VE C T O R AN A L YSIS
A TE XT-B O O K F O R T H E U SE O F
OF
M ATH E M ATICS
F O UN D E D
UP ON
JWIL L A R D
.
P rofe s s o r
E D WI N
TH E L E C T UR E S OF
LL D
G IB B S, P H D
.
f M a th e m a tic a l
o
B ID
AN D P H
S TU D E N T S
YS ICS
.
.
P by s z c s i n Ya le Un ive rs ity
WE L L WIL SO N
,
PH D
.
.
In s t ru c t o r i n M a t/i n n a t ic s in Ya le Un ive rs ity
B
N EW YO RK : C H ARLE S S CRI N ER ’ S S O N S
LO N DO N
:
EDWARD ARN O LD
1
901
B Y YA L E
P
U
u bl i s h e d ,
U
NIV E S I
R
De c e mb e r , 1 9 01
TY
.
.
NIV ITY PCAM B IDJOG N IL ON
AN D
ERS
SO N
H
RESS
R
W
E,
S
U S A
.
.
.
P RE FA CE B Y P R OFE SSOR G IB B S
the printing of a short pamphl et on the E le m e n ts of
Ve c t o r An a ly s is in the years 1 88 1 -8 4
never published but
somewhat widely circulated among those who were known t o
the desire has been e xpressed
b e interested in the sub ject
in more than one quarter that the sub stance of that t rea
tise perhaps in fuller form should be made accessib le t o
the pub lic
A s however the
ears passed without my fi ndi ng the
leisure to meet this want which seemed a real one I was
very glad to have one of the hearers of my course on Vec t or
A nalysis in the year 1 8 9 9—1 900 undertake the preparation of
a tex t-b ook on the sub ject
I have not desired t hat Dr Wilson should aim simply
at the reproduction of my lectures but rather that he shoul d
use his own judgment in all respects for the production of a
te x t b ook in which the subject should be so illustrated b y an
adequat e numb er of e x amples as to meet the wants of s t u
dents o i geometr and ph sics
SIN C E
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y
NI ER S ITY
YALE U
V
y
,
Se p t e
m
.
JWILLARD
.
be
r
,
1 9 01
.
ENG INEERIHG l
BB
GI
S
.
G EN E RA L P REFA CE
WH EN I undertook to adapt the lectures of P rofessor G ibbs
on VE CTOR A N A LY S IS for pub lication in the Yale B ic e n t e n
nial Series Professor G ibb s himself was already so fully
engaged upon his work to appear in the same series E le m e n t a ry
P r in c ip les i n St a ti s ti c a l M e c h a n ic s that it w a s understood no
material assistance in the composition of this book could b e
e xpected from him Fo r thi s reason he wished me to feel
entirely free to use my own di scretion ali ke in t h e selection
of the topics to b e treated and in t h e mode of treatment
It has been my endeavor to use the freedom thus granted
only in so far as was necessar for presenting his m e t hod in
te x t-book form
B y far the greater part of the material used in th e follow
ing pages has been taken from the course of lectures on
V ector A nalysis delivered annually at the Universit by
Professor G ib bs Some us e however has been made of the
chapters on V ector A nalysis in Mr O liver H e a vis ide s E le c
t ro m a gn e tic Th e o ry ( Electrician Series 1 8 9 3 ) and in P rofessor
Fop p l s lectures on D ie M a xwe ll s c h e Th e o ri e de r E le c t ric i t ci t
T
n s has
u
b
ner
e
M
previous
study
of
uaternio
Q
(
y
also b een of great assistance
Th e m aterial thus o b t ained has been arranged in the way
whi ch seems b est suited to easy mastery of the sub jec t
Those Arts whi ch it seemed b est to incorporate in the
te x t b u t which for various reasons may well b e omitted at
the fi rst readi ng have b een marked with an asterisk
Nu
m e ro u s illustra t ive e x amples have been dra wn from geometry
mechani c s and physics Indeed a large part of the te x t has
to do wi t h applications of the met hod These application s
have not been set apart in chapters b y themselves b u t have
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G E N E RAL
P RE FA CE
b een
distributed throughout t h e b ody of the book as fast as
the analysis has b een developed su ffi ciently for t heir adequate
treat men t It is hoped t hat b y t his means t h e reader may b e
G rea t care
b e t t er enab led t o mak e prac t ical use of the book
has b een t a k en in avoidi ng t h e introduction of unnecessary
ideas and in so illus t ra t ing each idea that is introduced as
to mak e its necess it y evident and its meani ng easy to grasp
Thus t h e b ook is not intended as a complete e x position of
the t heory of Vector Analysis b u t as a te xt-b o ok from whi ch
so much of t h e sub jec t as may b e required for practical a p p li
cations may b e learned Hence a summar inclu di ng a list
of t h e more important formul ae and a num b er of e x erc ises
have b een placed a t t h e end of each chapter and many less
e s sen t ial points in the te x t have b een indica t ed rather t han
fully worked o u t in t h e hope that t h e reader will supply th e
de t ails Th e sum mary may b e found useful in revie ws and
for reference
Th e s ub ject of Vector A nalysis naturally divides i t self into
three di stinc t part s Firs t t hat whi ch concerns addi t ion and
the scalar and vector produc t s of vectors Second that which
concerns the differential and i ntegral calculus in it s relations
to scalar and vector func t ions
Third that which contain s
the theory of the lin ear vector function Th e fir st part is
a necessary introdu c t ion to b oth other parts Th e second
and t hird are mutually independent Ei t her may be taken
up fi rst Fo r practic al purposes in mathema t ical physics the
second must b e regarded as more elementary than the t hird
B u t a studen t not prima ri ly in t ereste d in physics would nat
u ra ll
fi
pass
from
the
rst part t o the t hird which he would
y
pro b ab ly fi n d more attrac t ive and easy than the second
Follo wi ng thi s division of t h e sub ject the main b ody of
the b ook is divided into s ix chap t ers of which two deal wi t h
each of the three part s in the order named Chapters I and
II treat of addi t ion su b trac t ion scalar mul t iplica t ion and
t h e scalar and vector products of vectors
Th e e x po s i t ion
h a s b een made quite elementary
It can rea di ly b e under
stood b y and is especially suited for such readers as have a
k nowledge of only the elements of Trigonometry and An a
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lytic G eometr
Those who are w ell versed in Q uaternions
or allied sub jects may perhaps need to read only t h e s u m
m a ries Cha pters III and IV c ontain the treat ment of
those topics in Vecto r A nalysis which t hough of les s value
to the studen t s of pure mathematics are of the utmost impor
t a nce t o students of physics
Chapters V and VI deal wi t h
the linear vector function To s t udents of physics t h e linear
vector function is of particular importance in the m a t h e m a t i
cal trea t ment o f phenomena conn ected with non isotropic
media ; and to the student of pure ma t hemati cs t his part of
the b ook will prob ab ly b e the most interesting of all o win g
t o the fact tha t it lea ds to M ulti ple A lge b ra or the Theory
of Matrices A concluding chapter VII which con t ains the
development of cert ain hi gher parts of the theory a num b er
of applica t ions and a short sketch of imaginary or comple x
vectors has b een added
In the treatment of the integral calculus Chapter IV
questions of mathematical rigor arise Al t hough modern
theoris ts are devoting much time and thought to rigor and
al t hough they will doub tless cri t icise this portion of the b ook
adversely it has been deemed b est to give b u t li ttle attention
to t h e disc u ssion of this sub ject An d the more so for the
reason that whatever system of notation b e employed ques
tions of rigor are indissolub ly associate d wi t h t h e c a lc u lus
and occasion no n e w diffi cul t to t h e s t udent of V ector
A nalysis who must fi rst learn what the facts are and may
pos t pone until later the detailed consideration of the re s t ric
tions that are put upon those facts
N otwi t hstan di ng t h e e fforts which have b een made dur ing
more than half a century to introduce Q uaternions in to
physics the fact remains that t hey ha ve not fou nd wi de favor
O n the o t her hand there has b e en a growing tendency espe
c ia ll
y in the l a st decade toward the adoption of some form of
V ector A nalysis
Th e works of Heaviside and Fo p p l re
ferred to b efore may be cited in evidence A s yet ho wever
no system of V ector Analys is whi ch makes any claim to
completeness has b een publ is hed In fact Heavi side s ays :
I am in hopes that the chapter whi ch I now fi ni sh may
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“
G E N E R AL
xii
P R E FA C E
serve as a stopgap till r e gular vectorial treatises come to be
wri t t en suitab le for physicis t s b ased upon the vectorial treat
ment of vectors ( Elec t romagne t ic Theory Vo l I p
E ls e where in the same chapter Heaviside has set fort h the
claims of vector analysis a s against Q uaternions and others
have e xpre s s ed similar views
Th e k eynote then to any system of vector analysis must
This I feel con fi dent was P rofessor
b e its prac t ical utility
He us es it
G ib bs s po int of view in b uil di ng up his system
entirely in his cours es on Electricit y and M agne t is m and on
ight In writing this b ook I
Elec t romagne t ic Theory of
have t ried to present the sub ject from this prac t ical s t and
point and keep clearly before t h e reader s mind the ques
t ions : What comb inations or functions of vectors occur in
physics and geometry ? A n d how m a y these be represented
symb olically in t h e way b es t sui t ed to facile anal t ic manip
Th e t reatmen t of these que s t ions in modern b ooks
ula t io n ?
on phys ics has b een too much confi ned to the addi tion and
sub t raction of vectors This is scarcely enough It has
b een the aim here to give also an e x position of scalar and
vector products of the Operator v of divergence and curl
which have gained such universal recognition since the a p
p e a ra n c e of M a x well s Tre a t is e o n E le c tri c i ty a n d M a gn e tis m
of slope po t ential linear vector function etc such as shall
be adequat e for t h e needs of students of physics at the
present day and adapted to them
It has b een asserted by some that Q uaternions V ector
A nalysis and all such alge b ras are of little value for investi
ga t ing ques t ions l n mathematical physics Whe t her this
assert ion shall prove true or not one may s t ill maintain that
vectors are to ma t hematical phys ics what invariants are to
geometry A s every geometer must b e thoroughl y conver
sant with the ideas of invariants so every student of ph sics
shou ld be ab le to t hin k in terms of vecto rs A n d there is
no way in which he espe c ially a t t h e b eginning of his sci
e n t ifi c studies
can come to so true an appreciation of t h e
import an c e of vecto rs and of the ideas connected with t hem
as by working in Vector A nalysis and dealing directly with
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GE N E RA L
r
P RE FA
CE
i
x
ii
the vecto s themselves To those that hold these views the
success of P rofessor Fop p l s Vo rle s u n ge n itbe r Te c h n is c h e
M e c h a n i le ( four volum es Te u b n e r 1 8 9 7 —1 9 00 already in a
second edi t ion ) in which the t heory of mechanics is devel
oped b y means o f a vector analysis can be b u t an e n c o u r
aging sign
I t ake pleasure in tha nking my colleagues D r M B P o rt e r
and P rof H A B umstead for assist in g m e wit h t h e m a nu
script Th e good servi ces o f the latte r have been particularly
valuab le in arrangin g Chapters III and IV in their present
form and in suggesting many of the illustrations used in the
work I am also un der ob ligations to my father Mr E dwin
H Wilson for help in connec t ion both wi t h the proofs and
the manuscript Finally I wish to e xpress my deep indeb t
Fo r although he h a s been so
e dn e s s to P rofessor G i bb s
preoccupied as to be u nable t o read ei t her manuscript or
proof he h a s always b een ready to talk matte rs over with
me and it is he who has furnished me with inspiration suf
fi c ie n t to carr through the work
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ED
YA L E U
NI E S I
V R
TY ,
O c to b e
r
,
1 901
.
WIN
B ID
WE LL WIL S O N
.
TA B LE O F C O N TE N TS
REFAC E BY ROFE SSOR GIBBS
G ENERA L REFAC E
P
P
P
CHA P TE R I
AD D ITION A ND
S
M ULTIP LICATION
S C ALAR
CAUALALR SANDANDNULLECTOR
S
VECTOR
S
P OINT OF VIEW OF T I S C AP TER
CADDITION
L AR MU L TI PL ICATION
NE
G ATIV E S IGN
P A A LL E L OGRAM L AW
UAWB TRACTION
S G OVERNING T E FOREGOING O P ERATION S
OM PONENT
S OF VECTOR S
ECTOR
E
UATION
S
T
RE
E UNIT VECTOR S
PPL IC A TION S
S UNDRY P RO B L EM S IN G EOMETRY
E CTOR RE L ATION S INDE P ENDENT OF T E ORIGIN
ENTERU S ES OFOF GRAVITY
ARYC
E NTRIC CO RDIN A T E S
V
E CTOR S TO DENOT E AREA S
U MARY OFON CC AAPPTERT E R I I
V
EQ
TH
S
H
E
S
TH
.
A
TH
.
E
H
E
R
H
L
C
.
TH
E
H
V
i,
A
Q
jk
,
To
.
H
V
C
B
.
TH
O
E
S
H
H
M
E xE R CI SE S
CH AP TE R II
D IRE C T
2 7 28
TH
E
2 9—3 0
TH
E
3 1 —3 3
TH
E
3 4—3 5
TH
E
36
TH
E
AN D S KE W P R OD U C TS OF VECTORS
DIRECT
S CA LA R OR DOT P RODUCT OF
DIS SEWTRIBVECTOR
UTIVE L AWOR CRO
A ND APPL ICATION S
SS PRODUCT OF
DITRIS TRILEB UTIV
E L AW A ND A PPL ICATION S
P RODUCT
B
,
,
K
,
P
,
A
°
C
Two
Two
VE CTOR S
VECTOR S
55
58
60
63
67
C ON TE N TS
x vi
CA
AR
TRI
PLE P RODUCT
B
OR
VECTOR
TRI
PL E P RODUCT
B
RODUCT
S OF MORE T AN T REE VECTOR S WIT
CATION
S
RIGONOMETRY
ECI
P ROCA L S Y S TEM S OF T RE E V E CTOR S
OLAUTION
OF
S CA L AR AND V E CTOR E UATION S L INE A R IN
N
UN
NOWN
VECTOR
YINEMATIC
S TEM S OF FORCE S A CTING ON A RIGID B OD
S OF A RI G ID B ODY
ONDITION
S FOR E UI L I B RIUM OF A RI G ID B ODY
ETLATION
S B ETWEEN
OF
J
REE
P ER P ENDICU L AR UNIT VECTOR S
ROUMMARY
B L EM S IN G EOMETRY L ANAR CO RDI ATE S
OFONC C AAPTER
II
P ER II
S L
TH E
TH E
A
x0
[ A B C]
A X ( X C)
°
H
P
To
H
A P P LI
T
H
R
S
H
Q
K
S
Y
K
C
Q
T wo
R
R I G H TZH
Y ST E M S
AN DE D
H
P
S
H
H
E xE R C ISE S
O
P
.
N
T
C H AP TER III
TH E
5 5— 5 6
DIFFE REN TIAL C ALCUL U S
OF VE C TOR S
RIVATIVE S AND DIFFER
E NTI AL S OF VECTOR FUNCTION S
A
S CA L AR ARIA B L E
URVATURE
A ND TOR S ION OF GAUC E CURV S
INEMATIC
S OF A P ARTIC LE
ODOGRA
P
IN
S TANTANEOU S
OF
ROTATION
NTE
G RATION W T APPL IC A TION S
INEM
A TIC S
CALARECTOR
FUNCTION
S OF PO S ITION IN S PACE
DIFFERENTIATING
O
P ERATOR
S CA L A O P ERATOR
ECTOR FUNCTIONS OF POSITIO PA E
NTER
P RETATION OF T E
NTER
PR E TATION OF T E CUR L
AWSPOFARTIAO PERATION
OF
L A PPL ICATION OF
OF
A
VEC
TOR FUNCTION A NALOGOU S AYLOR S EOREM
DIFFERENTIATING
O
P ERATOR S OF T E S ECOND OR E R
G EOMETRIC INTER PRETATION OF A PLA C E S O P ERATOR
T
E
DI
S PE R S ION
UMMARY OFONC C AAPTER
III
PT E R III
DE
W
C
E SP EQ
.
V
To
K
.
TH
H
TH E H
E
H
A x rs
E
I
I
H
To
K
S
TH
E
TH
E
V
A jz
R
o
V
N
H
H
I
I
L
E
78
TH
V
o
.
T
’
H
E
L
V V
Q
Vx
VX
V V
V E x P A N SIO N
To
77
§
D IV E R G ENQ E
,
TH
IN
As
S
Ex E R CISE S
H
H
H
TH
.
D
’
1 59
1 66
C ON TE N TS
XVII
CH AP TE R IV
A
INTIONE INTEGRA
L S OF VECTOR FUNCTION S WIT APPL ICA
S
G AU SS S
EOREM
ONVER
S E OF TO E S S EOR E M WIT APPL ICATION S
RAN S FORMATION S OF LINE S URFACE AND VOL UM E
EMAR
S ON M U L TI PL E VA L UED FUNCTION S
OTENTIA
L
INTEGRATING
O
P ERATOR
OMMUTATIVE
PRO P ERTY OF
AND
EMARINTES GURATIN
P ON T E FOREGOING
G O P ERATOR S
E LATION S B OETWEEN
T
E
INTEGRATING
AND
DIFF
ER
P ERATOR S
P OTENTIA L
A S O L UTION OF OI SS ON S
UATION
OFUNCTION
L ENOIDA L AND IRROT A TIONA L P ART S OF A VECTOR
ERTAIN
O
P ERATOR S AND T EIR INVER S E
M UTUA L OTENTIA L S
EWTONIAN S APL ACIAN S AND
ERTAIN
B OUND A RY VA L UE T EOR E M S
UMMARY
OF
C
A
P TER IV
E RCI S E S ON C A P TER IV
TH E INTEGRAL
C ALC UL U S
OF VE CTORS
P
H
L
TH
’
C
S
TH
’
K
T
H
,
R
-
K
P
TH
.
“
E
C
H
K
TH
POT
V
P OT
R
IN
,
“
E
N E W,
”
“
LA P ,
”
“
MAx
H
R
E N TIA T I N G
TH
“
E
POT
”
’
P
Is
EQ
9 3 94
S
‘
.
C
H
P
N
,
,
L
,
M A X WE L L IA N S
C
S
H
H
H
EX
CHA P TE R V
LINE AR VE
C TO R F UN C TIONS
INE
A R VECTOR FUNCTION S DEFIN E D
D YADIC S DEFINED
L IN E AR VECTOR FUNCTION MAY B E RE PR E S ENTED
B Y A DYADIC
RO
P ERTIE S OF DY A DIC S
NONION
FORM
OF
A
DYADIC
DYAD
OR
INDETERMINATE
P RODUCT OF
VEC
TOR
S I S T E MO S T GENERA L
UNCTIONA
L P RO P ERTY
OF
T
E
S CA L AR AND VECTOR P RODUCT S
RODUCT
S OF DYADIC S
D EGREE S OF NU LL IT OF DYADIC S
IDEMF ACTOR
L
AN Y
.
TH
E
TH
E
P
Two
H
.
H
P
Y
TH
E
F
GE
II
C ON TE N TS
0 0 .
XVI
ECI
P ROCA L DY A DIC S
OWER
S A ND ROOT S OF DYADIC S
ONS ELUFGCON
ATE U GDYADIC
S
E
L F CON U G ATE AND ANTI
ATE
P ART S OF A DYADIC
NTI S ELF UADRANTA
CON UGATEL DYADIC S
VECTOR PROD
EDUCTION
OF
DYADIC
S
NORMA
L FORM
D OU B L E MU L TI PL ICATION OF DYADIC S
S ECOND AND T IRD OF A DYADIC
ONDITION
S FOR DIFFERENT DEGREE S OF NU LL ITY
ONION
FORM
D ETERMINANT S
NVARIUATION
A NT S OF A DYADIC
H A MI L TON A Y LE Y
UMMARY
OF
C
A
P TER V
ERCI S E S ON C A PTER
R
C
J
A
-
UCT
.
S
.
J
-
J
-
1 1 2 -1 1 4
P
.
J
-
TH
.
Q
E
V E R SO R S
R
To
TH
H
E
C
N
.
I
TH
.
C
-
E
EQ
S
H
H
EX
v
CH AP TE R VI
ROTATION S AN D S TRAIN S
OMOGENEOU
S S TRAIN RE PRE S ENTED B Y A DY A DIC
OTATION
S A OUT A FI ED POINT
VECTOR
S EMI TANGENT OF VER S ION
BIYCUADRANTA
L
AND T EIR P RODUCT S
L IC DYADIC S
IGONICTS TENANDSOR S
EDUCTION OF DYADIC
S TO CANONICA L FORM S TONIC S
S IM PL E AND
S EARER S
UMMARY OF C A PTER VI
H
R
X
B
TH
.
