilde; O&aring; +N
1237. ~&ecirc;&Aring;&euml;&aacute;&aring; &ntilde; = &ntilde; +
+
+K+
+ K I=
O⋅P O⋅ Q ⋅R
O ⋅ Q ⋅ SK (O&aring; )(O&aring; + N)
&ntilde; &lt; N K=
=

π 
&ntilde; P N ⋅ P&ntilde; R
N ⋅ P ⋅ RK (O&aring; − N)&ntilde; O&aring; +N
+
+K+
+ KI
1238. ~&ecirc;&Aring;&Aring;&ccedil;&euml; &ntilde; = −  &ntilde; +
O 
O⋅P O⋅Q ⋅R
O ⋅ Q ⋅ SK (O&aring; )(O&aring; + N)

&aring;
&ntilde; &lt; N K=
=
1239. ~&ecirc;&Aring;&iacute;~&aring; &ntilde; = &ntilde; −
(− N) &ntilde; O&aring;+N &plusmn; K I= &ntilde; ≤ N K=
&ntilde;P &ntilde;R &ntilde;T
+ − +K+
P
R T
O&aring; + N
&aring;
=
1240. &Aring;&ccedil;&euml;&Uuml; &ntilde; = N +
&ntilde;O &ntilde;Q &ntilde;S
&ntilde; O&aring;
+
+
+K+
+ K=
(O&aring; )&gt;
O&gt; Q&gt; S&gt;
=
1241. &euml;&aacute;&aring;&Uuml; &ntilde; = &ntilde; +
&ntilde;P &ntilde;R &ntilde;T
&ntilde; O &aring; +N
+
+
+K+
+ K=
(O&aring; + N)&gt;
P&gt; R&gt; T&gt;
=
=
=
=
=
315
CHAPTER 11. SERIES
11.12 Binomial Series
=
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&aring;I=&atilde;=
o&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&ntilde;=
`&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml;W= &aring; ` &atilde; =
=
=
&aring;
1242. (N + &ntilde; ) = N + &aring;`N&ntilde; + &aring;` O &ntilde; O + K + &atilde; ` &aring; &ntilde; &atilde; + K + &ntilde; &aring; =
=
&aring;(&aring; − N)K[&aring; − (&atilde; − N)]
1243. &aring; ` &atilde; =
I= &ntilde; &lt; N K=
&atilde;&gt;
=
N
1244.
= N − &ntilde; + &ntilde; O − &ntilde; P + K I= &ntilde; &lt; N K=
N+ &ntilde;
=
N
1245.
= N + &ntilde; + &ntilde; O + &ntilde; P + K I= &ntilde; &lt; N K=
N− &ntilde;
=
&ntilde; &ntilde; O N ⋅ P&ntilde; P N ⋅ P ⋅ R&ntilde; Q
1246. N + &ntilde; = N + −
+
−
+ K I= &ntilde; ≤ N K=
O O⋅Q O⋅Q ⋅S O⋅Q ⋅S⋅U
=
&ntilde; N ⋅ O&ntilde; O N ⋅ O ⋅ R&ntilde; P N ⋅ O ⋅ R ⋅ U&ntilde; Q
1247. P N + &ntilde; = N + −
+
−
+ K I= &ntilde; ≤ N K=
P P⋅S
P⋅S⋅V
P ⋅ S ⋅ V ⋅ NO
=
=
=
11.13 Fourier Series
=
f&aring;&iacute;&Eacute;&Ouml;&ecirc;~&Auml;&auml;&Eacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;W= &Ntilde; (&ntilde; ) =
c&ccedil;&igrave;&ecirc;&aacute;&Eacute;&ecirc;=&Aring;&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;&euml;W= ~ M I= ~ &aring; I= &Auml;&aring; =
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W=&aring;==
316
CHAPTER 11. SERIES
∞
~
1248. &Ntilde; (&ntilde; ) = M + ∑ (~ &aring; &Aring;&ccedil;&euml; &aring;&ntilde; + &Auml;&aring; &euml;&aacute;&aring; &aring;&ntilde; ) =
O
&aring; =N
=
π
N
1249. ~ &aring; = ∫ &Ntilde; (&ntilde; )&Aring;&ccedil;&euml; &aring;&ntilde; &Ccedil;&ntilde; ==
π −π
=
π
N
&Auml;
=
&Ntilde; (&ntilde; )&euml;&aacute;&aring; &aring;&ntilde; &Ccedil;&ntilde; ==
1250. &aring;
π −∫π
=
=
317
Chapter 12
Probability
=
=
=
=
12.1 Permutations and Combinations
=
m&Eacute;&ecirc;&atilde;&igrave;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;W= &aring; m&atilde; =
`&ccedil;&atilde;&Auml;&aacute;&aring;~&iacute;&aacute;&ccedil;&aring;&euml;W= &aring; ` &atilde; =
t&Uuml;&ccedil;&auml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;W=&aring;I=&atilde;=
=
=
1251. c~&Aring;&iacute;&ccedil;&ecirc;&aacute;~&auml;=
&aring;&gt; = N ⋅ O ⋅ PK(&aring; − O)(&aring; − N)&aring; =
M&gt; = N =
=
1252. &aring; m&aring; = &aring;&gt; =
=
&aring;&gt;
1253. &aring; m&atilde; =