VE R SO R S
-
E
H
V E R SO R S
Q
C
H
R
T
C Y C L O TO N I C S
R
,
C o M P L Ex
C Y C L O TO N I C S ,
S
H
H
CH AP TE R VII
MIS C E LL AN EOU S AP P LICATION S
1 36 1 42
1 4 3 -1 4 6
1 4 7— 1 4 8
1 4 9 —1 5 7
1 5 8 -1 6 2
UADRIC
S URFACE S
P RO PAG ATION OF IG T IN CRY S TA L S
ARIA
B L E DYADIC S
URVATUR
E OF S URFACE S
H A MONIC I B RATION S AND
Q
TH
L
E
H
V
C
R
V
B
IV E C T O R S
,
c
i Cf
/
t
VE C TOR A N AL YSI S
,
Th e p o s i ti ve
typ ic a l s c a l a rs
a n d n e ga
tive
n u m be rs o
f
lge bra a re the
alge b ra is called
o rdi n a ry a
this reason th e ordinary
t o distinguish it from the
s c a la r alge b ra when necessar
algeb ra or analys is which is the sub ject of thi s book
Fo r
.
y
ve c to r
.
Th e typ ic a l ve c to r is the disp la
C onsider fi rst a point P ( Fig
f
c emen t o
L et
.
t ra
n s la
tio n i n
sp a c e .
P b e di splaced
in a
straight line and take a new position P
This change of position is represented by the
Th e magnitude of the di splace
1 line P P
ment is the lengt h of P P ; the direction of
it is the direction of the li ne P P fro m P to
N e x t cons ider a displacement not of one
P
F G 1
but of all the points in space
et all the
points move in s t raight li nes in t h e same direction and for the
same dista nce D This is equivalent to shifting space as a
rigid b ody in t hat direction through t h e dis t ance D without
ro t ation Such a displacement is called a tra n s la tio n It
possesses di rection and magnitude When Space undergoes
a translation T each point of Space undergoes a displacemen t
equal to T in magnitude and d rection ; and conversely if
t h e displacement P P which any one particular point P s u f
fers in t h e translati on T is known then that of any other
point Q is a so known : for Q Q must b e equal and parall el
to P P
Th e translation T is represented geometrically or graphically
by an arrow T ( Fig 1 ) of which the magn itu de and direction
are equal to those of the translation Th e ab solute position
of th is arrow in Spa ce is entirely immate rial Technically the
arrow is called a s tro ke Its tail or initial point is its o rigi n ;
and its he a d or fi nal point its te rm in u s In the fi gure the
origin is designated by 0 and the terminus b y T Thi s geo
metri c quanti t y a stroke is used as the mathematical sym b ol
for all vectors jus t as the ordinary positive and negative num
bers are used as the symbols for all scalars
’
.
'
'
~
.
’
’
’
L
.
I
.
.
.
,
.
.
.
.
i
,
’
l
,
’
’
.
.
.
.
.
,
.
.
,
,
,
.
A DDI TION
AND
3
S CA LAR M UL TIP LI CA TI ON
ex amples of scalar quantities mass ti me den
O thers are di s
s ity and temperature have been mentioned
tance volume moment of inertia work etc Magnitude
however is b y no means the sole propert y of these quantities
E ach has its
Each implies somethi ng b esides m a gni t ude
own d stingu ishing characte ristics as an e xample of which
its dim e n s i o n s in the sense well known to physicists may
be cited A distance 3 a time 3 a work 3 etc are ver
differen t Th e magnitude 3 is ho wever a prope rty common
to them all — perhaps the only one O f a l l scalar quanti
tities p u re n u m be r is the Simplest It implies nothing but
magnitude It is the scalar p a r e xc e lle n c e and consequentl
it is used as the mathemat ical symbol for all sca ars
A s e x amples of vector quantiti e s force displacement velo
city and acceleration have been given Each of these has
other characteristi cs than those which b elong to a vecto r pure
and Simple Th e concept of vec to r involves t wo ideas and
two alone — magnitude of the vector and direction of the
vector B u t force s more complicate d When it is applied
to a rigid bod the lin e in which it a cts must b e taken into
consideration ; magnitude and di rection alone do not suf
An d in cas e it is applied to a non -rigid bo dy the p o i n t
fice
of application of the force is as important as the magnitude or
Such is frequen t ly true for vector quantit ies other
direc t ion
tha n force M oreover t h e question of dimensions is present
as in the case of scalar quantities Th e mathematical vector
t h e s t roke which is the primary object of consideration in
t hi s b ook a b s t racts from all directed quantities their m agni
tude and di rection and nothing but t hese ; jus t as the mat he
m a ti c a l scalar pure numb er abstracts the magni t ude and
that alone Hence one must b e on his guard lest from
analogy he attribute some properties to the mathematical
vector which do not belong to it ; and he must be even more
careful lest he Obta in erroneous results b considering the
As
,
,
.
,
,
,
,
,
.
,
,
.
i
.
,
,
.
,
,
,
.
y
,
,
.
y
.
l
.
,
.
,
.
,
.
y
.
i
.
.
.
.
.
,
,
,
,
,
.
y
4
VE C TOR A N AL YSIS
vector quantities of physics as possessing no properties other
than th o se of the mat hemat ical vector Fo r e x ample it would
never do to consider force and its e ffects as unaltered b y
s h ift ing it parallel t o it s elf This warnin g may not be
necessary yet it may pos s ib ly s ave some confusion
Inasmuch as taken in its entirety a vector or stroke
is b u t a Single concept it may appropriately be designated b y
one let ter O wing however to the fundamental di fference
be tween scalars and vecto rs it is necessary to dis t inguish
caref u lly t h e one from the o t her Sometimes as in mathe
m a t ic a l physics t h e distinction is furnished b y the physica
int erpreta t ion Thus if n b e the inde x of refraction it
must b e scalar ; m the mass and t the time are also
scalars ; b u t f t h e force and a the acceleration are
vectors When however t h e letters are regarded merely
symb ols with no particular physical signi fi cance some
as
t pographical difference must b e relied upon to distinguish
vectors from scalars Hence in this b ook Clar e n do n typ e is
used for sett ing up vectors and ordinary type for scalars
This permits t h e use of t h e same letter di fle re n t ly printed
to represent the vector and its scalar magnitude } Thus if
G b e t h e electric current in magni t ude and direction 0 may
b e used to represent the magnitude of tha t current ; if g b e
the vector acceleration due to gravity g may be the s calar
value of that acceleration ; if v b e the veloci t y of a moving
mass v may b e the magni t ude of that velocit y Th e use of
Clarendons to denote vecto rs makes it possi b le to pass from
directed quan t i t ies to t heir scalar magnitudes by a mere
change in t h e appearance of a letter wi t hout any confusing
change in the lette r itself
De n i t io n
Two vectors are said to b e e qu a l when they have
the same magnitude and t h e same direction
.
.
.
,
,
,
,
.
,
l
,
.
,
.
,
,
,
,
,
,
,
.
,
,
,
y
.
.
,
,
,
.
fi
i
m
.
by n o e a n s i n va ria bl y fo llo we d In
in s t a n c e s it wo ul d p ro ve j us t a s u n d e s i ra bl e a s it is c o n ve n i e n t in o t h e rs
c h i e fly va l ua bl e in t h e a ppl i c a t i o n o f ve c t o rs t o p h y s i c s
1
Th i s
.
c o n ve n t o n
,
h o we ve r, is
.
.
.
so
m
e
It is
A DDI TI ON A N D
y of two vectors
equali t
usual Sign
Th e
Thus
5
S CA LAR M UL TIP LI CA TI ON
A
and
A
B
B
is denoted by the
.
y
Evidently
a vector or stroke is not a ltered b shi fting it
ab out p a r a lle l to itself in space Hence any vector A P P
( Fig 1 ) may b e dr a wn from any a s signed point 0 as origin
for t h e segment P P may b e moved parallel to itself unt il
the point P fal ls upon the poin t 0 and P upon some po nt T
’
.
.
’
i
’
.
Th e n
y
this wa all vectors in space may b e replaced by directed
segments radiating from one fi x ed point 0. E qual vectors
in space will of course coincide when placed with their ter
mini a t the same point 0 Thus ( Fig 1 ) A P P and B Q Q
b o th fall upon T
0T
Fo r the numerical determination of a vector th re e scalars
are necessary These may be chosen in a variet of ways
0 b e polar co ordinates in Space any vector r drawn
If r
wi t h its origin at th e origin of co ordinates may be represented
by the three scalars r (I) 9 which determine the terminus of
the vector
7
S
In
,
’
’
.
,
.
,
.
y
.
.
,
’
,
,
(
.
q,
,
if x y z be Cartesian coordi na te s in space a vector r may
be considered as given b y the differences of the co ordinates as
y z of its term inus and those at y z of its origin
Or
,
,
’
,
’
’
,
r
- co
,
y
,
,
r
-
.
y
,
z
r
i
in particul ar the origin of the vector co ncide with the
origin of c o Ordi n a t e s the vector will b e represented by the
three co ordin ates of its termi nus
If
,
l
(m
r
,
y"
2
1
)
r
When tw o vecto s are equal the three scala s which re p e
sent them must b e equal respecti vely each to each Hence
one vector equalit implies th ree scalar equali t ies
y
r
.
.
6
fi
VE C TOR A N AL YS I S
q l
vector A is said to be e ua to z e ro when its
magnit ude A is z ero
Such a vector A is called a n u ll or z e ro vec to r and is writte n
equal to naught in the usual manner Thus
De
A
n i ti on
.
.
A = O if A = 0
.
i
null vectors are regarded as equ al to each other w thout
any considerat ions of direction
In fact a n u ll vec t or from a geometrical standpoint woul d
that is to
b e represented b y a linear segment of length z ero
say b y a point It consequently woul d have a W holly inde
terminate direction or what amounts to t h e same thing none at
all If however it b e regarded as the limit approached b a
vector of fi n i te length it might be considered to have that
d rection which is the limit approached by the direction of the
fi ni t e vector when t h e lengt h decreases inde fi ni t ely and a p
T
ero
as
a
limit
h
ju
t
i
fi
cation
for
sregar
ng
r
a
h
z
e
s
di
d
i
s
o
c
e
p
this direction and looking upon all n ul l vectors as equal is
tha t when they are added ( Art 8 ) to other vectors no change
occurs and when mul t iplied ( Arts 2 7 3 1 ) b y o t her vectors
the product is z ero
In e x tending to vectors the fundamental Opera t ions
of algebra and arithmetic namely addition sub traction and
multiplication care must be e x ercised not only to avoid self
contradictor de fi nitions b u t also to lay down u seful ones
B oth these ends may be accompl ished most naturally and
easil y b y looking to physics ( for in that science vectors con
t in u a ll y present themselves ) and b y observing how such
quantiti es are treated there If then A be a given displace
ment force or velocity what is t wo three or in general 3:
t imes A ?
Wha t the negative of A ? An d if B be ano t her
what is the sum of A and B ? That is to say what is the
equivalent of A and B taken together ? Th e obvious answers
to these questi ons suggest immedi ately the desired de fi nitions
All
.
,
.
.
y
,
,
,
,
i
,
,
.
.
.
,
.
y
,
,
,
,
,
.
.
,
,
,
,
,
,
,
,
.
A DD I TI ON
fi
A ND
Sc a la
7
S CA LAR M UL TIP LI CA TI ON
r
M u ltip lic a t io n
y
vector is said to be mul tiplied b a
ositive scalar when its magni t ude is mul tiplied by that scalar
and its direction is left unaltere
times
Thus if V be a velocity of nine knots E ast b y N orth
v is a velocity of t went -one k nots with the di rection still
O r if f be the force e x erted upon the scale
E ast b y N orth
pan b y a gram weight 1 000 times f is the force e x erted b y a
kilogram Th e direction in both cases is verti cally down
ward
If A be the vector and a: the scalar the p oduct of ac and A is
denoted as usual by
a: A or A cc
De
p
n i tio n
A
d
.
y
,
.
,
.
r
.
.
y
is however more customar to place the scalar multiplier
before the multipli cand A This multiplicati on by a scalar
is called s c a la r m u ltip lic a tio n and it follows the assoc iati ve law
It
,
,
.
,
y
as in or di nar algebra and arithmetic This statement is im
me di atel ob vious when the fact is taken into consideration
that scal a r mul tiplication does not alter directi on but merely
multiplies the length
De n i tio n : A u n i t vector is one whose magnitude is uni t
A n y vector A may be looked upon as the product of a unit
vector a in its di rection b y the positive sca ar A its magni
tude
y
.
fi
y
.
l
.
,
.
A
A
a
unit vector a may similarly be written as the product of
A by l /A or as the uotient of A and A
Th e
q
.
8
fi
VE C TOR A N A L YS I S
negative Sign
pre fix ed to a vector
re ve rs e s its di rection b u t leaves its magni tude un changed
Fo r e x ample if A b e a displacement for two feet to t h e right
A gain if the
A is a di s placement for two feet to the left
s t roke A B b e A the stroke B A whi ch is of the same length
as A B b u t whi ch is in the direction from B to A instead of
from A to B will be A A nother ill ustration of the use
of t h e negative S gn ma y b e taken from N ewton s th ird la w
of motion If A denote an action
A will denote t h e
reaction
may be prefi x ed to a vec
Th e positive Sign
tor to call particular at tention to the fact that t h e direction
has not been reversed Th e two signs and when used
in connection wi t h scalar m ul tiplication of vectors foll ow the
same laws of operation as in ordi nary algebra Thes e are
symboli cally
De
n i tio n :
Th e
,
.
,
.
,
,
,
i
.
’
”
“
,
.
”
.
,
.
.
—
Th e
interpretation is ob vious
A ddi tio n
A)
.
.
a nd
Su btra c ti o n
addi tion of t wo vectors or strokes may be treated
mos t simply b y regarding them as de fi n ing translatio n s in
space (Art
et S b e one vector and T the other
et P
be a poin t of space ( Fig
Th e trans
lat ion S carries P into P such that the
li ne P P is equal to S in magni tude and
direction Th e transformation T will then
”
carr P into P
the line P P b eing
parallel to T and equal to it in magnitude
Consequently the result of S follo wed by
Fm 2
T is to carry t h e point P into the point
P
If now Q be any other point in space S will carry Q
in t o Q such that QQ S and T will then carry Q into Q
Th e
L
.
L
.
.
’
’
y
.
’
’
”
.
.
,
”
.
,
’
’
’
”
10
s
VE C TOR A N A L YS IS
q y p
e uentl the oints
r
and P lie at the ve tices of
a parallelogram
Hence
”
is equal and par
P P
allel to P P
Hence S
T fol
carries P into P
lowed b y S therefore car
ries P into P through P
whereas 8 followed b y T
carries P into P thr ough
Th e fi nal result is in
This ma be designated s mbolicall
P, P
’
,
P
’”
”
,
.
”
’
’
.
”
’”
‘
.
’
”
,
”
FI G 3
.
.
either case the same
b y writing
.
y
y
y
is to be noticed that S P P and T P P are the two sides
of the parall elogram P P P P which have the point P as
common origin ; and that R P P is the diagonal dr awn
through P This leads to another ver common wa of
stating the defi nition of the sum of two vectors
If two vecto s be drawn from the same origin and a parall elo
gram be constructed upon them as sides their sum will be that
diagonal which passes through their com mon origin
parallelogram law according to
Thi s is the well-known
whi ch the physical vector quantities force acceleration v o loe
ity and angular velocity are compounded It is important to
note that in case the vectors lie along the same line vector
addi tion b ecomes equivalent to algebraic scalar addi tion Th e
lengths of the t wo vectors to b e added are added if the vectors
have the same direction ; but subtracted if the have Oppo
site directions In either case the sum has the same d rection
as that of the greater vector
After the de fi n ition of the sum of two vecto s has
b een laid down the sum of several ma be found b y adding
t o gether the fi rst two to this sum the third to this the fourth
and so on until all the vectors have been combined into a sin
’
’
It
”
’
’”
”
y
.
y
r
.
,
.
”
“
,
,
.
,
.
y
.
r
.
y
,
,
i
,
,
A DDI TION AND
ALAR
SC
M
P
UL TI LI CA TI ON
11
gle one Th e fi nal result is the same as tha t obtained by placing
th e origin of each succeedin g vector upon the terminus of the
precedin g one a n d then drawing at once the vecto r from
the origin of the fi rst to the term inus of the last In cas e
these two points coincide the vecto s form a closed polygon
and t heir sum is z ero Interpreted geometrically this s t ates
that if a number of dis placements R S T are such that the
strokes R S T form the sides of a closed polygon taken in
order then the effect of carr ing out the displacements is n il
E ach point of space is brought b ack to its starting point
In
t e rp re t e d in mecha nics it state s that if an number of forces
form the sides of a closed polygon
a c t at a point and if the
taken in order then the resulta nt force is z ero and t h e po int
s in equi librium under the action of the forces
Th e order of sequence of the vectors in a sum is of no con
sequence Thi s may b e shown by proving that any two a dja
fect ng the resu t
c e n t vecto s may be interchanged w thout af
To show
.
r
.
.
,
,
,
y
,
,
.
y
y
i
.
,
.
.
r
L et
hen
L et now
conse quently
i
i
T
Then 6 B 0 1)
B O = D
”
”
.
0 D =C
’
.
l
.
p r ll l r
is a a a e og am and
Hence
i
y
wh ch proves the statement Since an two adjacent vectors
may be interchanged and since the sum may be a anged in
an
y order by successive interchanges of adjacent vectors the
order in whi ch the vectors occur in the sum is immaterial
De n i t io n : A vector is said to be subtrac ted when it
s added afte r reve sa of di rection S mbolically
.
rr
,
,
fi
i
By
i
.
rl
A
r i
.
B
A
th s means subt act on is reduce
y
,
B)
.
d to ad
di tion
and needs
VECTOR ANA L YSIS
12
no special consideration There is however an interestin g and
import an t way of represen t ing t h e di fference of two vec t ors
C omplete
geome trically
et A 0A B O B ( Fig
the parallelogram of whi ch A and B
are t h e sides Then the di agonal
O O = C is the sum A + B of t h e
two vectors N e x t complete the
parallelogram of which A and B
O B are the sides
Then the di
agonal OD D will be t h e sum of
B u t the
A and the negative of B
Fro 4
segment O D is parallel and equal
to B A Hence B A ma be taken as the difference to the two
vectors A and B This leads to t h e foll o wing rule Th e di ffer
ence of two vectors whi ch are drawn from the same origin is
the vec t or drawn fro m the terminus of the vecto r to be s u b
tracte d to t h e te rminus of the vector from which it is s u b
tracted Thus the two diagonals of the parallelogram which
is constructed upon A a n d B as sides give the sum and di f
ference of A and B
In the foregoing parag raphs addition sub traction and
scalar multiplication of vectors have been de fi ned and inte r
To make the development of vector algeb ra ma t he
p re t e d
m a t ic a l l y e x act and systematic it wo ul d now become necessary
to de mo ns tra te that these three fundamental operations foll o w
the same formal laws as in the ordi nary scalar algebra a l
though from the standpoint of the physical and geometrical
interpretati on of vectors this may seem super uous These
laws are
.
L
.
.
,
.
.
’
.
.
.
,
y
.
.
,
.
,
.
,
,
.
,
fl
I
:
m
(n
A)
I,
II
Il l a
III ,
III,
’
°
m
(A + B )
m A + m B,
—
A
—
B
.
.
ADDI TI ON A ND
S
CALAR
M
P
UL TI L I CA TI ON
13
is the s o -called law of association and commutation of
the scalar factors in scalar multiplication
Ib is the law of association for vectors in vector addi tion It
states that in addi ng vectors parentheses may be ins erted at
an points without altering t h e result
II is the commuta t ive law of vector addition
III is the di stri b utive law for scalars in scalar multipli
cation
III, is the distri b utive l a w for vecto rs in scalar mul tipli
cat ion
III is the distribu t ive law for the negative sign
Th e proofs of these laws of operation depend upon those
propo sitions in elementary geometry which have to deal wi t h
the fi rst properties of the parallelogra m and si m ilar triangles
They wil l not b e given here ; b u t it is suggested that the
reader work them o u t for the sake of fi x ing the fundamental
ideas of ad di tion sub traction and scalar multiplication more
clearly in mind Th e res ul t of the laws may b e summed up
in the statement
Ia
.
.
y
.
.
,
.
.
.
,
.
,
,
.
Th e
l a ws
wh ic h go ve rn
m u l tip lic a tio n
p
o
e ra
t io n s i n
f ve c to rs
o
a
ddi tio n ,
s u btra c ti o n ,
a nd
sc a
la
r
ide n t i c a l wi th tho s e go ve rn in g th e s e
l a r a lge bra
a re
o rdin a ry s c a
.
is precisely thi s identity of formal laws which justi fi es
the e x tension of the use of the familiar signs
and
of ari t hmetic to the alge b ra of vectors and it is also this
which ensures the correctn ess of results ob ta ined b y operat
ing with t hose signs in t h e usual manner On e caution only
need b e mentioned Scalars and vectors are en t irely di fferent
Fo r t his reason they can never b e equated
sorts of quantit
to each other e x cept perhaps in the trivial case where each is
Fo r the same reason they are not to be added to gether
z ero
So long as t his is borne in mind no difii c ul t y n e ed b e antici
p ated from dealing wi th vectors much as if they were scala rs
Thus fr om equations in which the vecto s enter lin early with
It
0
.
y
.
.
.
.
r
.
VECTOR A NA L YSIS
14
r
y
scalar coeffi cients unknown vecto s ma be el iminated or
found b y solution in the same way and wi th the same limita
tions a s in ordinary algebra ; for the eliminations and solu
tions depend solely on the scalar coe ffi cients of the equations
and not at all on what the variab les represent If for
instance
.
then A
t hree
as
B
,
,
C,
or
D
y
ma be e xpressed in terms
—
1
—
Cl
vector equations such as
An d two
2 A+ 3 B =F
and
yield by the usual processes the solution
s
A= 3 E
B = 3 E
Co mp o n e n ts
—
4 F
Z E
~
.
f Ve c to rs
o
are said to be collinear when
they are parallel to the same line ; coplanar when parallel
to the same plane Two or more vecto rs to which no line
can be dra wn parallel are said to be non -collinear Three or
more vectors to which no plane can be drawn parallel
said to be non -coplanar Obviously any two vecto s are
00 planar
An y vector b collinear with a may be e xpressed as the
product of a and a positive or negative scalar which is the
ratio of the magnitude of b to that of a Th e sign is posit ive
when b and a have the same directi on ; negative when they
have op p osite directions If then 0A a the vector r drawn
Def
in itio n
:
Vectors
,
.
.
r
.
.
.
,
.
,
ADDI TI ON
AN D S
y
CALAR
M
P
UL TI LI CA TI ON
from the origin 0 to an point of the lin e
either direction is
r
x
a
OA
15
produced in
1
( )
.
i
be a variable scalar parameter th s equation may there
fore b e regarded as the ( vector) equation of all points in the
line O A
et now B b e any point not
upon the line 0A or that lin e produced
in either direction ( Fig
et O B b Th e vector b is surely
not of t h e form se a D raw thr ough B
F c 5
a line pa ra llel to 0A and let R be a n y
point upon it Th e vector B B is collinear with a and is
consequentl y e xp essible as m a
Hence the vector drawn
from O to R is
If x
L
.
L
.
.
.
r
r
.
.
.
.
0R = OB + B R
r
=b +
x a
This
equation may be regarded as the ( vector) equation of
all the points in the line whi ch is parallel to a and of which
B is one point
14 ]
An y vector r coplanar with two non -collinear vecto rs
a and b may be resolved into two components parallel to a
and b respectively This resolution may
b e accomplished b y construc t ing the pa r
6
of
wh
i
ch
the
sides
are
a ll e l o ra m ( Fi
)
g
g
parallel to a and b and of which the di
agonal is r O f these components one is
m a ; the other y b
5c and y are re s p e c
t ive ly the scalar ratios ( taken with the
proper sign) of the lengt hs of these components to the lengths
of a and b Hence
r x a yb
.
.
.
.
.
,
.
y
is a typical form for an vector 00plan ar with a and b If
several vecto s r 1 r 2 r 3
may b e e xpressed in this form as
r
,
,
.
VECTOR A NA L YSIS
16
their sum r is then
Th
is
the well -kno wn theorem that the compon ents of a
sum of vectors are the sums of the components of those
vecto rs If the vector r is z ero each of its components must
th e
one vector e uation r O is
Conse uentl
b e z ero
eq u ivalent to the two scalar equations
is
q
.
.
y
q
t
= 0
3
.
y
vector r in space ma be resolved into three
components para llel to any t hree given non -coplanar vec t o rs
et the vectors b e a b
and c Th e resoluti on
may then b e a c c o m
p li s h e d b y constructing
the parallelopiped ( Fig
7 ) of which t h e edges
are parallel to a b and
c and of which the di
agonal is r Thi s par
a l l e l o p ip e d m a y b e
>
a
i
drawn
easily
b
pass
ng
F c 7
y
three planes parallel re
s p e c t ive l y to a and b b and c c and a through the origin 0
of t h e vector r ; and a similar set of t hree planes through its
t erminus B
These s ix planes w
then b e parallel in pairs
An y
L
.
,
,
.
.
,
\
.
r
.
.
,
.
,
ill
,
VE C TOR
18
A
= fl,
r
x a
r
NAL YSIS
+ gb +
z
q
’
r
cc
9:
'
g
,
y
z
z
’
,
—
’
—
cc
x
O,
z
y
—
y
’
z
w
.
fl
b
O,
z
’
- z
z
w
.
z
this wo ul d not be true if a b and 0 were coplanar In
t hat case one of the th ree vectors could be e xpressed in terms
of the other two as
B
ut
=m
c
sc a
’
n
r
a
+ yb +
’
+ y b +
’
[(
z
i
+
h
n
c
z
"
b,
c
)
[a w e
m
x
a
.
,
,
’
(
(m
z
r
c
—
a,
i
ss b = 0
r
Hence the indiv dual componen t s of
a and b ( supposed difi e re n t ) are z ero
m + m e
x + m z
ence
H
i i
.
in the d rect ons
’
r
.
’
y
’
l
this b no means necessitates cc y z to be equa re s p e c
In a similar manner if a and b were col
t ive ly to x y z
linear it is impossible to infer that their coe ffi cients vanish
in d viduall
Th e theorem ma perhaps be stated as follows :
B
ut
,
’
,
i
'
,
.
y
y
.
In
or
'
t wo
c a se
two
e qu a
co rre sp o n di n g s c a
y t ru e
c op
la
na r
( Ar ts
.
l
non-collinear
c e s s a r il
18
la
r
ve c to rs a re e xp re s s e d
ve c to rs ,
fi
coe
h
e
w
o
t
t
f
i
This
.
,
or
c ie n t s
ve c to rs
th re e
a re
be
i n te rm s
f
o
non-coplanar
e qu a
l
B
.
c o lli n e a r
ut
ve c to rs ,
this is
th e
not n e
th e th re e ve c to rs ,
or
;
o n e ve c to r ,
principle will be used in the appli cations
et
Th e Th re e Un i t Ve c to rs i,
j
,
k
.
the foregoing paragraphs the method of e xpress
ing vecto rs in term s of three given non-coplanar ones has been
explained Th e simplest set of three such vecto s is the rect
In
.
r
recta ngular syst em may however be either of t wo very distinct
1
—
t pes In one case ( Fig 8 fi rst part) the Z axis lies upon
that side of the X Y plane on which rotati on through a right
angle from the X-axi s to the Y-ax is appears c o u n te rc l o c kwis e
or p o si t ive a ccordin g to the convention adopted in Trigo n o m e
If the X
tr
Thi s relation may be stated in another form
ax is be di rected to the right and the Y-ax is vertically the
ax is will be d irected toward the observer O r if th e X
Zaxis point toward the observer and the Y-ax is to the right
St
the Z -axis will point upwar
another method of state
y
.
.
y
,
.
.
,
.
d
ill
.
,
L ft h
- a nde d
e
FIG 8
.
l y
.
ment is common in mathematica ph sics and engineering If
a right -handed s c re w b e turn ed from th e X-ax is to the Y
axis it will advance along the ( p o s i tive ) Z —a sc is Such a s s
tem of ax es is call ed right-ha nded positive or c o u n t e rc l o c k
2
wise
It is easy to see that the Y ax is li es upon that side of
the Z X-plane on which rotation from the Z -ax is to the X
ax s is coun tercl o ckw se ; and the X-ax is upon that side of
t
Th X Y
B y th X Y
Z — x i t h p it iv h lf ft h t x i i
t h pl
wh i h
pl
th X
t h pl
0
t i
d Y- x i i
.
y
.
,
,
-
.
i
1
i
m i ig d d y m
i
l y
il bl
i
m
b fi fi g
d g
ig
d
m
b
fig
d
f m
p l m p di l
if
dfi g
b
d
p lm
ig
g
l
fi fi g
ig
d d y m f m d b
fig
k
d
m
b fi fi g
dfi g
ea ns
A
o fthe
fi
rs t
e
se c o n
,
er
be
n
er
s
n
n
s t re t c
he
be
er
.
s
c
h t-h a n
rs t
or
e
os
e
con a ns
e
e
s
o ut
ro
e
wa r
y
th e
t he
the
n
a
fin
wh
e rs
c
e r o ft he r
a
a
ta
t
en
r
en
a
h is
a
.
c
.
an
in t h e
ea n
s a va
to
a
.
e c o ns s t s
Ift h e t h u
.
to e a c h
es
e
.
a ne 2
e
,
wa
h t ha n
cu a r
ht
s
s
s,
a
an
pe r
a
a
o
a nd o ne
st e
e r, a n d se c o n
e n t o ve r t o
is
s te
a
a ne
e
c o n ve n e n t r
thu
n
se c o n
han
or
e
a ne
3
m
,
o
rs t
o r e r th u
a nd
the
t h e r, a n d
n
,
e r, a r
rs t
n
ht
e r,
VEC TOR A NAL YSIS
20
the YZ -plane on which rotation from the Y axis to the Z
a xis is counterclockwise Thus it appears that t h e relation
between the three ax es is perfec t ly symmetri cal so long as the
same cyclic order X YZ X Y is o b served If a righ t -handed
screw is turned from one axis toward the ne xt it advances
along the third
In t h e other case ( Fig 8 second part ) the Z -a x is lies upon
that side of the X Y-plane on which rotation through a right
angle from the X-axis to the Y-axis a ppears c lo c kwis e or n e g
a ti ve
Th e Y-a xis then lies upon that side of the Z X plane
on whi ch rotation from the Z -a xis to the X a x is appears
clockwise and a similar s t a tement may be made concerning
the X-axis in its relation to the YZ —plane In this case too
the relation between the three a xes is symm etrical s o long
as the same cyclic order X YZ X Y is preserved b u t it is just
the Opposite of that in the former case Ifa left-handed screw
is t urned from one a xis toward the nex t it advances along
the third Hence this system is cal l ed left-handed negative
1
or clockwise
Th e two sys t ems are not superposable
They are sym
metric O n e is t h e image of the other as seen in a
mirror If t h e X and Y-ax es of the two difi e re n t syste m s be
superimposed the Z -ax es will point in opposite di rections
Thus one system may be obtained from the other by reversing
the direction of o n e of the ax es A little thought will show
that if two of the ax es be reversed in direction the system will
not b e altered but if all th re e b e so reve sed it will b e
Whi ch of the two syste m s be used m a t t e rs little B u t in
a s m u c h as the formul ae of geometr
and
mech
a nics dif
f
er
y
slightly in the matter of sign it is advisab le to settle once for
all which shall be adopte d In th s book the right-handed or
counterc lock wise system will be invariab ly employed
-
.
.
.
,
.
-
.
-
.
,
,
.
,
.
,
.
.
.
.
,
.
.
r
,
.
,
,
.
1
o ne
m
mg
le ft-h a n d e d s ys t e
wa s fo r e d b y t h e ri
A
ma y b e
ht
.
m
.
i
fo r e d by t h e l e ft
.
ha n d
j us t a s
i
a r
g
h t -ha n d e d
fi
ADDI TI ON A ND
S
CALAR
j
L TI P LI CA TI ON
21
MU
three letters i k will be reserved to de
note three vectors o f un it length drawn respe ctively in the
directions of the X Y and Z ax e s of a right-handed re c ta n
gular system
In terms of these vectors any vecto r ma be e x pressed as
De
Th e
n i ti o n
,
,
y
.
,
6
( )
coe ffi cients x y z are the ordi nary Cartesian coordinates
of the term inus of r if its origin be situated at the origin of
co ordinates Th e components of r parallel to the X Y a n d
Z -a x es are respec t ively
Th e
,
,
.
t
a
y
fi
z
k
.
j
rotations about i fro m jto k about from k to i and
about k from i to are all pos itive
B y means of these vectors i
k such a correspondence is
established between vector analysis and the analysis in Ca r
t e s ia n co ordi nates that it becomes possible to pass a t will
from either one to the other Th ere is nothi ng c o n t ra dic
tor between them O n the contrar it is often desirab le
or even neces sar to translate t h e fo rm ul m ob tained b y
vector methods into Cartesian co ordinates for the sake of
comparing t hem with results a lready kno wn and it is
still more frequently convenient to pass from Cartesian
analysis to vectors both on accoun t of the b revit y thereby
obta ined and b ecause the vector e xpressions show forth the
intrinsic meaning of the fo rmul ae
Th e
j
,
,
y
j
.
,
y
.
y
,
.
.
Ap p lic a tio n s
Problems in plane geometry may frequently be solved
easily b y vector methods A n y two no n -c ollinear vecto s in
the plane may be taken as the fundamen t al ones in terms of
which all others in that plan e may be e x pressed Th e origin
may also be selected at pleasure O ften it is possib le t o
r
.
.
.
VE C TOR A NAL YSIS
22
make such an advanta geous choice of the origin and fun da
mental vectors that the analytic work of solution is ma teriall y
simplifi ed Th e adapta b ility of the vector me thod s about
the same as that of ob lique Cartesian co ord nate s with difi e r
x es
upon
the
two
a
en
a
es
c
s
l
t
Th e line which joins one vert e x of a parallelo
E xa mfi dg
;
d
l
r
a
m
mid
e
point
of
an
opposite
side
the
diag
fi
f
h
e
t
n
s
e
t
s
c
g
onal (Fig
et A B OD be the parallelogram B E the lin e joining t h e
verte x B to the middle point E of the side
A D R the point in which this line cuts the
diagonal A C
To show A B is one th rd of
A O Ch o o s e A as origin A B and A D as t h e
F G 9
two fun damenta l vectors 8 and T Then
A O is the sum of S and T Le t E
B ) To show
i
.
i
.
.
’
’
L
’
'
.
,
i
,
.
.
I
.
,
.
.
M
/
.
.
l
where
a;
is
the rati o of E R to
EB
R= y
S
(
An d
’
where y
r
—
an
l
unkn own sca ar
T),
is the scalar atio of A g to
AO
q l
to be sho wn e ua
to
Hence
;
T+
n
(S
Hence e quating correspondi g coef cient
,
n
fi
s
.
r
A
( t
.
ADDI TI ON
From
AN D S
CALAR
M UL TIP LI CA TI ON
23
which
i
i
as a: s also 1 the line E B must be t r secte d a s
well as the diagon al A O
E
aea mp le
If
through
an
p
o int within a triangle lines
D
?
<
be drawn parallel to th e sides the sum of the ratios of these
lines to their correspo ndi ng S1 de s is 2
et A B C be the trian gle R the point within it Choos e
A as origin A B and A O as the two fundamenta l vectors 8
an d T
et
Inasmuch
y
.
‘
a
/
fl
L
.
’
,
.
L
.
,
( )
a
i
m S is
y
A
the fract on of A B which is cut o ff b the line th rough
R parallel to A O Th e remainder of A B mus t be the frac
t on 1 m 8 Conse uentl b simi ar t ian gles the ratio of
the l ine parallel to A O to the line A C itself is ( 1 m )
Similarly the ratio of the line parall el to A B to the l ine A B
itse f is ( 1 n ) N e x t express B in te rms of S and T S the
th rd side of the tri angle Evidently from ( a )
i
.
.
q
yy l r
’
.
l
i
.
.
.
—
S)
.
y
Hence ( m n ) S is the fraction of A B whi ch is cut o ffb the
line through R parallel to B 0 Conse uently by similar tri
angles the ratio o f th s n e to B O itse f is ( m n ) A dding
the three a t ios
i li
r
l
.
q
.
(1
and the the o rem
proved
E xa mp le 3
If from any poin t withi n a p a ra ll e l o gra m lines
be drawn parallel to th e sides the diagonals of the pa allelo
grams thus formed intersect upon the di agonal of the given
parallelogram
et A B CD be a parallelogram R a o int within it KM
an L N two lines
to A B and
e l resp ect vel
f
F
1s
.
-f
‘
r
,
L
d
.
’
,
p
i y
,
VE C TOR A NAL YSIS
24
the point s
A D,
D
B O, C_
re s
pe
c
ti
l lelograms KE N D a n d
of the two p araon A C T
Ch o o s e A as origin A B and E as the two funda
mental vectors 8 and T
et
LM
fl
a
v
or
Ann
L
.
c
p
c
w
4
.
R
flu
..
I
3
.
,
AR
m S
n
T,
and let P be the point of inte rsection of KN wi th L M
Then
— n
P =A P =A
LM =
K+
x
.
)T
,
KM
I
(
P = A P = A L + y LM
E
quat ng
i
c o e fii c ie n t s ,
n
By
soluti on
,
+
se
l
(
n
so
m +
n —
— n
)
-r
1
m
m +
n
—
1
either of these solutions in the e x pression
m +
n
which shows that P is collinear with A O
Problems in three dimensional geometr ma be
solved in essentially the same manner as those in two dim e n
sions In this case there are three fundamental vec t ors in
terms of whi ch all others can be e xp essed Th e method of
solution is ana logous to that in the simpler case Two
y y
.
.
r
.
.
VE C TOR ANA LYSIS
26
manner
’
A B
z
x
0+ y
/2 D
2
— B
)
.
— B
)
O = 1 + k2
x
2
(l
= k 2 m,
=
k
y2
Henc
In
k,
e
P B
’
B B
’
n
1
.
l
—
kr
l
a
i y
the same wa y t ma be shown tha t
P C
'
"
c o
”
D
d
DD
ri
Addi ng t h e
’
F
‘
i
four at os the res ul t s
1
r
i
a lin e which passes th ough a g ve n
point and cuts two given lines in space
et the two lines be fi x ed respec t ively by two poin ts A
and B O and D on each
et 0 be the g ven point Choose
it as orig n and let
E xa mp le 2
To fi n d
:
L
.
,
i
.
A = OA ,
An y
An y
po int P
o fA
po int Q of
If th e
points
a re c o llin e a r
.
.
P
B
CD
L
B = OB
y
i
,
.
D = 0D
d
.
ma be expresse as
—
A)
—
O)
i
may likew se be written
and Q li e in the
That is
P =
z
sa
i
r
me l n e th ough
Q
.
0,
d
P an Q
A DDI TI ON AND
ALAR
SC
M
P
UL TI LI CA TI ON
27
efore it is possib le to equate coeffi cients one of the
vectors must be expressed in te rms of the other three
et
Then
P = A + ze ( B — A)
B
L
.
Henc
l
e
—
x
ac
H enc
y l,
m
y ,
z
z
m
e
a
:
l + m
9
1
ituting in P and
Subst
Q
ither of these may be taken as de ing a
fin
E
and cutting
AB
and
dr
a
wn
OD
’
.
Vec to r Re la tio ns i n dep e n de n t of th e Origin
20 ]
m
n
.
Choose
y
divide a line
To
E xa mp le 1
F
i
( g
L
0
.
P = 0P = 0A +
That
is
,
i
in a g ven
.
i
r
AB
ry
an arbit ar po int 0 as
origin
et OA A and O B B
To fi n d the vecto r P
O P of which
the terminus P d vides A B in the
ati o m n
.
from
m
m +
A B =A+
Fro
.
10
.
O
VE C TOR ANA L YSIS
28
r
component s of P parallel to A and B a re in inve se ratio
t o the segm ents A P and P B into which the line A B is
di vided b y t h e point P If it should so happen t hat P divided
t h e line A B e x ternally the ratio A P / P B would be nega
t ive and t h e signs of m and n would be oppos ite b u t t h e
formula would hold without change if this difference of sign
in m and n be taken into account
To fi n d the point of intersection of the me di ans
E xa mp le 2
of a triangle
et A B C be the given
Choose the origin 0 at ran dom
et OA A O B B and O C G
et
t riangle
C
b e respectively the middle po ints of t h e sides opposi t e t h e
vertices A B 0
et M be the point of intersection of the
medians and M OM the vec t o r drawn t o it Then
Th e
.
,
,
,
.
.
L
.
,
.
.
,
,
,
L
L
’
L
.
.
(B
—
A)
—
B
2
(C
)
2
Assuming
that 0 has been chosen outside of the plane of the
triangle so that A B C are non-COp l a n a r corresponding c o e ffi
eleu t s ma be e uated
y q
,
,
,
.
x
= l — y,
Hence
Hence
5
(A
B
C)
.
A DDI TI ON AND
ALAR
M
SC
P
UL TI LI CA TI ON
29
vector drawn to the median point of a triangle is equal
to one third of the sum of the vecto rs drawn to t he vertices
In the pro b lems of which t h e solution has j u st b een given
the orig in co ul d b e chosen arb itrarily and the result is in
dependent o f that choice Hence it is even possible to dis re
gard the origin entirely and replace the vectors A B 0 etc
Thus the points themselves
b y their termin i A B C etc
become the subjects of analysis and the formul ae read
Th e
.