=
(&aring; − &atilde; )&gt;
=
1254. _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml;=`&ccedil;&Eacute;&Ntilde;&Ntilde;&aacute;&Aring;&aacute;&Eacute;&aring;&iacute;=
&aring;
&aring;&gt;
&aring;
=
` &atilde; =   =
 &atilde;  &atilde;&gt; (&aring; − &atilde; )&gt;
=
1255. &aring; ` &atilde; = &aring; ` &aring; −&atilde; =
=
1256. &aring; ` &atilde; + &aring; ` &atilde; +N = &aring; +N ` &atilde; +N =
=
318
CHAPTER 12. PROBABILITY
1257. &aring; ` M + &aring; `N + &aring; ` O + K + &aring; ` &aring; = O&aring; =
=
1258. m~&euml;&Aring;~&auml;∞&euml;=q&ecirc;&aacute;~&aring;&Ouml;&auml;&Eacute;=
=
o&ccedil;&iuml;=M=
=
=
=
=
=
=
o&ccedil;&iuml;=N=
=
=
=
=
=
N=
o&ccedil;&iuml;=O=
=
=
=
=
N=
=
o&ccedil;&iuml;=P=
=
=
=
N=
=
P=
o&ccedil;&iuml;=Q=
=
=
N=
=
Q=
=
o&ccedil;&iuml;=R=
=
N=
=
R=
= NM=
o&ccedil;&iuml;=S=
N=
=
S=
= NR= =
=
=
=
N=
=
=
=
N=
=
O=
=
N=
=
P=
=
S=
=
Q=
= NM= =
OM= = NR=
12.2 Probability Formulas
=
b&icirc;&Eacute;&aring;&iacute;&euml;W=^I=_=
m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;W=m=
o~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W=uI=vI=w=
s~&auml;&igrave;&Eacute;&euml;=&ccedil;&Ntilde;=&ecirc;~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;W=&ntilde;I=&oacute;I=&ograve;=
b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil;=&icirc;~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=uW= &micro; =
^&aring;&oacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ecirc;&Eacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;W= ε ==
p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=&Ccedil;&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring;W= σ =
s~&ecirc;&aacute;~&aring;&Aring;&Eacute;W= σ O =
a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;W= &Ntilde; (&ntilde; ) I= &Ntilde; (&iacute; ) =
=
=
1259. m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=~&aring;=b&icirc;&Eacute;&aring;&iacute;=
&atilde;
m(^ ) = I==
&aring;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&atilde;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;I==
&aring;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&iacute;&ccedil;&iacute;~&auml;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K=
=
319
=
=
=
N=
=
R=
=
=
=
=
=
N=
=
S=
=
=
=
=
=
N=
=
=
=
=
=
=
=
N=
CHAPTER 12. PROBABILITY
1260. o~&aring;&Ouml;&Eacute;=&ccedil;&Ntilde;=m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=s~&auml;&igrave;&Eacute;&euml;=
M ≤ m(^ ) ≤ N =
=
1261. `&Eacute;&ecirc;&iacute;~&aacute;&aring;=b&icirc;&Eacute;&aring;&iacute;=
m( ^ ) = N =
=
1262. f&atilde;&eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute;=b&icirc;&Eacute;&aring;&iacute;=
m( ^ ) = M =
=
1263. `&ccedil;&atilde;&eacute;&auml;&Eacute;&atilde;&Eacute;&aring;&iacute;=
m(^ ) = N − m(^ ) =
=
1264. f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=b&icirc;&Eacute;&aring;&iacute;&euml;=
m(^ L _ ) = m(^ ) I==
m(_ L ^ ) = m(_ ) =
=
1265. ^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;=o&igrave;&auml;&Eacute;=&Ntilde;&ccedil;&ecirc;=f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=b&icirc;&Eacute;&aring;&iacute;&euml;=
m(^ ∪ _ ) = m(^ ) + m(_ ) =
=
1266. j&igrave;&auml;&iacute;&aacute;&eacute;&auml;&aacute;&Aring;~&iacute;&aacute;&ccedil;&aring;=o&igrave;&auml;&Eacute;=&Ntilde;&ccedil;&ecirc;=f&aring;&Ccedil;&Eacute;&eacute;&Eacute;&aring;&Ccedil;&Eacute;&aring;&iacute;=b&icirc;&Eacute;&aring;&iacute;&euml;=
m(^ ∩ _ ) = m(^ ) ⋅ m(_ ) =
=
1267. d&Eacute;&aring;&Eacute;&ecirc;~&auml;=^&Ccedil;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;=o&igrave;&auml;&Eacute;=
m(^ ∪ _ ) = m(^ ) + m(_ ) − m(^ ∩ _ ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
^ ∪ _ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&igrave;&aring;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&Eacute;&icirc;&Eacute;&aring;&iacute;&euml;=^=~&aring;&Ccedil;=_I==
^ ∩ _ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aacute;&aring;&iacute;&Eacute;&ecirc;&euml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;=&ccedil;&Ntilde;=&Eacute;&icirc;&Eacute;&aring;&iacute;&euml;=^=~&aring;&Ccedil;=_K=
=
1268. `&ccedil;&aring;&Ccedil;&aacute;&iacute;&aacute;&ccedil;&aring;~&auml;=m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=
m(^ ∩ _ )
m( ^ L _ ) =
=
m(_ )
=
1269. m(^ ∩ _ ) = m(_ ) ⋅ m(^ L _ ) = m(^ ) ⋅ m(_ L ^ ) =
320
CHAPTER 12. PROBABILITY
1270. i~&iuml;=&ccedil;&Ntilde;=q&ccedil;&iacute;~&auml;=m&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=
&atilde;
m(^ ) = ∑ m(_ &aacute; )m(^ L _ &aacute; ) I==
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= _ &aacute; =&aacute;&euml;=~=&euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=&atilde;&igrave;&iacute;&igrave;~&auml;&auml;&oacute;=&Eacute;&ntilde;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute;=&Eacute;&icirc;&Eacute;&aring;&iacute;&euml;K==
=
1271. _~&oacute;&Eacute;&euml;∞=q&Uuml;&Eacute;&ccedil;&ecirc;&Eacute;&atilde;=
m(^ L _ ) ⋅ m(_ )
m(_ L ^ ) =
=
m(^ )
=
1272. _~&oacute;&Eacute;&euml;∞=c&ccedil;&ecirc;&atilde;&igrave;&auml;~=
m(_ ) ⋅ m(^ L _ &aacute; )
m(_ &aacute; L ^ ) = &atilde; &aacute;
I==
∑ m(_ &aacute; ) ⋅ m(^ L _ &aacute; )
&acirc; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
_ &aacute; =&aacute;&euml;=~=&euml;&Eacute;&iacute;=&ccedil;&Ntilde;=&atilde;&igrave;&iacute;&igrave;~&auml;&auml;&oacute;=&Eacute;&ntilde;&Aring;&auml;&igrave;&euml;&aacute;&icirc;&Eacute;=&Eacute;&icirc;&Eacute;&aring;&iacute;&euml;=E&Uuml;&oacute;&eacute;&ccedil;&iacute;&Uuml;&Eacute;&euml;&Eacute;&euml;FI=
^ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&aacute;&aring;~&auml;=&Eacute;&icirc;&Eacute;&aring;&iacute;I==
m(_ &aacute; ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&aacute;&ccedil;&ecirc;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&aacute;&Eacute;&euml;I=
m(_ &aacute; L ^ ) =~&ecirc;&Eacute;=&iacute;&Uuml;&Eacute;=&eacute;&ccedil;&euml;&iacute;&Eacute;&ecirc;&aacute;&ccedil;&ecirc;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&aacute;&Eacute;&euml;K=
=
1273. i~&iuml;=&ccedil;&Ntilde;=i~&ecirc;&Ouml;&Eacute;=k&igrave;&atilde;&Auml;&Eacute;&ecirc;&euml;=
p

m &aring; − &micro; ≥ ε  → M =~&euml;= &aring; → ∞ I==
 &aring;

p

m &aring; − &micro; &lt; ε  → N =~&euml;= &aring; → ∞ I==
 &aring;

&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
p&aring; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&igrave;&atilde;=&ccedil;&Ntilde;=&ecirc;~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;I=