.
,
,
,
,
,
.
,
.
,
n
A + m B
m +
n
M
y
This
y
is t pical of a whole class of problems soluble b vector
methods In fact a n y p u re ly ge o m e tri c relati on b etween t h e
difi e re n t parts of a fi gure must neces sarily be independent
of the origin assumed for the analytic demonstration In
some cases such as those in Ar t s 1 8 1 9 the position of th e
origin may be specializ ed with regard to some crucial point
of the fi gure so as to facilitate the computation ; b u t in many
other cases the generality obtained b y leaving the ori gin u n
Speciali z ed and undeterm ined leads to a s y mm e try w hi ch
renders the results just as eas y to compute and more easy
to rememb er
Th e o rem : Th e necessar and su f
fi cient condition that a
vector equation represent a relation independent of the origin
is that the sum of the scalar coeffi cients of the vectors on
one side of the sign of equalit is equal to the sum of the
coe ffi cients of the vectors upon the other side Or if all the
terms of a vector equation b e tra nsposed to one side leaving
z ero on the other
the sum of the scalar coe ffi cient s must
be z ero
et the e uation wri tte n in the latte r form he
.
.
,
.
,
,
y
.
y
.
,
L
.
q
0
.
VE C TOR ANA L YSIS
30
C ha nge
the origin from O to O b y addi ng a constan t vector
Th e equation
R
O O to each of the vectors A B C D
then becomes
’
’
,
,
,
o
this is to be independent of the origin the coeffi cient of
mus t vanish Hence
If
B
.
That
thi s condition is ful fi lled in the two e xamples cited
is obvious
.
If
m
necessary and su ffi cient condition that two
vectors sa t isfy an equation in which the sum of the scalar
coeffi cients i s z ero is that the vectors be equal in magnitude
and in direction
Th e
,
,
.
Firs t l e t
and
a
A+ bB = O
a
+ b
0
z
.
is of course assumed that not both the coeffi cients a and b
vanish If they did the equation would mean nothing Su b
s t it u t e the value of a obtained from the second e uation into
It
q
.
t h e fi rs t
Hence
.
A
B
.
.
A DDI TI ON
Secon dl
q
y if
A
an
ALA R
q
31
i
are e ual in magni tude and d recti on
d B
e uati on
th e
M UL TIP LI CA TI ON
AN D SC
A
0
B
subsists between them Th e sum of the coeffi cient s is z ero
fi cient condition tha t th ree vecto s
Th e necessar and s u f
sa tisfy an e uation in wh ich the sum of the scalar coeffi cients
is z ero is that when drawn from a common origin the te rmi
)
nate in th e same s t raight line
q
y
r
.
.
y
,
,
E mH %
a
A + bB +
md
a
+ b +
C= 0
c
c
=fi
i
q
dq
all the coeffi cients a b c van sh or the e uations
wo ul d b e mean ingless
et c be a non-vanis hing coe ffi cient
Substitute the value of a obta ned from the secon e uation
No t
L
.
in t o t h e fi rs t
,
,
,
i
.
.
c
(C
—
i
B
i
)
.
d
Hence the vector wh ch jo ns the e xtremiti es of C an A is
collinear w th that which joins the e x tremities of A and B
Hence those three points A B C lie on a line Secondl y
suppose three vectors A O A B O B C 0 0 drawn from
the same origin 0 termin ate in a stra ight line Then the
vecto s
i
.
’
,
.
,
,
r
are co llinear
i
,
.
A B =B
.
—
A
a nd
q
A O= C— A
Hence the e uati on
id i
subs sts Th e sum of the coeffi cients on the t wo s es s
t h e same
Th e necessar and su f
fi cient condition that an e uati on
in which the sum of the scalar coe ffi cients s z ero subs ist
.
.
1
Ha
m
y
mm igi
V e c t o rs wh i c h ha ve a c o
”
“
ilt o n l a min a c o lline a r
.
on o r
n a nd
mi
te r
na te
q
i
in
o ne
,
,
lin e
a re c a
ll e d by
VECTOR
32
A
NAL YSIS
between four vectors is that if drawn from a common
1
they terminate in one plane
,
.
Firs t l e t
and
L et
of a
be a non-van ishi ng coe ffi cient Sub sti t u te
obtained from the last equation into the fi rst
d
t he
.
.
(D
d
—
C)
.
line A D is coplanar with A B and A 0 Hence all four
termini A B C D of A B C D li e in one pla n e Secondl
suppose that the termini of A B C D do lie in one plane
— A an d A B
Then A D = D
B — A a re c c
planar vectors O n e o f them may b e ex pressed in terms of
the other two This leads to the equation
Th e
y
.
,
,
,
,
,
,
.
,
,
.
,
z
,
.
.
l
where
—
(B
-
A)
0,
and n are certain scalars Th e sum of the c o e ffi
c ie n t s in th s equation is z ero
B etween any fi ve vectors there e x ists one equation the sum
of whose coeffi cients is z ero
et A B C D E be the fi ve given vectors Form the
differences
l, m ,
L
i
.
.
.
,
,
,
.
,
E _ A,
_B,
E
E _ C,
E
—
D
.
of these may be e x pressed in terms of the other three
— o r what amounts to the same thing there must e xist an
equation be t ween them
On e
.
70 ( 13
Th e
1
by
—
11 ) + l (n
—
B)+ m
(E
q
i
sum of the coeffi cients of th is e uation s z ero
m
Ve c t o rs wh ic h ha ve a c o
H a ilto n te rmin o —
c o mp la
mm i gi
on or
n a r.
"
n a nd
m
te r in a t e
in
o ne
.
p la n e
a re
ca
lle d
VECTOR A NAL YSIS
34
r
However the points E C and A lie upon the same st aight
line Hence the equation which connects the vectors E C
and A must be such that the sum of its coe ffi cient s is z ero
n
This determines x as 1
,
,
,
.
.
.
Hence
By
E
— n
C= D—
n
B =
(1
— n
)A
.
another rearrangement and similar reasoning
Subt
ract the
fi rst
equation from the second
—
This
n
)A
.
vector cuts B C and A G It must therefore b e a
multiple of F and such a multiple that the sum of the c o e ffi
c ie n t s of the equat ons whi ch conn ect B C and F o r G A
and F shall b e z ero
.
i
,
,
,
,
.
Hence
B + C
Hence
2
d
and the theorem has been proved Th e proof h a s covere
considera b le space because each detail of the reasoning has
In realit however the actual anal sis h a s c o n
b een given
sisted of just four e uations ob tained simply from the fi rst
E xa mp le 2 : To determine the equations of the line and
plane
et the line be fi xed b y two points A and B upon it
et
P b e an
poin t of the line Choose an arbitrar origin
Th e vecto s A B and P terminate in the same ne
enc e
.
L
y
q
.
y
,
,
.
.
y
r
,
,
a
Therefo
r
li
.
A + bB + p P = 0
=
a + b +
Q
p
a
A + bB
e
a
b
y
.
.
H
L
.
i
A DDI TI ON
ALA R
T ON
35
M UL TIP LI CA I
AN D SC
d fferent point s P the scalars a a n d b have different
values They ma be replaced by x and y which are used
more generall to represent variables Then
Fo r
y
y
.
,
.
x
A+ yB
+ y
x
L et a plane determined by th ee poin s
and
L et be any point of the pla e ho se an arbitrary origin
vecto rs
P t rmi ate in one p lane Hence
be
P
n
Th e
A, B , C,
an
d
e
a
As
a
,
b,
customa
ry
c,
0
,
o
n
.
.
.
A + bB +
+ b +
a
y
C
.
A, B
t
r
C
c
c
i
var for di fferent point s of the plane it s more
to write in th eir ste ad x y z
,
,
x
,
A+ yB +
x
+ y +
.
z
C
z
line whi ch joins one ver te x of a com
i
quadrilateral
to
the
ntersection
of
two
d
agonals
l
t
e
e
p
divides the opposite sides har
E xa mp le 3
Th e
:
i
i
F
( g
A , B , O, D
m o n ic a ll y
L et
be four vertices
of a quadrilat ral L et
meet
.
e
AB
.
in a fi fth verte x E and A D
meet B C in the six th vertex F 5
et the t wo di agonals A C and
F G 12
To show
B D inte rsect in 0
that F G intersects A B in a poin t E and CD in a point E
such that the lines A B and CD are divi ded inte rnall at
E and E in the same ratio as they are d vided e xte rna y
b E That is to show that the cross rat os
CD
,
L
.
I
.
.
.
’
’
y
”
'
i
.
(A B
EE
’
)
( OD E E
o
”
)
y
i
—
r
.
”
ll
36
VE C TOR
ii
Choos e
A
the or g n at random
drawn from it to t h e points
plane Henc e
NAL YS S
Th e
.
A, B
r
r
fou vecto s A B
O D terminate in
,
,
,
,
C, D
o ne
.
Separate
q i
y
the e uat ons b transpos ing two terms
d
A+
C=
+
a
a
Di vi de
c
c
A+
bB + dD
C
b + d
In lik e m a n n e r
F
d
A + dD
a
(
(
Separate
bB +
c
b +
c
+ d
+ )G
a
a
+
c
>
(
+ )G
a
c
c
c —
q
d
+
a
i
C
c
C
C — dD
c
e
(
c
—
)
dD
— d
the e uations again and d vide
a
Hence
c
A + bB
a
i
c
C + dD
+ b
c
+ d
d vides A B in the ratio a : b and CD in
”
divides CD n the
c
d
B u t e uation ( a ) shows that E
Hence E and E divide OD internall and
c : d
ra tio
e x ternally in the same ratio Whi ch of the two d v sions is
inte rnal and which ex ternal depends upon the relat ve signs
of c and d If the have the same sign the internal point
of division is E ; if opposite sign s it s E In a similar wa
E and E may b e shown to div de A B harmonically
Exa mp le 4 : To di scuss geometric nets
B y a geometric net in a plane is meant a fi gu e composed
of points and straight lines obtained in the follow ng manner
Start w th a certain number of oin t s all of wh ch lie in one
E
q
.
”
ii
i
.
.
.
y
i
’
,
i
y
”
.
.
.
i
i
y
p
i
r
i
.
A DDI TI ON A ND
ALA R
M
SC
P
UL TI LI CA TI ON
37
plane Draw all the lines joining these points in pairs
These lines will intersect each other in a number of points
N e x t draw all the lines whi ch connect these points in pairs
Thi s second set of lines wil l determine a still greate r number
of points which may in turn be joined in pairs and so on
At each step
Th e co n struction m a y be kept up inde fi ni tel
the numbe r of points and li nes in the fi gure increases
Prob ab ly the mos t interesting case of a plane geometric net is
that in which fo u r poin t s are given to commence with
oining these there are s ix lines whi ch inte rsect in three
J
point s d fferent from the given four Three new lines ma
now be drawn in the fi gure Thes e cut out s ix new points
From thes e mo re lines may be obtained and so on
To trea t th s net an a ytically w te down t h e e uat ons
.
.
.
.
y
.
.
.
.
i
y
.
.
i
.
l
ri
q i
.
and
which subsist be tween the fo u r vectors drawn from an unde
t e rm in e d origin to the fo u r given points
From these t is
poss ible to obta n
i
a
A + bB
a
a
a
c
+
c
C= dD
c
C
9
+ d
bB + dD
b + d
A + dD
a
c
+ b
A+
a
li
i
.
bB +
c
b +
c
+ d
C
sp tting the equations into tw o parts and dividin g N ext
four vectors such as A D E F ma be chosen and the e ua
tion the sum of whose coe ffi cient s is z ero ma y be determ ned
This would be
by
,
i q
,
,
treating th s e uati on as ( c )
be obtained
By
.
y
wa s t
q
i
.
reated
ne
w
.
p int may
o
s
VECTOR ANAL YSIS
38
—d A + dD
_
2
+ d
a
a
+ b +
b
+
d
dD+
a
c
9
c
+d
e
+ b) E
(a
+ b + d
E quations
between other se t s of four vecto rs selected from
A B C D E F G may be found ; and from these more points
obt ained Th e process of fi nding more points goes forward
inde fi nitely A ful ler account of geometric nets ma y be
found in Hamilton s E le m e n ts of Qu a te rn io n s B ook I
As regards geometric nets in space jus t a word may be
said Five points are given From these new points may be
obta ined by fi n di ng the intersections of planes passed thr ough
sets of three of t h e given points w th lin es connecting the
remaining pairs Th e cons truction may then be carried for
ward with the points thus ob tain ed Th e anal tic treatment
is similar to th at in the case of plane nets There are
fi ve vectors drawn from an undetermined origin to the given
fi ve points
B etween these vecto s there e x ists an equation
the sum of whose coe ffi cients s z ero This equation may be
separated into parts as before and the new point s ma thus
be ob tained
,
,
,
,
,
,
.
.
’
”
,
.
.
.
i
y
.
.
.
r
i
.
y
.
.
and
a
A + bB
a
a
+ b
A+
a
c
c
C + dD +
c
C
+ d +
b + d +
E
e
E
e
bB + dD +
+ b
e
e
are two of the poin t s and others may be found in the same
wa y
N ets in sp a ce are als o s cus sed by Hami lton lo c c it
.
di
,
.
.
ADDI TI ON AND
S
CA LAR
Ce n te rs
Th e
M
P
UL TI L I CA TI ON
39
G
r a vi ty
f
O
y
f gr a vi ty of a s stem of particles may
c e n te r o
foun d very easily b y vec t or methods Th e two laws of
ph sics which will b e assumed are the following :
Th e center of gravity of t wo mas ses ( considered as
situated at points ) lies on the lin e connectin g the two mas ses
and divides it into two segment s which a re inversel pro
portional to the masses at the ex trem iti es
In fi n ding the center of gravity of two systems of
masses each system ma b e replaced by a sin gle m a ss equal
in magnitude to the sum of the masses in the system and
situated at the center of gravity of the system
G iven two masses a and b situated at two points A and B
Their cente r of gravit G is given b
be
y
.
y
.
y
.
y
y
.
A+ bB
a
9
+ b
a
y
where the vecto rs are referred to an origin whatsoever
This follows imme di ately from law 1 and the formula ( 7 )
for division of a lin e in a given ratio
Th e center of gravit of three masses a b c situated at the
three points A B C may be found b y means of la w 2 Th e
masses a and b ma be considered as equivalent to a single
mass a b situated at th e point
y
,
,
.
,
y
A + bB
+ b
a
G
(a
a
Hence
,
.
a
Then
.
a
+ b +
A+ bB +
a
b
c
c
c
C
40
E videntl
a
,
be
b,
c,
y
VE C TOR ANAL YSIS
th e
d,
found in
a
y
cente r of gravit of any number of masse s
si tuated at the points A B C D
si milar manner Th e esult is
r
.
,
,
,
,
r iy
lines which join the center of g a v t of a
triangle to the vert ces divide it into three tri angles whi ch
a re proportion a l to the masses at t h e o p
F
i
i
vertices
et
A
B
s
t
e
C
o
g
p
(
be the vertices of a tri angle weighted
with masses a b c
et C be the cen
ter of gravi t
oin A B C t o G and
J
Fro 1 3
produce the lines until they in te rsect
the opposite sides in A B C res ectivel
To show th at
the areas
Th e
Th e o re m 1
i
L
.
y
.
.
’
’
last proporti on between A B
from compounding the fi rst three
the demonstration
Th e
,
L
.
,
,
.
p
’
,
,
,
,
y
,
.
c comes
and a
b
It is however us eful n
C
,
.
i
,
.
Henc
In
AB C
A A
’
CB C
G A
’
AB C
e
A C
’
G A
a
+ b+
G B C
a s i mi lar man ner
CA
’
b +
c
a
c
a
B CA
a
G CA
CA B
r
+
CA
’
GAB
+ b +
c
6
a
+ b +
c
c
Hence the p oportion is proved
The o re m 2 : Th e lines wh ch join the center of gravi t of
a te t rahedron to the vertices d vide the tetrahedron into four
i
.
i
y
VECTOR ANA L YSIS
42
may therefore be looked upon as co ordi nate s of the
points P ins ide of the triangle A B C To each set there
corresponds a de fi nite point P and t o each po in t P t here
corresponds an in fi nite n umber of sets of quantities which
however do not di ffer from one another e xcept for a factor
of proportionality
To obtain the points P of the plane A B C which lie o u ts ide
of the triangle A B C one may resort to the conception of
negative weights or masses Th e cente r of gravity of the
masses 2 and 1 situated at t h e points A and B respectively
would b e a point G dividi ng the li ne A B e xte r n a lly in the
ratio 1 2 That is
a
,
b,
'
c
.
,
,
.
.
.
y
y
point of t h e line A B produced ma be represented b
a suitab le set of mas ses a b w hi ch difi e r in sign Similarly
any point P of the plane A B C may be represented by a
s u itab le set of m a sses a b c of which one will di ffer in sign
from the other two if the point P lies outs ide of the triangle
Inasmuch as only the ratios of a b and c are im
AB C
portant t wo of the quantities may always be ta k en positi ve
Th e idea of employi ng the masses situated at the vertices
as coordinates of t h e cente r of gravity is due to M ob ius and
was publis hed b y him in his book entitled B a ry c e n tris c h e
Ca lc i tl
in 1 8 2 6 This may be fairl regarded as the start ing
point of modern analyt ic geometr
Th e concept ion of negative masses which have no e xistence
in nature may be avoided b replacing t h e masses at the
vertices b y the areas of the triangles G B C G CA and
G A B to whi ch they are proportional
Th e co ordi nates of
a poin t P would then b e three numbe s proportional to the
areas of the three triangles of which P is the common ver t e x ;
and the sides of a given triangle A B C the bases Th e sign
of these areas is determined b the follo wing de fi nition
An y
,
,
.
,
,
.
,
.
”
,
y
.
“
y
.
y
,
,
r
.
y
,
.
.
fi
A DDI TI ON AND
S CA
LAR
M
P
UL TI LI CA TI ON
43
area A B C of a tri angle is said to be
positive when the vertices A B C fol low each other in the
positi ve or counterclockwise di recti on upon the circle de
scri bed thr ough them Th e area is said to be negative when
t h e poin t s foll ow in th e negative or clockwise direction
Cyclic permutation of the lette s t herefore does not alter
the sign of the area
De
Th e
n i tio n :
,
,
.
r
.
.
A B C = B CA = C A B
.
rl
Interchan ge
of two letters which amounts to a reve sa of
the c cli c order changes the sign
y
.
A CB = B A C = CB A
IfP be
an
y
—
AB C
.
point withi n the triangle the equation
P A B + P B C + P CA = A B C
mus t hold Th e same will also hold if P be outside of the
triangle provided the s ign s of the are as be t aken into con
sideration Th e areas or th ree uantities proportiona l to
them ma be regarded a s coo rdi nates of the point P
Th e e x tension of the idea of
ba ry c e n tric
co ordin ates to
space is imme di ate Th e four points A B C D situate d at
t h e verti ces of a tetrahedr on are weighted wi th mass a b c d
respectively Th e cent er of gravity G is represented b y
these uan tities or four others proportional to them To
obta in points outside of the tetrahedron negative masses
may be emplo ed O r in the light of theorem 2 page 4 0
the masses may be replaced by the four tetrahedra which
are proportional to them Then the idea of negative vol
umes ta k es the place of that of negative weights As thi s
idea is of considerab le importa nce later a brief treatment of
it here may not be out of place
Th e volume A B CD of a tetrahedron is said
Defin i tio n
to be positive when the triangle A B C appe a s positive t o
.
q
y
.
.
”
“
,
.
,
,
,
q
,
,
.
.
y
.
,
.
.
,
.
:
r
,
44
VE C TOR
y
A
NAL YSIS
i
i
the e e situated at the point D Th e volume s n e gat ve
if the area of the tri angle appear negative
To make the discussion of the sig n s of the vario u s
tetrahedra perfectl clear it is a lmo s t n e c e ss a ry to have a
solid mode A p la n e draw ng is scarcely s u ffi cient It is
di ffi cult to see from it wh ch triangles appear posi t ive and
which negative Th e following relat ons wi b e seen t o
hold if a model be ex amined
Th e in terchange of two letters in the te trahedron A B CD
changes the sign
.
.
y
l
i
.
i
.
i
.
ll
.
.
A C B D = C B A D = B A C D = D B CA
= A D CB = A B D C = — A B CD
'
.
sign of the tetra hedron for any given one of t h e p o s
sible twent -four arran gements of the letters ma be obtained
by reducing that arrangement to the order A B C D b
means of a number of successive inte rchanges of t wo ette s
If the number of interchanges s even the sign is the sam e
as that of A B CD ; if odd opposite Thus
Th e
y
y
l r
i
,
CA D B
If P
—
.
.
C A B D = + A CB D
is any point in ide of
quat on
y
s
—
A B CD
the tetrahedron
.