&aring; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&eacute;&ccedil;&euml;&euml;&aacute;&Auml;&auml;&Eacute;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K=
=
1274. `&Uuml;&Eacute;&Auml;&oacute;&euml;&Uuml;&Eacute;&icirc;=f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;=
s(u )
m( u − &micro; ≥ ε ) ≤ O I==
ε
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= s(u ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=uK=
321
CHAPTER 12. PROBABILITY
1275. k&ccedil;&ecirc;&atilde;~&auml;=a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
( &ntilde; −&micro; ) O
−
N
O
ϕ(&ntilde; ) =
&Eacute; Oσ I==
σ Oπ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&ntilde;=&aacute;&euml;=~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;K=
=
1276. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=k&ccedil;&ecirc;&atilde;~&auml;=a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
O
N − &ograve;O
&Eacute; =
ϕ(&ograve; ) =
Oπ
^&icirc;&Eacute;&ecirc;~&Ouml;&Eacute;=&icirc;~&auml;&igrave;&Eacute;= &micro; = M I=&Ccedil;&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring;= σ = N K=
=
=
=====
=
Figure 210.
=
1277. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=w=s~&auml;&igrave;&Eacute;=
u−&micro;
w=
=
σ
=
1278. `&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute;=k&ccedil;&ecirc;&atilde;~&auml;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&ntilde;
−
N
c(&ntilde; ) =
&Eacute;
∫
σ Oπ −∞
( &iacute; −&micro; ) O
OσO
&Ccedil;&iacute; I==
322
CHAPTER 12. PROBABILITY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&ntilde;=&aacute;&euml;=~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I==
&iacute; =&aacute;&euml;=~=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;K=
=
 α −&micro; β−&micro;
1279. m(α &lt; u &lt; β ) = c
 I=
 − c
 σ   σ 
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=
u=&aacute;&euml;=&aring;&ccedil;&ecirc;&atilde;~&auml;&auml;&oacute;=&Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil;=&ecirc;~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I=
c=&aacute;&euml;=&Aring;&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;I==
m(α &lt; u &lt; β ) =&aacute;&euml;=&aacute;&aring;&iacute;&Eacute;&ecirc;&icirc;~&auml;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K=
=
ε
1280. m( u − &micro; &lt; ε ) = Oc  I==
σ
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
u=&aacute;&euml;=&aring;&ccedil;&ecirc;&atilde;~&auml;&auml;&oacute;=&Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&Eacute;&Ccedil;=&ecirc;~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I=
c=&aacute;&euml;=&Aring;&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute;=&aring;&ccedil;&ecirc;&atilde;~&auml;=&Ccedil;&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K=
=
1281. `&igrave;&atilde;&igrave;&auml;~&iacute;&aacute;&icirc;&Eacute;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&ntilde;
c(&ntilde; ) = m(u &lt; &ntilde; ) = ∫ &Ntilde; (&iacute; )&Ccedil;&iacute; I==
−∞
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;=&aacute;&euml;=~=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;=&ccedil;&Ntilde;=&aacute;&aring;&iacute;&Eacute;&Ouml;&ecirc;~&iacute;&aacute;&ccedil;&aring;K=
=
1282. _&Eacute;&ecirc;&aring;&ccedil;&igrave;&auml;&auml;&aacute;=q&ecirc;&aacute;~&auml;&euml;=m&ecirc;&ccedil;&Aring;&Eacute;&euml;&euml;=
&micro; = &aring;&eacute; = I= σ O = &aring;&eacute;&egrave; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&aring; =&aacute;&euml;=~=&euml;&Eacute;&egrave;&igrave;&Eacute;&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=&Eacute;&ntilde;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&aring;&iacute;&euml;I==
&eacute; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;=&ccedil;&Ntilde;=&Eacute;~&Aring;&Uuml;=&Eacute;&ntilde;&eacute;&Eacute;&ecirc;&aacute;&atilde;&Eacute;&aring;&iacute;&euml;I=
&egrave; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I= &egrave; = N − &eacute; K=
=
1283. _&aacute;&aring;&ccedil;&atilde;&aacute;~&auml;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=======
 &aring;
&Auml;(&aring;I &eacute;I &egrave; ) =   &eacute; &acirc; &egrave; &aring; − &acirc; I==
&acirc;
323
CHAPTER 12. PROBABILITY
&micro; = &aring;&eacute; I= σ O = &aring;&eacute;&egrave; I=
&Ntilde; (&ntilde; ) = (&egrave; + &eacute;&Eacute; &ntilde; ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&aring;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&iacute;&ecirc;&aacute;~&auml;&euml;=&ccedil;&Ntilde;=&euml;&Eacute;&auml;&Eacute;&Aring;&iacute;&aacute;&ccedil;&aring;&euml;I=
&eacute;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;I=
&egrave;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I= &egrave; = N − &eacute; K=
=
1284. d&Eacute;&ccedil;&atilde;&Eacute;&iacute;&ecirc;&aacute;&Aring;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=
m(q = &agrave;) = &egrave; &agrave;−N&eacute; I==
N
&egrave;
&micro; = I= σ O = O I==
&eacute;
&eacute;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
q=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;&aacute;&ecirc;&euml;&iacute;=&euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&Ntilde;&igrave;&auml;=&Eacute;&icirc;&Eacute;&aring;&iacute;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&euml;&Eacute;&ecirc;&aacute;&Eacute;&euml;I=
&agrave;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Eacute;&icirc;&Eacute;&aring;&iacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;I=
&eacute;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&iacute;&Uuml;~&iacute;=~&aring;&oacute;=&ccedil;&aring;&Eacute;=&Eacute;&icirc;&Eacute;&aring;&iacute;=&aacute;&euml;=&euml;&igrave;&Aring;&Aring;&Eacute;&euml;&euml;&Ntilde;&igrave;&auml;I==
&egrave;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;=&ccedil;&Ntilde;=&Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;I= &egrave; = N − &eacute; K=
=
1285. m&ccedil;&aacute;&euml;&euml;&ccedil;&aring;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=
λ&acirc; −λ
m(u = &acirc; ) ≈ &Eacute; I= λ = &aring;&eacute; I==
&acirc;&gt;
O
&micro; = λ I= σ = λ I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
λ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&ecirc;~&iacute;&Eacute;=&ccedil;&Ntilde;=&ccedil;&Aring;&Aring;&igrave;&ecirc;&ecirc;&Eacute;&aring;&Aring;&Eacute;I=
&acirc;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&aring;&igrave;&atilde;&Auml;&Eacute;&ecirc;=&ccedil;&Ntilde;=&eacute;&ccedil;&euml;&aacute;&iacute;&aacute;&icirc;&Eacute;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;&euml;K=