A B CD t h e
i
e
A R OP
—
B CD P + C D A P
ill i
i
—
DA B P
i
z
A B CD
r
holds good It st s true f P be w thout the tetrahed on
provi ded the signs of the volumes be taken into considera
tion Th e equation ma b e put into a form more s y m m e t ri
cal and more easil remembered b transposing all the terms
to one number Then
.
y
.
y
y
.
A B CD + B CD P +
r
proportion in theorem 2 page 4 0 does not hold t u e
f the signs of the te trah edra be rega rded It should read
i
Th e
,
,
.
A DDI TI ON AND
S
CALAR
P
UL TI L I CA TI ON
M
45
the point G lies ins ide the te tr ahe dron a b c d re p re
sent quantities proporti onal to the masses wh ich must be
l o cated at the vertices A B C D respectivel if G s to be t h e
center of gravit If G lies outside of the te trahedron the may
still be regarded as masses some of wh ch are negati ve — o r
perhaps better merel as four numb ers whose ratios determine
the pos it on of t he poin t G In this manner a set of ba ry
c e n tri c
coo rdi nates is establis hed for Space
Th e vector P dra wn from an indeterminate orig n t o an
poin t of th e pla ne A B C is ( page 3 5)
If
,
,
y
,
,
y
,
i
y
i
.
y
i
,
“
.
i
.
xA
+ yB +
x
+ y +
ring this with the e pression
Co mpa
z
C
c
C
y
z
x
A + bB +
a
a
+ b +
c
it will seen that the quantities are in reality nothing
mo re nor less than the ba y centric co rdinates of the point
with respect to the tri angle
like ma er from
e q uation
be
x,
y
z
P
o
r
AB C
.
x
,
In
nn
A+ yB +
z
C+ wD
+ y +
z
+ w
x
y
y
dr
y
ri
a
A+ bB +
c
C+ dD
+ b +
c
+ d
which expresses an vector P awn from an indete m nate
origin in term s of four given vectors A B C D drawn from
the same origin it ma be seen b comparison w th
,
r
a
i
,
,
,
r
i
y
y
tha t the fou quantit es x y z w a e pre cis el the bar
centri c co ordi nate s of P the terminus of P with respect to
the te trahedr on A B CD Thus the vector methods in wh ch
“
the origin is undeterm ned and the methods of the B a ry
o
u
l
u
s
are
pract
cally
c
e xtens ve
C
a
l
c
c e n t ric
It was m e nt one be fo e a n d t m a y b e w e to epe a t he r e
,
,
,
i
i d
i
,
,
.
r
i
i
i
.
ll r
VECTOR ANAL YSIS
46
that the origin may be left wholly out of consideration and
Th e vector e uat ons
t h e vectors replaced b y their termini
then become point e uations
q
x
q i
.
A + yB +
z
C + wD
This
step b rings in the points themselves as the objects of
alysis and leads still nearer to the B a ry c e n tri s c he Ca lc ul
of M ob iu s and the A u s de h n u n gs le h re of G rassmann
”
“
.
fi
Us e
The
f Ve c to rs
O
to de n o t e A re a
y
s
area l ing in one plane M N and
b ounded b y a continuous curve P QR which nowhere cuts
itself is said to appear positive from the point 0 when the
lett ers P QR follow each
other in the counterclockwise
or
positive
order
negative
;
N
when the follow in the
nega tive or clockwise order
De
An
n i t io n
y
Fi
( g
It
,
i
.
is ev dent that an area
can have no dete rmined sign
r se
n
l
e
but
y
in
reference
o
Fro 1 4
p
to that directi on in which its
boundar is supposed to be traced and to some point 0 out
side o f its plane Fo r the area P R Q is negative relative to
P Q R and an area viewed from O is negative relative to the
same area viewed from a point 0 upon the side of the plane
opposite to O A circle l ing in the X Y-plane and desc ri b ed
in the positive trigonometric order appears positive from ever
point on that side of the plane on which the positive Z -ax is
lies but negative from all points on the side upon whi ch
,
,
.
y
.
.
,
y
’
y
ADDI TI ON AND
S
CALAR
M
P
UL TI LI CA TI ON
47
the negative Z ax is lies Fo r this reason the point of view
and the di rection of description of the bo undary must be kept
clearly in mind
A nother method of stating the de fi nition is as follows : If
a person walking upon a plane traces out a closed curve the
area enclosed is said to b e positive if it lies upon his left
hand side negative if upon hi s right It is clear that if t wo
persons b e considered to trace out together the same curve b y
walking upon opposite sides of t h e plane the area enclosed
will lie upon the right hand of one and the left hand of the
other To one it will consequen t ly appear positive ; to the
other negative That side of the plane upon whi ch the area
seems positive is cal led the positive side ; the side upon
which it appears negative the negative side This idea is
familiar to students of electricit and magnetism If an
electric current flo w around a closed plane cu rve the l ines of
magnetic force through the circ ui t pass from the negative to
the positiv e side of the plane A positive magnetic pole
placed upon the positive side of the plane w l be repelled b
the circuit
A plane area may b e looked upon as possessing more than
positive or negati ve magnitude It may be considered to
possess direction namely the direction of the normal to t h e
posi t ive side of the plane in whi ch it lies Hence a plane
area is a vector quantity Th e follow ng theore m s concerning
areas when looked upon as vectors are important
If a plane area be denoted by a vector whose
The o re m 1
magn itude is the numerical value of that area and whose
direction s the normal upon the positive side of the plane
then the orthogonal projection Of that area upon a plane
wi ll be represented b y the component of that vector in the
direction normal to the plane of projection (Fig
et the area A lie in the plane M N
et it be projected
’
’
o thogonall upon the plane M N
et M N and M N inter
-
.
.
,
.
,
.
.
,
y
,
.
.
il
.
y
.
.
,
,
i
.
.
.
i
r
L
,
y
’
.
L
.
L
.
’
VECTOR A NAL YSIS
48
l
sect in the lin e l and let the di edra angle between these
two planes b e x C onsider fi rst a rectangle P OR S in M N
whose sides P O R S a n d QR SP are respectively pa rallel
and perpendicular to the line I This will project into a
’
in M N Th e sides P Q and R S
rectangle
’
will b e equ al to P O and R S ; but the sides Q R and S P
will be equal to OR and SP multipl ied by the cosine of x
the an gle between the planes C onsequently the rectangl e
.
,
,
,
.
’
’
’
’
’
’
’
.
’
,
.
QR S cos
P
FIG
Hence
i
.
15
x.
.
i y
rectangles of wh ch the sides are respect vel
parallel and perpendi cular to l the line of intersection of the
two planes project in t o e ctangles whose sides are likewise
espectively pa allel and perpendicul ar to l and wh o se a re a is
r
,
,
e qu a
l t o th e
c o s i ne
f th e
o
r
r
a re a
a n le
g
From
f
o
t he
,
o rigi n a
re c ta n gle s
l
be twe e n the p la
n es.
m u ltip l ie d by t he
i
y
y
this it follows that a n y area A is projected nt o an
area which is equal to the g ven a ea mul tiplied b the cosine
of t h e angle between the planes Fo r an area A ma be di
vide d up into a large number of small rectangles by drawing a
series of l ines n M N pa a e an perpend c u la r to the lin e I
i
r
.
i
r ll l d
y
i
.
VE C TOR ANA L YSIS
50
Th e
The o re m 2
y
vector which represen t s a c lo s e d pol hedra l
surface is z ero
This may be proved by means of certain co n siderations of
hydrostatics Suppos e the polyhedron drawn in a b od of
fluid assumed to be free from all e x ternal forces gravi t y in
cluded 1 Th e fluid is in equil ib rium under its own internal
pressures Th e portion of the fluid b ounded b y the closed
surface moves neither one way nor the o t her Upon each face
of the surface the uid e x er t s a de fi n ite force p rop o r tio n a l
to the a re a of the face and normal to it Th e resultant of all
these forces must be z ero as the fluid is in equ ilibrium Hence
the sum of all t h e vector are a s in the closed surface is z ero
Th e proof may be given in a purely geometric manner
Consider the orthogonal projection of the closed surface upon
any plane This consists of a do u ble area Th e part of the
surface farthest from the plane projects in to positive ar ea ;
the part nearest the plane into negative area Thus the
s urface projects into a certain portion of the plane which is
covered twice once with positive area and once with negative
These cancel e a ch other
Hence the total projection of a
closed surface upon a plane ( if taken with regard to sign) is
z ero
B u t b y theorem 1 the projection of an area upon a
plane is equal to the component of the vec t or representing
t hat area in the di rection pe rpendicular to tha t plane
Hence
the vec t or whi ch represents a closed surface has no component
along the l ine perpen di cular to the plane of projection This
however was a n y plan e wha tsoever Hence the vector is
z ero
Th e theorem has been proved for the case in which the
closed surface consists of planes In case that surface b e
.
y
.
,
.
.
fl
.
.
,
.
.
.
.
.
,
.
,
.
.
.
.
,
.
,
.
.
.
Suc h
a s ta
f i rs
mg
m
i d t o a ll pra c t ic a l p u rp o se s in t h e c a s e o f a
po lyh e dro n s us pe n d e d in t h e a t o s ph e re a n d c o n s e q u e n t ly s u bj e c te d t o a t o s
Th e fo rc e o f ra vi t y a c t s b ut is c o un t e rba la n c e d by t h e t e n s i o n
p h e ric p re s s u re
in t h e s us pe n din st ri n
1
t e o fa f a
.
g g
.
is
re a l z e
A DDI TI ON
A N D S CA LA R M
P
UL TI L I CA TI ON
51
curved it may be regarded as the li m it of a polyhedral surfa ce
whose number of faces increases without limit Hence the
vector which represents any closed surface polyhedral or
curved is z ero If t h e surface b e not closed b u t be curved it
may be represen t ed b y a vecto r just as if it were polyhedr al
That vector is the li mit 1 appro a ched by the vector which
represents tha t polyhedral surface of whi ch the curved surface
is the limit when the num b er of faces be comes inde fin i tel
grea t
.
.
.
y
q
SUMM AR Y
or
CHA PTER I
vector is a uantity considered as possessing magnitude
and di rection E qual vecto s possess the same magnitude
and the same direction A vector is not altered by shi fting it
arallel to itsel f A null or z e ro vector is one whose mag
n it u de is z ero
To multiply a vector by a positive scalar
multiply its length b y that sca ar and leave its direction
uncha nged To multiply a vector by a negative scalar mul
t ip l y its length b y that scalar and reverse its direct on
V ecto s add accordi ng to t h e pa rallelogram law To subtract
a vector reverse it s direction and add A ddition s u b tra c
tion and multipli cation of vec t ors by a scalar follow t h e same
laws as addi tion subtrac t ion and multipl ication in ordinary
algeb ra A vecto r may b e resolved into thr ee components
parallel to any three non-coplanar vectors Thi s esolution
can be accomplis hed in only one way
A
r
.
p
.
.
l
.
i
.
r
.
.
.
,
,
,
,
r
.
.
.
components of equal vectors parallel to three given
non-coplanar vecto rs are equal and conversely if the com
n e n ts
are
equal
the
vect
rs
are
equal
T
h
three
unit
o
e
o
p
vectors i jk form a ri ght-handed rectan gular sys tem In
Th e
,
,
,
.
,
1
t he
m
.
,
li i t e xi s ts a n d is u n iqu e
po lyh e dra l s urfa c e a ppro a c h e s t h e
Th i s
.
i n d e pe n de n t
c urve d s u rfa c e
It is
.
of
the
m
e th o
d
in whic h
VECTOR A NAL YSIS
52
rms of them any vector may be e xpressed
Cartesian co ordi nate s x y z
te
,
,
x
Applications
ratio
m
Th e
.
is given
:n
by
means of the
by
.
i + y j+
z
k
6
( )
.
point which divides a l ine in a given
the form ul a
n
A+ m B
m +
y
(D
n
necessar and su ffi cient con di tion that a ve c tor equati on
represent a rel ation independent of the origin is that the sum
of the scalar c o e fli c ie n t s in the equation be z ero B etween
any four vectors there e x ists an equation with scalar c o e fli
If the sum of the c o e fli c ie n t s is z ero the vectors are
c ie n ts
termino-coplanar If an equation the sum of whose scalar
coeffi cients is z ero ex ists b etween three vectors the are
termino-collinear Th e center of gravity of a number of
mas ses a b c
situated a t the te rmini of the vecto s
supposed to be drawn from a common o ri gin is
A B C
given b y t h e formul a
Th e
.
.
y
.
r
.
,
,
,
,
r i
vecto r may be used to denote an area If the a ea s
plane the magnit ude of the vector is equal to the magnit u de
of the area and the direction of the vector is the direction of
t h e normal upon the positive side of the plane
Th e vector
representing a c lo s ed surface is z ero
A
.
,
.
.
EXER CI SE S
on
the laws stated in Art 1 2
2
A tri an gle may be constructed whose sides are pa alle
and e ua l to the med ans of an given tri an gle
1
.
.
D emonstrate
CHA PTER I
q
.
i
y
r l
.
.
ADDI TI ON A ND
S
CA LAR
M
P
UL TI LI CA TI ON
53
po ints in which the thr ee di agonals of a com
1
meet
the
pairs
of
opp
site
sid
o
e s lie three
l
a
quadr
ngle
e
t
e
p
by three upon four straight lines
If two tri angles are so situated in space that the three
4
points of intersecti on of corresponding sides lie on a line t hen
the li nes joining t h e corresponding vertices pass through a
common point and conve sely
Find the mi ddle point
G iven a quadrilateral in space
5
of the line whi ch joins the middle points of t h e diagonals
Find the middle poin t of the li ne whi ch joins the middle
points of two opposite sides Show that these t wo points are
the same and coincide with the center of gravit of a system
of equal masses placed at the vertices of the quadrilateral
If two opposite sides of a quadr ateral in space be
6
di vided proportionally and if two quadri late rals b e formed by
joining the t wo poin t s of division then the centers of gra vi ty
of these two quadrilaterals lie on a li ne with the center of
grav t y of the original uadrilateral B y the center of gravity
is meant the center of gravity of four e ual masses pl a ced at
t h e vert ices
Ca n thi s theorem b e generaliz ed to the case
where the masses are not equal ?
Th e bisecto rs of the angles of a triangle meet in a
7
po int
If the edges of a h e x a h e dro n meet four b four in three
8
points the four di agonals of the he x ahedr on m eet in a point
In the special case in whi ch the he x ahedron is a parallelopiped
the three points are at an in fi n ite di s tance
Prove that the three straight lines through the mi ddle
9
points of the sides of any face of a te t rahedron each parallel
to the straight lin e connecting a fi x ed point P with the mid
dl e point of the opposite edge of the tetrahedron meet in a
3
Th e
.
s ix
.
.
,
r
.
.
.
.
y
.
il
.
.
,
i
q
q
.
.
.
y
.
.
.
,
.
.
,
,
m
pl q
g
f p i
i j i
p i
1
A
t h ro u h
wh c h
ete
co
o ur
o n
the
ua
g
d ra n l e
o n ts n o
i
c o n s st s o f t h e s ix s t ra
i
i
g
lli n e a r
o fpa i rs o f s id es
t h re e o fwhic h a re
o n t s o f n t e rs e c t o n
i
co
g
li n e s wh i c h m a y b e pa ss e d
Th e di a o na l s a re t h e l in e s
ht
.
.
'
VE C TOR A NAL YSIS
54
point E and that this point is such that P E passes
and is b isected by the center of gravit of the t e tra h
Show tha t wi tho u t e xc ep tio n there e x ists 0
10
equation with scalar coe ffi cients between an fo
vectors A B C D
Discuss the condi t ions imposed upon three
11
fi ve vectors if they satisfy t wo e uatio n s the sum 0
e fli c ie n t s in each of wh ch s z e o
y
y
.
,
,
,
.
.
i i r
q
.
,
C H AP TER II
DIR ECT A N D SKEW
P ro du c ts
PR OD UCTS
O F VE CTOR S
f Two Ve c to rs
o
operati ons of a ddition subtraction and scalar
mul tipli cation have been de fi ned for vec t ors in the way
suggested by ph sics and have been employed in a few
applications It now becomes necessar to introduce two
new comb inati ons of vectors These wil l be called p r o du c ts
b ecause the obey the fundamental law of prod u c t s ; i e the
dis tri bu tive law which states that the product of A into the
sum of B and C is e ual to the sum of the products of A into
B and A into 0
De n i tio n : Th e di rec t p r o du c t of t wo vectors A and B is
the s c a la r quantit ob tain ed b y multiplying the product of
the magnitudes of the vecto s b the c o s i n e of the an gle b e
tween them
Th e di rect product is denoted b wri ting the two vectors
w th a dot between them as
TH E
,
y
,
y
.
y
.
.
.
,
q
fi
.
y
r y
y
.
i
A B
.
This
is read A do t B and therefore may often be called the
dot product inste ad of t h e di rect product It is al s o called
the scalar product ow ng to the fact that it s value is s o a
lar If A be the magnitude of A and B tha t of B then b
de fi nition
B = A B cos
A-
i
.
.
Obv
,
iously the d rect product follows the commutative law
i
A B
B
A
.
y
VECTOR A NA L YSIS
56
i
ei t her vector be multiplied by a scalar the pr duct s
mult iplied b y that scalar That is
If
o
.
(
x
B)
A)
.
case the two vectors A and B are colli near the angle b e
tween the mbecomes z ero or one hundr ed and eighty degrees
and its cosine is therefore equal to unity with the pos itive or
negat ive sign
ence the scalar product of two parallel
vectors is num erically equal to the product of their lengths
Th e sign of the product is positi ve when the di r ec t ions of the
vectors a e the same negative when they a re opposite Th e
product of a vector b it self s therefore e u al to the s ua e
of its lengt h
In
H
.
.
r
y
i
,
q
A
A A
C onse
quently if
q r
.
2
( 3)
y
produc t of a vector b itself vanish the
vector is a null vector
In case the two vectors A and B are perpendicu a the
angle between them b ecomes plus or m inus ninety degrees
and the cosine vanishes Hence the product A B vanishes
C onversely if the scalar product A B van shes then
the
lr
.
i
.
.
,
cos (A B ) 0
Hence either A or B or cos ( A B ) is z ero and eith er the
vecto s are perpendic ul ar or one of them is null Thus the
A B
,
r
,
fo r
Th e
r j
i,
,
k
is A
o
0
B
f
vec t o rs , n e ithe r o
d
.
scalar products of the three fun amental unit
are evidently
r ly
more ge n e a l
product
If
f two
e n dic u la ri ty
th e p e rp
va n is he s ,
vecto s
,
.
c o n di t io n
wh ic h
.
a
and
a
b
b
are any two uni t vecto
008
(a
,
b)
.
r
s
the
VECTOR ANAL YSIS
58
i
scalar
or
d
rect product foll ows the
]
law of multi plication Tha t is
=
R
C
R
A
C
C
A
+
+
)
(
Th e
29
ive
di stri but
.
.
This
6
( )
L
may be proved b y means of projections et C be equal
to its magn itude C multiplied b a u ni t vector 0 in its dire c
tion To show
y
.
.
(A + B )
c +
B
-( C c )
c
.
is the projection of A upon 0 ; D c tha t of B upon 0 ;
u t the projection of the
B
that
of
upon
0
c
A
B
B
A
)
(
sum A B is equal to the sum of the projections Hence
the relation ( 6) is proved B y an immedi ate genera z ation
a
A
B
r
A
+
+
+
+
(
P + B (1 +
+ B A
c
c
o
,
.
,
li
.
.
i
scalar product may be used j u st as the product in ord
nar algebra It has no peculiar d ffi culties
If two vectors A and B are e x pressed in terms of the
three unit vecto rs i k as
Th e
y
i
.
,
j
B
:
,
_ A B
1
B
1
+ A2 B
+ A3 B
i
.
l
l
i
i
+
r
o
s
+ B
i + A1 B
j i + A2 B
o
k
o
j+ A 3 B
2
2
2
i
s
ky
j+
A1 B
j j+ A 2 B
o
k
.
j+ A 3
a
3
B
i
k
jk
o
a
n
k
,
means of ( 4) th s reduces to
By
A
r
If in
ir
(7)
particular A and B are unit vecto s the components
A 1 A 2 A 3 and B 1 B 2 B 3 are the dire cti on cosines of t h e
l nes A an B efe e t o X Y Z
i
,
,
d r rr d
,
,
,
,
.
,
P
DIRECT AND SKE W R OD UC TS OF VECTOR S
cos (A X)
cos (B X)
AI
B
1
,
,
,
,
i
cos (A
cos (B
A2
B
2
,
Y)
A8
,
Y)
B
,
,
cos (A Z)
c os (B Z)
,
3
c os
B
A
,
)
(
cos (A X ) cos
X)
(B ,
,
,
.