=
1286. a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;==
&aring;
&Auml;
m(~ ≤ u ≤ &Auml;) = ∫ &Ntilde; (&ntilde; )&Ccedil;&ntilde; ==
~
=
1287. `&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=r&aring;&aacute;&Ntilde;&ccedil;&ecirc;&atilde;=a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=
~+&Auml;
N
I==
I= &micro; =
&Ntilde;=
&Auml;−~
O
324
CHAPTER 12. PROBABILITY
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Ntilde;=&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ccedil;&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=&Ntilde;&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;K=
=
1288. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml;=a&Eacute;&aring;&euml;&aacute;&iacute;&oacute;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
&Ntilde; (&iacute; ) = λ&Eacute; −λ&iacute; I= &micro; = λ I= σ O = λO =
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;=&aacute;&euml;=&iacute;&aacute;&atilde;&Eacute;I= λ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;=&ecirc;~&iacute;&Eacute;K=
=
1289. b&ntilde;&eacute;&ccedil;&aring;&Eacute;&aring;&iacute;&aacute;~&auml;=a&aacute;&euml;&iacute;&ecirc;&aacute;&Auml;&igrave;&iacute;&aacute;&ccedil;&aring;=c&igrave;&aring;&Aring;&iacute;&aacute;&ccedil;&aring;=
c(&iacute; ) = N − &Eacute; −λ&iacute; I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&iacute;=&aacute;&euml;=&iacute;&aacute;&atilde;&Eacute;I= λ =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Ntilde;~&aacute;&auml;&igrave;&ecirc;&Eacute;=&ecirc;~&iacute;&Eacute;K=
=
1290. b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil;=s~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=a&aacute;&euml;&Aring;&ecirc;&Eacute;&iacute;&Eacute;=o~&aring;&Ccedil;&ccedil;&atilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
&aring;
&micro; = b(u ) = ∑ &ntilde; &aacute; &eacute;&aacute; I==
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;= &ntilde; &aacute; =&aacute;&euml;=~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I= &eacute; &aacute; =&aacute;&euml;=&aacute;&iacute;&euml;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K=
=
1291. b&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil;=s~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=`&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=o~&aring;&Ccedil;&ccedil;&atilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
∞
&micro; = b(u ) = ∫ &ntilde;&Ntilde; (&ntilde; )&Ccedil;&ntilde; ==
−∞
=
1292. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=b&ntilde;&eacute;&Eacute;&Aring;&iacute;~&iacute;&aacute;&ccedil;&aring;&euml;=
b(u + v ) = b(u ) + b(v ) I==
b(u − v ) = b(u ) − b(v ) I=
b(&Aring;u ) = &Aring;b(u ) I==
b(uv ) = b(u ) ⋅ b(v ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Aring;=&aacute;&euml;=~=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1293. b(u O ) = s(u ) + &micro; O I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&micro; = b(u ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Eacute;&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil;=&icirc;~&auml;&igrave;&Eacute;I===
s(u ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;K=
=
=
=
325
CHAPTER 12. PROBABILITY