,
Moreover A B s the cosine of the included an gle
the equation becomes
o
59
Henc e
.
cos ( A Y) cos ( B Y)
cos (A Z ) cos ( B Z )
,
,
,
,
.
case A and B are perpendi cular this reduces to the we l l
known relation
In
0=
cos ( A X ) cos ( B X)
,
cos ( A Y) cos ( B
cos ( A Z ) c o s ( B
,
,
,
i
be t ween the direct on cosines of th e
line A and the line B
If A and B are t wo sides O A
and O B of a tri angle 0A B the thi r
side A B is C B A ( Fig
Y)
,
,
Z)
.
d
,
FIG 1 7
.
.
.
A—
2
=
A + B
0
2
That
2 —
2 A B
c os
2AB c os
(AB )
.
is the square of one side of a t ri angle is equal to the
s u m of the squa es of the other t w o sides diminished b y tw c e
their product times the cosine of the angle b etween them
O r t h e square of one side of a tri angle is equal to the sum of
the squares of t h e other two sides di minished by twice the
projection of either of those sides upon the other the theorem
sometimes known as the generaliz ed P ythagorean theorem
If A and B are t wo sides of a parallelogram C
A
B
and D A B are the diagonals Then
,
r
i
.
,
,
.
,
.
D = (A — B )
D-
o
(A
—
B
)
=AA
—
-A +
B
2 AB + B
B)
.
o
B,
VECTOR A NAL YSIS
60
That
l
r
ll l
is the sum of the squares of the di agona s of a para e o
gram is equal to t wi ce the sum of the squa es of t wo sides
In like mann er als o
,
D= 4 A
CC — D2 —
0
That
D = 4 A B
2
.
B
o
c os
l
is the difference of t h e squares of the di agona s of a
parallelogram is equal to four times the product of one of the
sides b y the projection of the other upon it
If A is any vector ex pressed in terms of i
k as
,
.
,
j
,
if A be e xpress ed in terms of any thre e non -c oplan a r un it
vecto rs a b c as
B
ut
,
,
A=
+ bb +
a a
a
c
o
+ 2 b
c
b
c +
2
2
c a
c
cos
c a
o
(
This
h
s
c, a
)
.
y
formula is analogous to the one in Cartesian geometr
whi ch gives the distance between two points referred to
oblique axes If the points he x 1 f’l v z ] a n d x 2 y 2 x2 the
d stance squ ared is
i
2
D
fi
,
,
.
2
( x2
(3 2
x ) (3
12
1
2
(h
(z
2
(
(x2
x
z
1)
?
— z
1
2
“
’
I
l r)
l
“
z
) ( x2
S
I
19 0
a
(z
003
,
z
2
1
)
,
2
( X Y)
,
Y
cos
Z
)
(
l
)
x l ) cos ( Z X)
’
,
.
of the vector A in to
the vecto r B s the vec to r quant t 0 whos e di rection is the
norm a upon that side of the p an e of A and B on which
De
l
n i tio n :
i
Th e
s ke w
p ro duc t
iy
l
P
DIRECT AND SKE W ROD UCTS OF VECTOR S
61
rotation from A to B through an angle of less than one
hun dred and eight degrees appears positive or counter
clockwi se ; a n d whose magni t ude is obtained b y multiplying
the product of the magni tudes of A and B by the s i n e of the
angle from A to B
a so be de fi ned as that in
B ma
Th e direction of A
which an ordin ar right-han ded
screw advances as it tu ns so as C Ax B
t o carr A toward B ( Fig
Th e skew product is denoted by
a cross as the di rect pro uct was
b a dot It s written
Fm 1 8
y
.
y
y
x
y
r
y l
‘
:
.
d
i
.
.
.
C= A X E
and re a d A c ro s s B Fo r this reason it is often called the c ro s s
product More frequently however it is called the ve c to r prod
owing to the fact that it is a vector quantity and in con
uc t
tra st with the direct or scalar product whos e value is scalar
Th e vector product is b de fi nit on
.
,
.
,
,
y
i
.
( 9)
r
when A and B a e the magnit u des of A a n d B respec t ively and
where c is a unit vector in the di rection of C In case A and
B are u nit vectors the skew product A x B reduces to the
unit vector 0 multi plied by the sine of the angle from A t o B
Obviously also if either vector A or B is multipl ied b a scalar
x their product is multiplied b
that scalar
.
y
y
(
If A and B
xA
)
D= A>
<
.
.
(
xB
)
=
xC
.
are p a ra lle l the angle between them is either z ero
or one hund ed and eighty degrees In either c a se the sine
vanishes and consequently the vector pro uct A x B s a nu
vecto r An d con versely if A x B s z e ro
r
.
i
.
AB
s in
d
i
ll
VE C TOR ANA L YSIS
62
Hence
p
a ra
or B or sin ( A
A
,
lle lis m
t wo
f
o
ve c to rs
y
B)
is z ero
n e i th e r
Thus the
.
whic h
f
O
c o n di tio n
va n is h e s
y
fo r
x
is A
B
a corollar the vector product of an vector into
itself vanis hes
Th e vector product of two vectors w l appear wh e r
32 ]
ever the sin e o f the included an gle is of importance just as
the scalar product di d in the case of the cosine Th e two prod
They have been
u c t s are in a certa i n sense complementary
denoted by the two common signs of mul tip cation the dot
and the cross In vector an alysis they o ccupy the place held
by the trigonome t ric functions of scalar analys is They are
at the same time amenable to algebraic treatment as will be
seen later A t present a few uses of the vec tor product ma
be cited
IfA and B ( Fig 1 8 ) are the two adjacent sides of a parall el
ogram the vector product
0
.
As
il
.
,
.
.
li
,
.
.
y
,
.
.
.
(A B)
,
c
represent s the area of that parallelogram in magnitude and
di rection ( Art
This geometric representation of A
B
s of such common occurrence and importance that it might
well be taken as the de fi nition of the product From it the
tri gonometric de fi ni tion follows at once Th e vector product
appears in mechanics in connection with couples If A and
A are two forces forming a couple the momen t of the
couple is A X B prov ded only that B is a vector drawn from
a n y poin t of A to an
point of - A Th e product makes its
appearance again in considerin g the velocities of the indiv d
ua l particles of a body which is rotating wi th an a n u l a r ve
g
l o c it y given in magn tude and di rection b y A
If R be the
radius vecto r drawn from any point of the ax is of ro t ation A
the product A B will give the velocit o f the e xtremity of
R ( Art
This velocity is perpendi cular a ke to the a xis
of rotat on and to the radi us vector R
i
x
.
.
.
.
i
y
,
i
.
'
i
x
.
i
.
y
.
.
li
KE W PR OD U
vector produc s x
DIRE C T
AN D S
T
VECTOR S
C S OF
63
and B x A are not the
same They are in fact the negatives of each other Fo r if
rotation from A to B appear positive on one side of the plane
of A a n d B rotation from B to A wil l appear positive on the
other Hence A B is the normal to the plane of A and B
upon that side opposite to the one upon which B A is the
normal Th e magnitudes of A B and B A are the s ame
Hence
Th e
A
t
B
.
.
'
x
,
.
.
A
Th e
f
if the
Th
in
a c to rs
a
f the
s ign O
x
ve c to r
p
B
B
ro du c
p ro du c t be
t
x
x
ca n
x
x
.
A
( 1 0)
be i n te rc h a n ge d
f
i
a nd o n l
y
re ve rse d.
is is the rst i stance in w
the laws of operation in
vector analysis differ essentially from those of scalar a n a l y
s s It may be that at fi rst this change of sign whi ch mus t
accompa ny the interchange of factors in a vector product will
ive
ri
s
e
to
some
di
f
fi
culty
and
co
n
fusion
C
hanges
similar
to
g
this are however ver familiar No one would think of inter
changing t h e order o fx and y in the e xpression sin ( x y )
without p re fi xin g the negative sign to the resul t Thus
i
fi
hi c h
n
.
,
y
,
.
.
.
sin ( y — x) — sin ( x y )
although the sign is not required for the case of the cosine
,
.
cos ( y — x) c o s ( x — y)
A gain if the cyclic order of the lette s A B C in the area of a
tri angle be changed th e area will b e changed in sign ( Art
.
r
"
,
.
AB C
—
A CB
r
the same man ner this reversal of sign which occu s
when the order of the factors in a vector product is reversed
will appear af t er a little practice and ac uaintance just a s
natural and convenient as it is necessar
Th e distributive law of multiplicati o n holds in t h e
case of vecto r products just as in ordi nar algebra — e xc ep t
In
,
y
q
.
y
,
VECTOR ANAL YSIS
64
f
orde r o
th a t the
the
f
mu s t
a c t o rs
u lly
f
be
c a re
m a in t a in e d
(1 1 )
0
A very simple
proof ma y be given by making use of the ideas
developed in Art 26 Suppose that C
is not coplanar with A and B
et A
and B be two sides of a triangle ta ken
in order Then ( A + B ) wi ll be the
Form the prism
th ird side (Fig
of which thi s triangle is the ba se and
of which 0 is the slant height or edge
Th e areas of the lateral faces of th s
pri sm are
L
.
.
.
.
.
FIG
.
19
,
i
A x c,
Th e
.
B x c,
areas of the bases are
x
A
B
(
- A x B
— 3
and
(
)
)
2
.
the sum of all the faces of the prism is z ero ; for the
prism is a closed surface Hence
B
ut
.
l
2
(1 1 )
i
relation is therefore proved in case 0 s non-coplanar
with A and B Shou ld C be coplanar with A and B choose D
any vector out of that plane Then C D also w ll lie out of
that plane Hence b y ( 1 1 )
Th e
i
.
.
,
,
.
Since
the thr ee vectors in each set A C D and B C D and
A
B C D will be non -c oplanar if D is properly chos en the
products may be e xpanded
,
,
,
,
,
,
,
,
,
.
VECTOR A NA L YSIS
66
This
may be wri t ten in the form of a determi nant as
Ax B
i
j
k
AI
A,
A3
formul ae for the sine and cosine of the sum or dif
ference of two angles follow immedi at ely from the dot a nd
cros s produc t s
et a and b be two unit vectors lying in the
plane If x b e the angle that a makes with i and y the
ij
angle b makes wit h i then
Th e
L
.
,
.
,
cos x
a
i +
s in x
j
'
,
— x
)
a
,
a
cos (y —
x
)
b =
’
c os x
c os
y
cos
i
y
s in
—
s in
+
y
y
s in x
y
s in x.
.
j
,
a
Hence
cos ( y
+
x
)
c os
y
s in
c os x —
)
y)
— x
s in x c o s
—
Hence
sin (y — x)
s in
Hence
sin ( y
s in
+
)
x
y
c os x —
y
c os x
+
s in x c o s
y
s in x c o s
y
,
.
.
.
and l m n are the direction cosines of two
unit vectors a and a referred to X Y Z then
If l, m ,
n
’
’
’
,
,
’
,
cos ( a
,
,
,
as has already been shown in A rt 2 9 Th e familiar formula
for the square of the sine of the angle between a and a ma
be found
.
.
’
.
y
RECT
DI
x
a
a
sin ( a a )
’
P
KE W R OD U CTS
AN D S
’
,
e
=
m
(
’
l
m
(
n
-
’
(
(
sin
a
x
a
a
) (
2
(m
This
a)
x
n
’
to
sin ( a a )
’
m
u
)
z
+
(
l
n
and a
a
e
,
’
l
’
l m ) k,
2
’
n
67
’
where e is a u ni t vector perpen di cular
’
VECTOR S
OF
sin ( a
2
e
'
n
’
l)
’
2
+(l m
,
’
l m) 2
'
leads to an easy way of establishing the useful identity
’
(m n
m
’
m
2
n
?
u
z
(
)
) (l
’2
P ro du c ts
’
n
l
m
’2
n
n
’
l)
’2
2
l
m
(
’
l m )2
’
W
(
)
mm
’
n n
’
)
2
.
e
h
a
t
r
M
o
t
n
T
w
o
V
ec
o rs
f
O
to
this
point
nothing
has
been
said
concerning
]
products in which the num b er of vecto rs is greater than
If three vectors are com b ined in t o a product the result
t wo
is called a t rip le product N e x t to the simple products
A B and A>
<B the triple products are the most important
All higher products may b e reduced to them
Th e simplest triple product is formed b y multiplying the
scalar product of two vectors A and B into a thi rd C a s
36
Up
.
.
o
.
.
A
( B)c
.
This
in reality does not di ffer essentially from scalar m ul ti
Th e scalar in t hi s case merely happens to
pli cat ion (A rt
M oreover
b e the scalar product of the two vectors A and B
inasmuch as t wo vectors cannot stand side by side in the
form of a product as B C wit hout either a dot or a cross to
uni t e them the parenthesis in ( AB ) C is superfluous Th e
e x pression
A R0
.
.
,
.
0
cannot be interpreted in any o t her way than as the product
of the vector 0 b y the scalar A B
1
.
La t e r
( Ch a p V )
g
m m
p ro du c t B C wh e re n o s i n e i t h e r do t o r c ro s s o c c u rs
will b e d e fi n e d
B u t it wi l l b e s e e n t h e re t h a t (A B ) C a n d A ( B C) a re id e n t i c a l
a n d c o n se qu e n t l y n o a
bi u i t y c a n a ris e fro t h e o is s i o n o f t h e pa re n t h e s is
1
.
.
.
th e
mg
,
o
,
o
.
VECTOR A NAL YSIS
68
second triple product is the scalar product of
two vec t ors of which one is itself a vector product as
or
X
B
C
A
B
X
C
A)
)
(
(
Thi s sort of product h as a scalar value and consequently is
oft en called t h e scalar triple prod
uc t
It s properties are perhaps most
easily deduced from its commonest
geometri cal inte rpretation
et A B
and C be any three vectors drawn
from t h e same origin ( Fig
FI G 2 0
Then B x C is t he a re a __o f the par
T
h
e
a ll e l o ra m of which B and C are two adjacent sides
g
Th e
,
,
.
.
.
L
,
A
,
.
.
.
__
.
A( B x C)
8 03 1“
r
( 1 4)
v
will therefo e be the volume of t h e parallelopiped of whi ch
Se e A rt 2 8
B>
<C is the base and A the slant height or edge
This volume v is positive if A and D x C lie upon the same
side of t h e B C plane ; but negative if they lie on opposite
sides In other words if A B 0 form a right-handed or
positive system of three vectors the scalar A ( B x C) is posi
tive ; but if they form a left-handed or negative system it
is negative
In case A B and C are coplanar this volume will be
neit her posit ive nor negative b u t z ero A n d conversely if
the volume is z ero the three edges A B C of t h e parallelo
piped must lie in one plane Hence th e n e c e s s a ry a n d s uffi
.
.
.
-
.
,
,
o
,
.
,
,
.
‘
,
,
.
c ie n t
c o n di tio n
f
or
th e
c o p la n a ri ty O
f
th re e ve c to rs A, B , 0
y
n one
a corollar the scalar
triple product of t hree vecto rs of which two are equal or
collinear must van sh ; for any two vecto rs a re coplanar
Th e two products AB
C
and
X
A
B
x
)
(
(
) C are equal to the
same volume v of the parallelopi p ed whose concurrent edges
are A B C Th e sign of the volume is the same in b oth
f wh ic h
va n is h e s
o
is
i
,
,
.
0
.
As
.
P
DIRECT AND SKE W R ODUCTS Of VE C TOR S
Thi s equalit may b e state d as a rule of Operation
Th e
y
a nd
th e
wi tho u t
c r o ss
a
”
.
in
a
l te ri n g the
sca
la
va
lu e
trip le p
r
f th e
p ro du c t
O
It m a y als o be seen that the
muted cyc licly without altering
A( B x C)
ro du c t
n
do t
e
d
g
.
vecto rs A B
the product
,
may be per
C
,
.
( Cx A )
B
m a y be i n t e rc h a
69
1
5
( )
C
each of the e xpressions gi ves the volume of the same
pa rallelopiped and that volume will have in each case the
same sign b ecause if A is upon the pos itive side of the B C
plan e B will b e on the p o sitive side of the C A-plane and 0
upon the pos itive side of the A B —plane Th e triple product
may therefore have any one of s ix equivalent forms
Fo r
,
,
.
A( B x C)
x
A
C
)
(
x
B
C
(
) A
B
x
A
B
)C
(
x
C
A
)B
(
r
.
however the cyclic order of the lette s is changed the
product will change sign
If
.
A( B x C)
1
6
( )
is may be seen from the
Th
H ence
fi gure
or from the fact that
CxB
B xc
.
l t e re d by i n te rc h a n gi n g
th e do t o r the c ro s s o r by p e rm u ti ng c y c li c ly t h e o rde r of th e
ve c to rs , bu t i t is re ve rs e d in s ign if t he c y c lic o r der be c h a n ge d
:
A
sc a
la
r
trip le p r o duc t is
n ot
a
y
.
word is necessar upon the sub ject of parentheses
in this triple product Ca n they b e omitted without a m
b igu ity ? They can
Th e e x pression
A
.
.
y
Ao
C
can have onl the one interpretation
A
i
It is im p o s
the e xpression ( AB ) x C s meaningless
sible to form the skew product of a scalar AB and a vector
Fo r
.
VECTOR A NAL YSIS
70
y
i
A
B xC ma
Hence
as
there
is
only
one
way
in
wh
ch
C
b e interpreted no confusion can arise from omitt ing the
parentheses Furthermore owing to the fa ct that there are
and C which have the same
s ix scalar triple products of A B
value and are consequently generally not worth distinguish
ing the one from another it is often convenient to use the
symbol
.
,
.
,
,
,
[A B
y
to denote an one of the
[ A B C]
s ix
C]
equal products
AB xc
BCx A
AxB C
A
B x C-
B
A
C]
[
.
C AxB
o
B
CxA-
[A C B ]
.
j
scalar triple products of the three unit vectors i k
a l l vanish e x cept the two which contain the three different
vectors
Th e
,
,
.
[i
=
k
j ]
Hence if three vectors
as
B =
then
=
A
B
A
B
C
[
]
I
—
A1 B
A, B
a
1
7
( )
,
C
i +
be ex pressed in terms of
j
,
k
a j+ a n
,
C3 + B
,
C,
B
—
i,
1
C, A 3 + 01 A , B
3
1
C, A ,
,
—
CI A 3 B
.
Thi s
may b e ob tained b y actually performing the mul t iplica
tions which are indicated in the triple product Th e result
may be wri tten in the form of a determinant 1
.
.
Al A 2 A3
[A B
1
Th i s is t h e
d ro n o n e
e x e rc i se s
t e t ra h e
see
.
m l gi i
fo r
of
u a
wh o s e
ve n
C]
B
,
B
B
,
,
01 02 03
in
ve rt c e s
is
so
lid
at
the
g
m
y
g
m g
ly t i c
o ri i n
a na
eo
.
Fo r
e tr
a
o re
mm l
l e
e n e ra l f o r
fo r t h e
vo u
o
fa
u a
DIRECT
KE W PR OD UCT
AN D S
S OF
Ifmore generally A, B
non-coplanar vectors
vectors
,
where
a
s ta n t s ,
then
l
,
a
,,
a
s
C
,
b,
3,
A=
a
C=
C
;
I
I
a
+
a
a
+
C
1
ba
c
,
b1
—
0
“
[A B
b +
a
b +
C
2
2
and
b l , b, , b, ;
c
a
a
,
3
I
c
+
3
l
c
c
y
3
0
“
— c
a
,
l
3
“
“
2
3
o
2
c
,,
a
0
1
1
0
3
c
,
b 1 b, b8
C]
71
are e x pressed in terms of any three
c which are not necessarily unit
[ A B C]
—
VECTOR S
[a
3
a re
certain con
bs
a
b ,)
b
c
[
a
b
c
]
.
]
s
third t pe of tri ple product is the vector product
vectors of which one is itself a vector product Such
Th e
.
and
vector
is pe rpendi cular to A and to (B x C)
B u t ( B x C) is perpendicular to the plane of B and 0
Hence
being perpendicular to ( Dx C) must lie in the
plane of B and C and thus take the form
Th e
.
.
s
a
y c,
where x and y are two scalars In like manner also the
vec tor
being pe rpend cular to
must li e
in the plane of A and B Hence it will b e of the form
.
i
.
=mA
where m and v are two scalars
in general
is
.
From
n o t e qu a
n
B
i
this it s evident that
l to
parentheses therefore cannot be removed or inter
changed It is essential to know which cross product is
Th e
.
VECTOR A NA L YSIS
72
formed fi rst and which second This product is termed the
vector triple product in contrast to the scalar triple product
Th e vector triple product may be used to e xpress that c o m
n t of a vector B which is perpendicular to a given vector
n
e
o
p
This geome t ric use of the product is valuable not only in
A
itself but for the light it sheds
upon
the
properties
of
the
product
B
AX
et A ( Fig 2 1 ) be a give n ve c t o r
and B another vector whose com
n t s parallel and perpendic u lar
n
o
e
p
A to A are to be found
et the
components of B parallel and per
n dic u l a r to A be B
e
and
r
e
B
p
Draw A and B from a
s p e c t ive l y
common origin Th e product AxB
Th e product
is perpendi cular to the plane of A and B
It is furthermore
Ax ( AxB ) lies in the plane of A and B
perpendi cular to A Hence it is collinear with B
An
e xamination of the fi gure will show that the direct on of
Hence
AX ( AxB ) is opposite to that of B
.
.
.
L
.
~
.
.
L
”
’
.
.
.
.
”
i
.
.
”
.
c
where c is some scalar constant
Now
but
‘
i
B
”
—
c
,
.
sin (A B ) b
= 34 B
— c
B
”
2
,
in
(A
,
B) b
if b be a un t vector in the direction of B
”
Hence
c
Hence
A2
”
”
,
”
A
A-
.
AA
component of B perpendicular to A has be en e x pressed
in terms of the vector triple product of A A and B Th e
component B parallel to A was found in Art 2 8 to be
Th e
,
,
’
.
.
VE C TOR ANAL YSIS
74
Substituting
these values in
B C
A-
AC B
A>
<(Ex C)
( 24 )
.
relation is therefore proved for any three vectors A B 0
A nother method of giving the demonstration is as foll ows
It wa s sho wn that the vector triple produc t Ax ( B XC) was
of t h e form
Th e
,
.
,
.
.
Ax ( B XC)
Since Ax
it b y A is z ero
.
B
C
y
is perpendi cular to
Hence
A, t h e
x
A
and
Hence
o
Ax ( B x C)
where n is a scalar constant
Mult iply by B
.
0
AB
.
A
( CB
n
direct produc t of
A
C
y
B
AC
w y
.
AB C),
remains to show
It
1
71.
.
.
n
(A c B B
AB 0
o
o
scalar triple product allows an interchange of dot
cross Hence
Th e
and
.
if the order
A( B x C)xB
of the
AB
C
B
[
—
B C Ao B
o
AC B
and
1
n
A
factors ( B x C) and B be inverte d
A[B x ( Dx C) ]
Hence
—
From t h e
y
BB
.
0]
BB AC
o
.
J
ADC
.
( 24 )
three letters A B C b difi e re n t arrangements
four allied products in each of whi ch B and C a re included in
parentheses may b e formed These are
,
,
,
.
a vec t or product changes its sign whenever the order of
two factors is intercha nged the above products evidently
satisfy the equations
As
,
A>
<( B
)
xC
D
IRECT
KE W
AN D S
P
S
R OD U C T
VECTOR
OF
S
75
e xpan sion for a vector triple product in which the
paren t hesis comes fi rst may therefore be o b tained directly
from t hat already found when the parenthesis comes last
Th e
.
A B
C-
B A
C-
Th e
.
form u l ae then be come
= AC B
AB C
AC B
3 A
0-
and
These
reduction formul ae are of such constant occurrence and
great import ance that the should b e com m i t ted to memory
Their content may be stated in the followi ng rule
To e xp a n d
y
fi
.
.
mu l t ip ly th e
e xte rio r fa c to r i n t o th e
p
re m o te r t e rm in t h e p a r e n th e s is to fo rm a s c a l a r c o efii c ie n t fo r
the n e a re r on e th e n m u l t ip ly th e e xte ri o r fa c tor i n to th e n e a re r
a
vec tor t rip le
r o du c
t
rs t
,
t he
t e rm i n
pa
re n
th e s is
re mo te r o n e , a n d s u btr a c t
to
th is
fo rm
a
res u l t
sc a
la
f
r
th e
ro m
fi
c ie n t
fi
c oe
rs t
f
or
t he
.
far as t h e prac t ical applicati ons of vector analysis
are concern ed one can generall get along without any
formul ae more complicate d than that for the vector t riple
product B u t it is frequently more convenient to have at
hand other reduction formul ae of which all may b e derived
simply b y making use of the e xpansion for the t riple product
and of the rul es of operation with t h e triple pro
duc t AB xC
To reduce a scalar product of two vectors each of which
is itself a vector product of two vectors as
As
y
,
.
.
,
(AxB )
L et this be regarded as a scalar t
vecto rs
and
thus
ri
A, B
,
<D
C>
-
AxB 4 Cx D)
Inte rc ha nge
the dot and the cross
.
r
ple product of the th ee
VE C TOR ANAL YSIS
76
AxB ( CxD)
B x ( CxD)
A-
BC D
B B 0
Hence
This
y
in
D B C
A-
D
AC B-
A
B
x
) ( Cx D)
(
ma be written
.
.
determin antal form
2
( 5)
.
AD
D
A
B
O
X
x
)(
(
3 4)
and D b e call ed the e xtremes B and C the means A
and C the antecedents ; B and D the consequents in this
product according to t h e familiar usage in proportions then
the e xpansion ma be stated in words Th e scalar product
of two vector products is equal to the ( scalar) p fo du c t of the
antecedents times the ( scalar ) product of the conse uents
di minished b y the ( scalar) product of the means times the
scalar
product
of
the
e
x
tremes
)
(
To reduce a vector product of two vecto s each of whi ch
is i tself a vector product of two vecto s as
If A
y
,
.
q
r
.
r
Let
C>
<D
Th e
E
.
product becomes
AE B
Substituting
A
( CxD) B
F
AxB
B EA
.
the value of E back into the equati on
o
L et
,
Th e
.
( 2 6)
( B Cx D) A
o
produ ct then becomes
ED 0
A
B
x
D)C
(
q
FC
D
A
B
x
C) D
(
.
equating these two e ui valent results and transposing
all the terms to one side of the e uation
By
[B
C D] A
q
,
2
( 7)
Th is
is an equation with scalar coe ffi cients between the four
vectors A B C D There is in general o nl y one such equa
,
,
,
.
P
DIRECT AND SKE W R OD UCTS OF VECTOR S
77
y
tion because any one of the vectors can be e xpressed in onl
one way in terms of t h e o t her three : thus the scalar c o e ffi
c ie n t s of that equation which e xists between four vectors are
found to be nothing b u t the four scalar tri ple products of
th o s e vectors ta ken three at a time Th e equation ma a so
b e written in the form
,
yl
.
c
M ore
.
e x amples of reduction formul a of whi ch some are
import ant are given among t h e e xercises at the end of the
chapter In view of these it becomes fairly obvious th at
the com binat ion of an numbe r of vecto rs conn ected in
any legitimate wa b y dots and crosses or the prod uct of an
numb er of such combinations can b e ultimately reduced to
a sum of terms each of which conta in s onl y one cross at most
Th e proof of this theorem depends solely upon analyz ing the
possible combinations of vectors and sho wing that they all
fall under the reduction formul a in such a way that t h e
crosses m a y be removed two at a time until not more than
one remains
Th e formul a developed in the foregoing article have
interesting geometri c interpreta tions They also a fi o rd a
simple means of deducing the formul a of Spherical Trigo
n o me tr
T
hese
do
not
occur
in
the
vect
o
r
a
n
al
sis
proper
y
The ir place s taken by the two quadruple products
,
,
y
.
y
y
.
.
.
.
y
i
.
,
D
A0 B-
( AxB )
i
L
B e An
[A CD]
B
[B
[AB D]
c
A
B
C]
[
( 2 5)
e n] A
2
( 6)
wh ch are now to be inte rpr eted
et a unit sphere ( Fig 2 2) be g ven
et th e vectors
A B C D be unit vectors dr awn from a common origin the
centre of t h e Sphere and terminating in the surface of the
Sphere at the points A B
Th e great circular arcs
C D
,
i
.
.
,
,
.
L
,
,
,
,
,
.
VECTOR ANAL YSIS
78
AB
A
A
,
.
and C
,
r
etc give the angles between the vecto s A a n d B
etc Th e points A B C D determine a quadrilateral
upon the sphere A C and B D are one
pair of opposite sides ; A D and B C the
other A B and CD are the diagonals
,
,
,
.
,
,
.
,
.
.
AC B D
AB B C
o
sin ( A B ) CxD sin ( 0 D)
Th e angle b e t ween A>
<B and CXD is the
angle
b e t ween the normals to the AB
Fu r 22
and CD planes This is the same as
the angle between the planes thems elves
et it be denoted
by x Then
AxB
,
,
.
,
.
.
-
L
.
.
.
sin (A B ) sin ( C D) cos x
A
B
x
(
)
angles ( A
circular arcs A B
Th e
B
,
,
y
.
,
,
ma b e replaced by the great
which measure them Then
C
(
)
,
,
CD
D)
.
sin A B sin C D cos cc
cos A C cos B D cos A D cos B
,
AC B D — AD BC
o
O
.
Hence
s in
AB
s in
CD
c o s cc
c os
A C
o
os
B D
—
cos
AD
cos
B C
.
words : Th e product of the cosines of t wo opposite sides
of a spherical quadr ilateral less the product of the cosin es of
the o t her two Opposite sides is equal to the product of t h e
sines of t h e diagonals m ul tiplied b y the
cosine of the angle b etween them This
theorem is credited to G auss
et A B C ( Fig 2 3 ) b e a spherical tri
an gle the sides o f which are arcs of great
by a b c
circles
et
the
sides
be
denoted
F G 23
respectively et A B C b e the unit vectors
drawn from the center of the sphere to the points A B 0
Furt hermore let p p p be the great circular arcs dr opped
In
.
L
.
,
L
,
.
I
.
.
,
.
.
L
,
,
,
, ,
,,
,
,
,
,
.
P
DIRECT A ND SKE W R OD UCTS OF VECTOR S
perpendicularly from the vertices A
Interpret t h e formula
B
,
C
,
sin ( A B )
s in c ,
,
Then
a
,
b,
c
.
BA
C A-
AC B A
o
to the sides
79
o
.
sin (0 A)
sin 0 sin b cos x
A
x
C
)
(
sin
,
b
.
,
where
is the angle b et ween A><B and C><A This
an gle is equal to the angle b etween the plane of A B and t h e
plane of C A It is ho wever not t h e interior angle A which
is one of the angles of the triangle b u t it is t h e e x terior
angle 1 8 0 A as an e x amination of the fi gure w l show
Hence
sin 0 sin b cos ( 1 80 A )
sin c sin b cos A
Bcos b cos c cos a 1
AA
C BC AA
B y equating the results and transposing
cos a c o s b c o s c — s in b s in c c o s A
cos b cos c cos a sin e sin a cos B
cos c cos a cos b s rn a s i n b cos 0
Th e last t wo may b e obtained by cyclic permutation of t h e
letters or from t h e iden t i t ies
x
.
,
,
,
.
,
il
:
°
,
.
°
.
,
.
BA CB
A,
C-
B AC
C-
BC
o
y
N ex t
.
inte rpret the identit
in the special
cases in which one of the vecto rs is repeated
.
L et the three vectors a
B x C, C>
<A, Ax B
B
A
A
C
[
]
.
be unit vectors in the
respec t ively Then
,
c
di rec t
ion of
.
Ax B =
c s in c ,
—
Ax C
b
s in
b
sin 0 sin b A sin 0 sin b sin A
C
c C sin 0
cos ( 9 0 -p ) sin c
c
[ A B C]
b,
xb
°
~
=
A
B
A
Sin
C
[
]
c
c s in
p;
80
By
VECTOR A NAL YSIS
q
e uatin g the results and cancelling the common factor
,
siu p
sin p
sin p ,
sin b sin A
sin 0 sin B
sin a sin
e
a
y
last two may b e obtained b y c clic permutation
le t t ers Th e formul a give t h e sines of the altitudes of the
tri angle in terms of the sines of the angle and sides
wri te
Th e
.
.
A
B
A
C
[
]
B
A
B
C
[
]
[0A B ]
Hence
sin c sin b sin A
sin a s in c s in B
sin b sin a sin 0
.
A
B
C
[
]
.
i
0
:
B
[ C A]
:
[C A B ]
.
q l
express ons [ A B C] [ B CA] [ C AB ] are e ua
the results in pairs and the formul a
Th e
,
s in
,
b s in A = s in
s m c s rn
sin a sin
s
Cz
a
s in
.
B
in b s in C
s in c s in
A
vectors is z ero for the triangle is a clos ed polygon
From this equation
as
.
VECTOR ANAL YSIS
82
where a b
M ultiply b y
,
are thre e scalar constants to be determined
c
,
.
b
xc
.
t)
b xc
r
a
a
[
In
a
o
r
b bb xc
b xc
h
o
]
[a
a
b
c
c c
]
b xc
.
like manner b y multiplying the equation by
b the coeffi cients b and c may be found
x
c
.
= b b
[
[
]
[r a b ]
re a
=
[r b c ] a
a
b
c
[
]
Hence
[
c
c a
[ b c a]
denominators are all equal
equation
Th e
[a
b
c
]
b]
c a
.
c
a
[ b]
Hence this
c
]r
which
must e xist b etween the four vectors r a
Th e equation may also be written
,
r
~
b >
<
[a
r
.
c
c
]
b x
c
b
'
r
c >
< a
[
b
a
c
]
c
>
<
[
b
b
r
o
[a
a
b
c
]
a
c
]
b,
x b
a
b
c
c
]
a
x b
[
b
.
[a
,
a
c
]
c
.
x
a and
DIRECT A ND
SKE PV
PR OD UCT
OF VECTOR S
S
83
which are found b y di vidi ng the three vector products b x c
c op la n a r vectors a b c b
c x a a x b of t hree n o n y the scalar
product [ a b c ] is called t h e re c ip ro c a l s y s te m to a b c
If a b 0 were c o
Th e word n o n -c op la n a r is impor t ant
plan ar the scalar triple product [ a b 0] would va ni sh and
consequently the frac t ions
c x a
b x c
a b
,
,
,
,
,
,
.
,
.
,
x
[
a
b
c
]
[
b
a
[
]
c
b
a
c
]
would all b ecome meaningless Thr ee coplanar vectors have
no reciprocal system This mus t b e carefully rememb ered
Hereafte r When the term re c ip ro c a l s y s te m is used it will b e
understood that the t hree ve c to rs a b c are not coplanar
Th e system of three vectors reciprocal to s stem a b 0
wil l be denote d b y primes as a b c
b x c
a x b
c x a
.
.
.
,
,
’
’
[
Th e
e xp re s s io n
b
]
c
fo r
[
b
a
c
.
,
,
'
,
,
a
y
,
[
]
a
b
c
]
r reduces then to the very simple form
’
r a
a
’
r c c
3
r
r b b
0
( )
The vector r may be e x pressed in terms of the re ciprocal
system a b 0 ins t ead of in terms of a b c In the fi rst
place it is necessary to note that if a b c are non-coplanar
a b c which are the norm als to the planes of b and c
c and a a and b must also b e non-coplanar
Hence r may
be e xpressed in terms of them by means of proper scalar
coefficients a y z
’
’
,
’
o
’
,
,
’
,
,
.
,
,
,
’
.
’
,
,
,
.
,
,
.
b
or
c
a
r
[
]
Multipl successively b y
b
a
[ 0] N i
y
[
]
[a b c ]
a
H ence
b
r
c
o
a
,
o
x
b,
o
b
[
c
c
This
.
a]
,
’
r a a
z [a b c ]
r
o
b b
’
-
33
r
ro c
gives
,
r a.
=
y
ro b
2
re c
’
r c c
,
.
VECTOR ANAL YSIS
84
a b
c
the
s
stem
reciprocal
to
the
y
]
scalar product of any vector of the reciprocal system in to the
correspon ding vec t or of t h e given system is unity ; b u t
the product of two n o n -corresponding vectors is z ero
’
If a
44
b
,
’
’
c
,
be
,
,
.
’o
a
a
’
o
b
’
a
o
’
a
b b
c
’o
’o
b
c
’
o
b
a
1
c
o
c
c
’
c
a
o
’o
0
b
.
This
may b e seen most easily by e xpressing a
terms of themselves according to the formul a ( 3 1 )
r
Hence
a
b
o
Since
ro a a
’
’
a
’
b
’
’
c
a
aa
o
o
ro b
’
a
o
’
’
a
’
c
b b
bb
’
b b
'
o
’
o
b
,
c
’
in
.
’
o
’
b
’
a a
’
,
’
’
r c c
’
b
'
’
a
c
c c
o
’
b
’
'
o
c c
c c
’
’
’
.
are non-coplanar the corresponding c o e ffi
c ie n ts on the two sides of each of these three equations mus t
b e equal
Hence from the fi rst
a
’
,
b
’
c
,
’
.
From
the second
Fro m the t hird
’o
1
a
0
Na
0
’o
c
O
a
This
a
1
’o
’
c
o
O
b
o
b
0
a
’
a
O
b
c
0
.
b
1
b
’
’
o
’o
c
c
.
proves the relati ons
They may also
directly from the de fi n i tions of a b c
.
,
b >
<
c
[
]
b
a
c
b >
<
c
[
]
a
b
c
-a
a
b
c
]
b x
c
o
h
[
a
proved
,
c
[
be
’
’
’
.
b
c
b
[
o a
[
b
a
]
c]
0
]
[a
b
c
]
and so fort h
Conversely if two sets of three vectors each say A
and a b c satisfy the relations
.
,
,
,
,
Aa
o
A
0
3
A
0
0
=B h
o
B
o
a
Cc
“
2
B
0
0
z
1
03
0
Cb = O
o
,
B
,
C,
VEC TOR A NA L YSIS
86
Th eo re m
’
Ifa
b
,
’
c
,
scalar triple products [ a
reciprocals That is
’
the
and a
’
’
b
]
y
reciprocal s stems
c ] are numeric a l
b
a
[
be
c
an d
’
c
b,
,
.
[
a
[
c
b
x
[
’
a
<a
c>
b
b
’
] [
’
c
b
x
[
b
a
]
c
3
a
b
c
c
c
]
[b x c
1
]
0
x
c >
<a
[
b
]
[
c
xa a x b ]
b
c
a
c
]
.
xb ]
a
[
a b c
] c
[
xb ]
a
x b
a
b
a
c
b
x
[
a
a
1
Hence
b
[
a
c
3
]
[
a
b
o
c
c
]c
[
xb
a
2
]
.
b
a
c
]
2
.
1
_
[
b
a
c
]
means of t his relation b etween [a b c ] and [ a
is possible to prove an i m portant reduc t ion formul a
’
By
’
’
b
c
] it
,
A
P-
P B
PC
A
Q-
Q B
C
Q-
BA
B B
B C
o
o
whi ch replaces the two scalar triple produc t s by a sum of
nine terms each of which is the product of three direct pro
ducts Thus the t wo crosses which occur in the two scalar
products are removed To give the proof l e t P
R be
e xpressed as
.
.
P
,
=P A A
~
B
P-
’
B
A A + QB B
Q = Q’
=
B
B A A
B B B
’
.
[P e n ]
:
o
o
'
P C C
o
’
610
’
PA
P B
PC
A
Q-
a s
we
RA
B
B -
RC
I
[A
B
I
o
C
J[A M T
I
1
’
0
'
c
'
.
Q
,
DIRECT
Hence
f
s y s te m o
[
k
B
P-
A
Q-
Q B
B A
B B
i
u n it r e c t o r s
,
j
k is
,
r e c ip ro c a
i
1
xj
k
[1 1 k]
Fo r thi s
87
o
i
k
1 ]
1
t hre e
VE C TOR S
OF
PA
[ A B C]
[P Q R ]
jx
P
KE W R OD UCTS
AN D S
l
35
.
reas on the primes i j k are not needed to denote
k
Th e primes will
a syste m of vectors reciprocal to i
therefore b e us ed in the fut ure to denote anot her set of rect
angular ax es i k jus t as X Y Z are used to denote a
set of a xes difi e re n t from X Y Z
’
’
j
,
,
,
j
’
a re
onl
y
ri h t -h a
g
t he
Let
fth re e
sy s te m s o
ve c t o rs
whi c h
n
de d
l eft -h a
a nd
j
,
n de d
o
wn
re c i
p
s y s te
ms
f
O
,
00
B B
o
1
Hence the vectors are all unit vectors
AB
o
A
t h e ir
,
32
( )
AA
Hence
a re
ro c a
ls
th r ee
j
u n it
is the system i k and the system i — k
C b e a set of vecto rs which is its own reciprocal
A, B ,
Then b y
.
.
That
r e c to rs .
’
,
,
,
Th e
’
,
,
.
,
is perpendicular to
BA
Hence B is perpendicular to
A
C-
e
and
BC
A
a nd
.
.
.
0
Ac
B
.
,
.
C
.
0
.
C
.
B
C-
0
.
Hence C is perpendicul ar to A and B
k
Hence A B C must be a system like i k or like i
A scalar equation of the fi rst degree in a vector r is
an equa t ion in each term of which r occurs not more than
once Th e value of each term must be scalar As an e xam
ple of such an equation the following may be given
.
,
,
,
j
,
,
.
j
.
.
a
a
b xr
c
fr
o
cl
O,
,
.