1294. j~&ecirc;&acirc;&ccedil;&icirc;=f&aring;&Eacute;&egrave;&igrave;~&auml;&aacute;&iacute;&oacute;=
b(u )
m(u &gt; &acirc; ) ≤
I==
&acirc;
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&acirc;=&aacute;&euml;=&euml;&ccedil;&atilde;&Eacute;=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1295. s~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=a&aacute;&euml;&Aring;&ecirc;&Eacute;&iacute;&Eacute;=o~&aring;&Ccedil;&ccedil;&atilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
[
]
&aring;
σ O = s(u ) = b (u − &micro; ) = ∑ (&ntilde; &aacute; − &micro; ) &eacute;&aacute; I==
O
O
&aacute; =N
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
&ntilde; &aacute; =&aacute;&euml;=~=&eacute;~&ecirc;&iacute;&aacute;&Aring;&igrave;&auml;~&ecirc;=&ccedil;&igrave;&iacute;&Aring;&ccedil;&atilde;&Eacute;I=
&eacute; &aacute; =&aacute;&euml;=&aacute;&iacute;&euml;=&eacute;&ecirc;&ccedil;&Auml;~&Auml;&aacute;&auml;&aacute;&iacute;&oacute;K=
=
1296. s~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=`&ccedil;&aring;&iacute;&aacute;&aring;&igrave;&ccedil;&igrave;&euml;=o~&aring;&Ccedil;&ccedil;&atilde;=s~&ecirc;&aacute;~&Auml;&auml;&Eacute;&euml;=
[
] ∫ (&ntilde; − &micro;) &Ntilde; (&ntilde; )&Ccedil;&ntilde; ==
σ = s(u ) = b (u − &micro; ) =
O
O
∞
O
−∞
=
1297. m&ecirc;&ccedil;&eacute;&Eacute;&ecirc;&iacute;&aacute;&Eacute;&euml;=&ccedil;&Ntilde;=s~&ecirc;&aacute;~&aring;&Aring;&Eacute;=
s(u + v ) = s(u ) + s(v ) I==
s(u − v ) = s(u ) + s(v ) I=
s(u + &Aring; ) = s(u ) I=
s(&Aring;u ) = &Aring; O s(u ) I=
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;=&Aring;=&aacute;&euml;=~=&Aring;&ccedil;&aring;&euml;&iacute;~&aring;&iacute;K=
=
1298. p&iacute;~&aring;&Ccedil;~&ecirc;&Ccedil;=a&Eacute;&icirc;&aacute;~&iacute;&aacute;&ccedil;&aring;=
[
]
a(u ) = s(u ) = b (u − &micro; ) =
=
1299. `&ccedil;&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;=
&Aring;&ccedil;&icirc; (uI v ) = b[(u − &micro;(u ))(v − &micro;(v ))] = b(uv ) − &micro;(u )&micro;(v ) I==
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
u =&aacute;&euml;=&ecirc;~&aring;&Ccedil;&ccedil;&atilde;=&icirc;~&ecirc;&aacute;~&Auml;&auml;&Eacute;I==
s(u ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=uI==
&micro; =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&Eacute;&ntilde;&eacute;&Eacute;&Aring;&iacute;&Eacute;&Ccedil;=&icirc;~&auml;&igrave;&Eacute;=&ccedil;&Ntilde;=u=&ccedil;&ecirc;=vK=
O
326
CHAPTER 12. PROBABILITY
=
1300. `&ccedil;&ecirc;&ecirc;&Eacute;&auml;~&iacute;&aacute;&ccedil;&aring;=
&Aring;&ccedil;&icirc; (uI v )
I==
ρ(uI v ) =
s(u )s(v )
&iuml;&Uuml;&Eacute;&ecirc;&Eacute;==
s(u ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=uI==
s(v ) =&aacute;&euml;=&iacute;&Uuml;&Eacute;=&icirc;~&ecirc;&aacute;~&aring;&Aring;&Eacute;=&ccedil;&Ntilde;=vK=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
327
=
i&ccedil;&ccedil;&acirc;=&Ntilde;&ccedil;&ecirc;=&ccedil;&iacute;&Uuml;&Eacute;&ecirc;=&Uuml;~&aring;&Ccedil;&Auml;&ccedil;&ccedil;&acirc;&euml;=~&aring;&Ccedil;=&euml;&ccedil;&auml;&icirc;&Eacute;&Ccedil;=&eacute;&ecirc;&ccedil;&Auml;&auml;&Eacute;&atilde;=&Ouml;&igrave;&aacute;&Ccedil;&Eacute;&euml;=~&iacute;=
&iuml;&iuml;&iuml;K&atilde;~&iacute;&Uuml;-&Eacute;&Auml;&ccedil;&ccedil;&acirc;&euml;K&Aring;&ccedil;&atilde;K=
=
=
=
=
```