VECTOR ANAL YSIS
88
where a b c d e f are known vec t ors ; and a b c d known
scalars O bviously an scalar e q uation of the fi rst degre e in
an unknown vector r ma b e reduced to the form
,
,
,
,
y
,
.
,
,
,
,
y
A= a
r
where A
is a known vector ; and a a known scalar To a c
compl ish th is result in the case of the given equation p rocee
a s follows
.
a
a
{
xb
a
a
r
b
fr
c
b
xb
d
.
,
r
f} -
c
0
d
o
d
.
more complicate d forms it may be ne c essary to make use
of various reduction formul a before the equati on can be mad e
to take the desired form
In
.
,
A
a
r
.
a vector has th ree degrees of freedom it is clear that one
scalar equation is in s u fii c ie n t to determine a vector Three
scalar equa t ions are necessary
Th e geometric in terpretation of the equa
ti on
As
.
.
A
r
( 36 )
a
L
is interesting
et r be a variable vecto r
et
( Fig 24 ) drawn from a fi x ed origin
A b e a fi xed vector drawn fr om the same
e uation then becomes
.
.
FIG 24
.
origin
.
.
Th e
.
q
7
A
r
cos ( r A)
‘
COS
a
3
,
,
,
if r be the magnitude of r ; and A that of A
.
r
is
L
cos ( r
,
Th e
r
e xp ession
A)
the projection of r upon A Th e equation therefore states
that the projection of r u pon a cer t ain fi x ed vector A must
.
VECTOR ANAL YSIS
90
From
q
four scalar e uations
ro A
=
ro B
= b
C=
r
I 'D
a
c
=d
li
the vector r may be entirely eliminated To accomp sh this
solve three of t h e e q u a t io n s and su b s t itute the value in the
fourt h
r
.
’
.
a
a
'
A D
o
B
D
O
[
]
bB
’
o
D
[ C A D]
b
c
’o
d
C Dz
=
D
d
A
B
]
[
c
[ A B C]
( 4 0)
.
vector equati on of the fi rst degree in an unknown
vector is an equat ion each term of which is a vector quantity
con t aining the unk n own vector not more than once Such
an equation is
A
.
D Er
n
r
F
where A, B , C, D, E, F
0,
are known vectors n a known scalar
and r the u nknown vector O n e such equation may in gen
eral b e solved for r That is to say one vector equation is in
general suffi cient t o determine the unknown vector which is
con ta ined in it t o the fi rst degree
Th e me t hod of solving a vector equation is to multiply it
wi th a dot successively b y three arb itrary kno wn non-00planar
vectors Thus three scalar equations are Obtained These
may be sol ved b y the methods of the foregoing article In the
fi rst place let the equation b e
,
,
.
.
,
.
.
.
.
Aa
where
o
r
r + C c
+ B b-
~
r
-
D,
a b c are known vectors No scalar c o e ffi
c ie n ts are wri t t en in the ter m s for th ey may be incorporated in
t h e vectors
Multiply the equation successively b y A B C
It is understood of course that A B C are non00planar
A, B , C, D,
,
,
.
,
.
,
,
,
,
’
’
’
.
DIRECT A ND
PR ODU
SKE IV
r
b-
’
DA
’
DB
c r
DC
a r
o
o
B
ut
a ar
’
’
DA a
’
r
Hence
r
T OF VECTOR S
91
C S
'
o
r
bB b
D-
’
b
’
’
c
c or
.
DC
’
’
c
’
.
solution is therefore accomplished in case A B C are non
oplanar
and
a
also
non
coplanar Th e special cases in
b
0
C
which either of t hese sets of three vectors is coplanar will not
b e discussed here
Th e mos t general vector equation of the fi st degree in an
unknown vec t or r contains terms of the t pes
Th e
,
,
,
,
.
r
.
y
A
-
I)
E xr,
n r,
a r,
.
That
is it will contain terms which consist of a known
vector multiplied b y t h e scalar product of ano t her known vec
tor and the unknown vector ; t erms which are scalar multi
ples of t h e unknown vector ; terms which are t h e vector
product of a known and the unknown vector ; and constan t
r may al ways b e reduced
Th e terms of the type A a t erms
to thr ee in numb er Fo r the vectors a b c -which are
multiplied into r may all be e x pressed in terms of t hree non
coplanar vec t ors Hence all the products a r br c r
may be ex pressed in terms of three Th e sum of all terms o f
r therefore reduces to an e x pression of thr ee
t h e t ype A a terms as
.
’
,
.
,
,
,
.
,
o
,
.
,
r
Aa-
y
terms of the t pes
in thi s form
Th e
r
B b-
r and
n
Cc
E>
<r
o
r
.
may also be e xpressed
.
n r
E xr
A dding
n a
’
Ex a
a r
o
’
+
a r
o
n
b
’
r +
b’
Exb b
n c
’
c or
c or
q
.
all these terms together the whole e uation reduces
to the form
r
KL
o
’
a
’
’
Km b
’
K
r
solution is in terms Oft hree non coplanar vecto s a b c
These form the system reciprocal t o a b c in terms of which
t h e prod u cts con t ai n ing the unknown vector r were e x pressed
-
Th e
,
'
,
’
,
’
.
,
.
SUN DRY APPLICA TIO N S
OF
P R OD U CTS
Ap p lic a tio n s to Me c h a n ic s
the mechanics of a ri gid b o dy a force is n o t a
vector in the sense understood in this book Se e Art 3
A force has magnitude and direction ; b u t it has also a line
of application Two forces whi ch are alike in magnitude
and direction but which lie upon different lines in the bod
do not produce the same e fi e c t N evertheless vectors are
su ffi ciently like forces to be useful in treating them
-a c t on a body a t the
If a number of forces f 1 £ 2 £ 3
same point 0 the sum of the forces adde d as vecto s is called
the re su lta n t R
48 ]
In
.
.
.
y
.
,
.
.
,
,
r
,
,
.
the same way if f 1 £ 2 f s do n o t act at the same point
t h e term resul t ant is still applied to t h e sum of these forces
added just as if they were vectors
In
,
,
.
( 41 )
a force does not differ from a vector
De n i tio n : Th e m o me n t of a force f about the poi n t 0 is
equal to t h e product of the force b y the perpendic u lar dis
tance from O to the line of action of t h e force Th e moment
however is b est looked upon as a vector quantity Its m a g
n it u de
is as de fi n ed above Its direction is usually taken to
fi
.
.
.
.
VECTOR ANALYSIS
94
f
T
h
e direction
M
o
his
is
the
magnitude
of
the
moment
T
{}
of dx f is t h e same as the direc t ion of the momen t Hence
t h e relation is proved
d
M O {f}
.
.
.
.
sum of t h e moments ab out 0 of a number of forces
ac t ing at t h e s a m e p o in t P is equal t o the moment
f1 f2
of t h e resul t ant B of the forces ac t ing at that point Fo r let
Then
d b e t h e vector from O to P
Th e
,
,
.
.
total moment ab out 0 of any numb er of forces f 1 f2
ac t ing on a rigid bo dy is equal to the to tal momen t of those
forc es ab out 0 increased b y the moment ab out 0 of the
result ant B o considered as acting at 0
Th e
’
,
,
’
.
M o {fl ’ £ 2
Let
M O {fv f2 9
,
M O {3
'
0
4
4
( )
2
be vectors dra wn from O to any point in
f1 f2
respectively
et (I d z
b e the vectors drawn
fro m O to the same points in f 1 £ 2
respecti vely
et 0
b e t h e vector from O to O
Then
d v dz ,
,
L
.
,
’
,
’
,
.
,
L
’
.
Mo
{f1
,
f2 ,
d1 >
<f1
M O {f1 , f2 ,
(l
'
d z x f2
’
f
x
l
l
(d1
( d2
n
B
is the vec or drawn from O to 0 Hence — C x f
is the moment ab out 0 of a force equal in magnitude and
parallel in direction to f1 b u t situate d at 0 Hence
ut
c
’
t
.
’
.
,
DIRECT AND SKE W
H ence
PR OD UCT
S
c
M 01 {f1 , f2 ,
MO
Ma
o
f
£
{ p 2
OF VE C TOR S
r
{R0 }
Mo
,
i
95
.
R
{ o
( 44 )
i
theorem is therefore proved
Th e resul tant R is Of course the same at a ll points
Th e
subs cript O is a t t ached merel t o sho w at wha t poin t it is
s up p o s e d to act when the moment about 0
is taken Fo r
the point of applicat ion of B affects the value of that moment
Th e scalar product of the total moment and the resultant
is the same no matter ab out what point the moment be taken
In other words the product of the total moment the result
ant and the cosine of the angle b etween them is invariant
for all points of space
Th e
.
y
.
’
.
.
.
,
,
.
£2
,
where O and O are any two points in space This important
relation foll ows immediately from the equation
’
.
MO
Fo r R Mo
’
f
{ l
a
r
§ f1
,
f2
f2
MO
Z
9
'
M o {f1
=B
,
M O {R 0 }
,
+ R
f2 ,
the moment of B is perpendicular to
the point 0 of application b e Hence
B
ut
.
B
.
-M O {B 0 }
,
.
no matt er what
.
R Mo
o
{R 0 }
'
0
and the relation is proved Th e variation in the total
moment due to a variation of the point ab out which the
moment is tak en is always perpendi cular t o the resulta nt
A poin t 0 may b e found such that the t otal moment
ab out it is parall el to the resultant Th e condi tion for
parallelism is
.
.
’
.
x
E t ra
B XM 0
,
if“ £ 2 ,
'
f
{ 1
,
£2 ,
= R>
<M o
0
0
( fl
+ R X MO
'
,
£2 ,
zRo
=
g O
VE C TOR ANA L YSI S
96
where O is any point chos en at random Replace
f
t
its
value
and
for
revity
omit
to
wri
e
the
f
b
1
2
by
Then
b races
.
,
B xM o
B B
in
,
0
}
the
.
problem is to solve thi s equa t ion for
Th e
/
0
B xM o
B xM o f
BI O {B
B
0
c
.
c R
0
.
is a known quantity M O is also supp osed to b e
kno wn
et c b e chosen in the plane through 0 p e rp e n
O and the e uation reduces to
dicular to R Then RC
Ne w B
L
.
.
q
.
RXM O
RB
O
0
B XM O
R
R-
l
chosen equal to th is vector the tota moment abo ut
the point O which is at a vector distance from 0 equal to c
M oreover since the scalar product of
will b e parallel to R
the total moment and t h e resultan t is constant and since the
res ul t ant i t self is cons tant it is clear that in the case where
they are parallel the numerical value of the total moment
will b e a mini mum
Th e to ta l moment is unchanged by di splacing the point
ab ou t which it is taken in the direction of the res ul tan t
IfC b e
’
,
,
.
,
.
.
M O § f1 , f2 ,
FOI
‘
M 0 {f1 , £ 2 ,
'
OXR
o
is parallel to B c
vanishes and the moment
ab out 0 is equal to that about 0 Hence it is possible to
fi n d n o t merely o n e poi nt 0 a b out which the total moment
is parallel to the resulta nt ; but the total moment about any
poin t in t h e line drawn through 0 parallel to R is parallel
to B Furtherm ore t h e solution found in equation for C is
t h e o n ly one which e xists in the plane perpendicular t o B
unl ess t h e result ant B vanishes Th e results that have been
ob tained may b e summed up as follows :
Ifc
’
0O
,
’
.
’
’
.
.
.
.
VECTOR A N AL YSIS
98
y
il
q
the ax is Th e velocit of a n y po int in its c rc e is e ual
to the product of the angular velocit and the ra d us of the
circle It is therefo e e ual to the product of the angular
velocit a n d the perpend c ul ar d s
tance from the point to the axis
s
Th e direction of the velocit
x is and to
a
rp e n dic u Ia r to t h e
e
p
the radi u s of the circ e described
b the point
et a ( Fig 2 5) be a vector dra wn
along the axis o f rotation in that
direct on in which a right-handed
screw woul d a d vance f turned in
the
direction
in
wh
i
ch
the
bo
y
is
F o 25
rotating
et the magnitude of a
be a the angular velocity Th e vecto r a m a y be taken to
represent the rotation of t h e body
et r be a rad us vector
drawn from any point of the ax is of rotation to a point in th e
Th e vector product
b ody
.
y
r q
.
y
i
i
i
yi
l
y
L
.
.
i
r
i
L
L
.
.
.
.
,
.
.
d
i
.
(
a r s in
a >
<r
)
a, r
y
is equal in magnitude and direction to the velocit v of t h e
terminus of r Fo r its directi on is perpen di cular to a and r
and its magnitude is the product of a and the perpendicular
distance r sin ( a r) from the point to the line a That is
.
,
.
v
a >
<r
.
y
If the
( 4 5)
body be rotating simul taneous l about several ax es
a
3
a
which pass through the same point as in t h e
]
2
3
cas e of the g roscope the velocities due to the var ou s
o t ati ons are
v
a >
<r
,
r
,
y
i
,
1
l
1
v2
v8
a
r
x
s
3
DIRECT A ND
KE W PR OD UCT
S
r
S
OF VECTOR S
99
where r 1 r 2 t a
a e the radi i vectores drawn from points
on the ax is a l
to the same point of the body Le t
ea
the vecto rs r l r 2 t a
be drawn from the common point of
intersect on of th e ax es Then
,
,
,
,
i
,
,
.
,
,
.
=r
'
o
o
y
This
shows that the bo d moves as if rotating with the
angular velocity which is the vector sum of the angular
velociti es a l 8 2 a 3
Th is theorem is sometimes known
a s the parallelogram law of an g ul ar velocities
It will be shown later ( Art ) 6 0 that the motion of any
rigid body one point of which is fi xed is at each ins tant of
time a rotation ab out some axis drawn through that point
Th s ax is is called the instantaneous a x is of rotation
Th e
ax is is not the same for all time but constantly changes its
pos ition Th e motion of a rigid body one point of whi ch is
fi x ed is th ere fore epresente d by
,
,
,
.
i
.
.
,
r
.
v
a
4
5
( )
xr
where a is the instantaneous angular velocity ; and r the
radi u s vector drawn from the fi xed point to any point of the
body
Th e most general motion of a rigid body no point of which
is fi x ed may b e treated as follows Choose an arb itrar
poin t 0 At any instant this point will have a velocit v
Re l a ti ve t o th e p o in t 0 the b ody will have a motion of rota tion
ab out some axis drawn through 0 Hence the velocit v of
any point of the body may b e represente d b y the s um of
of O and e xr t h e velocity of that po int
V the velocit
relative to 0
v v a xr
( 46)
,
.
y
.
.
y
.
y
0
.
0
.
y
0
.
VECTOR A NALYSI S
1 00
case V is p arallel to a t h e b ody moves around a and
along a simultaneou sly This is precisely t h e motion of a
scre w advancing along a In case V is perpendicular to a it
is possib le to fi n d a point given b y the vector r such that
its veloci t y is z ero That is
In
,
0
.
,
0
.
,
,
.
Thi s
a
y
xr
ma be done as follows
M ul tipl
.
y by
xa
.
vo x a
-
Le t
vo x a
At a
a a r
r be chosen perpendicular to a
Then
.
vo
aoa r
x
xa
v0
.
-is z ero and
a r
a
-
a a
y
point r thus determined has the propert that its ve l o c
ity is z ero If a line b e drawn through t hi s point parallel to
a the mot ion of the b ody is one of instan taneous rotati on
ab out t his n e w ax is
In case v is neither parallel nor perpendicular to a it m a y
b e resolved into two components
Th e
,
,
.
,
.
0
V0
V0
'
V0
i
whi ch
are respectively parallel and perpend cular to a
v
v
v
a ><r
A point may now be found such that
v
a xr
et the different points of the b ody referred to thi s point be
denot ed b y r Then the equat ion b ecomes
’
”
o
0
”
L
.
,
’
.
v
vo
’
a
Th e
’
xr
.
motion here e xpressed consists of rotation about an a xis
a and t ranslation along that a x is
It is therefore seen that
t h e most general motion of a rigid b ody is at any instant
.
1 02
As
VECTOR ANAL YSIS
l
f is paral el to a the scalar product [ a
’
dxi
a-
a
o
i
van shes
df
.
”
dxf
.
the other hand the work done by f is equal to the work
done by f during the displacement Fo r f being parallel to
If h be the common
a is perpendic ul ar to its line of action
vecto r perpendicular from the line a to the force f the work
done by f during a rotation of angular veloci t a for time
t is appro ximatel
On
’
.
.
y
y
W
:
"
hf
t
a
y
A
"
nxf t
.
y
vector (1 drawn from an point of a to any p oint of f ma
be broken up into three components of which one is h another
is parallel to a and the third is parallel to f In the scalar
triple product [ a d f o nl y that component of d which s
perpendic ular ali ke to a and f has an e fi e c t Hence
Th e
,
i
,
y
”
W a h xf
:
.
’
’
a dxf t
t
a
If a
dx ft
.
rigid body upon whi ch the forces f 1 f2
act be dis
placed b y an angular velocity 9 for an infi ni tesimal time t
and if d l d2
be the vectors drawn from any point 0 of
a to any points of f
f
respectively then the work done
1
2
by the forces fl f2
will be a pp ro xim Me ly
W= ( a d1 >
<f1
a d >
t
2 <f2
,
,
.
,
,
,
,
,
,
,
o
a
a
i q
If the
d
( l
o
(l
1
M o {fl , £ 2 ,
f
x
g
2
t
t
.
body be n e u ilibrium thi s work must be z ero
Hence
aM o if] £2
t
0
Th e scalar product of the angular ve oc t 9 and the tota l
moment of t h e forces fl f,
about an point 0 must be
z ero
A s a may be any vector whatsoever the moment itse f
must vanish
.
,
,
,
,
o
:
.
l iy
y
.
.
M o { f1 , f2 ,
O
.
l
DIRE C T
KE W PR OD U
AN D S
y
T OF VE C TOR S
C S
1 03
y
necessar conditions that a rigid bod he in e q u ili b
riu m under the action of a system of for ces is that the resul t
ant o f t hose forces and the total moment about any point in
spa ce sha ll vani sh
Conve sely if the resultant of a s s tem of forces and the
moment of those forces about any one particular point in s ace
vanish simultaneously the bod will be in e q uilib rium
0 then for an di splacement of tra ns lation D
If R
Th e
r
y
.
,
,
p
y
y
.
DR=O
o
.
W= I) f1
f, +
D-
y
r
and the tota l work done is z ero when the bod suffe s any
d s placement of tran slation
be z ero for a g ven point 0 Then for
et M O { i n £ 2
an other point 0
i
,
L
y
,
’
ut
.
’
M O { i n fa
B
i
.
y
,
by h p othes is
Henc
‘
R
a
e
y
o
MO
h
is also
MO
z
ero
f
£
{ p 2,
:
MO
£29
.
’
Hence
{ B O}
0
4t
where a is an vector whats oever B u t this e xpression is
equal to the work done by the forces when the body is rotated
for a time t with an angular velocit a about the line a
passing through the po in t O Th s work is z ero
An y displacement of a rigid bod may be regarded as a
t ranslation through a d stance D combined with a rotation
for a time t wi t h angu lar velocity a about a s u itable line a in
space It has been proved that the total work done by the
forces during thi s di splacement is z ero Hence the forces
mus t be in e uilibrium Th e theo e m s proved
.
’
.
i
i
y
y
.
.
q
r i
.
.
.
VECTOR ANAL YSIS
1 04
Ap p lic a t io n s t o G e o m e try
Relation s b e t ween
two right-handed systems of th ree
Le t i
mutually perpen di cular uni t vectors
k and i
k
b e t wo such syste ms They form t heir own reciprocal systems
Hence
,
.
j
’
,
,
j
’
,
.
.
scalars a l a 2 “3 5 bl
direc t ion cosines of i ; ;
Tha t is
Th e
,
j
,
’
a
1
b1
c
cos
cos (j i )
cos ( k i)
same manner
(i
’
,
i)
In t h e
ijj
+
’
k jj
’
’
'
,
jj
u
k kl
l
0 , 0
2
3
,
,
’
,
’
,
’
’
i k k
j
o
’
k k
’
’
’
k k k
i i
'
k
’
3
c
;
,
o
o
l
are respectively the
wi t h respec t to i k
bz , b
j
j
j
cos (i )
cos (j )
cos ( k )
2
0
2
,
+
’
b2
,
’
1
,
’
a
’
’
1
a
a
a
1
1
a
r;
,
3
0
1
2
,
’
,
’
0
3
i
’
a
,
61
3
b3
i
“
z
z
i b2
“
0
2
.
cos ( i k )
cos (j k ) ( 4 8 )
cos (k k)
,
j+
’
i + bl
2
2
a
’
,
’
’
1
j
2
2
s
’
ba
’
j
“ 2
3
O3
C
3
2
2
c
l
0
2
o
s
.
k
’
k
’
k
’
VE CTOR ANAL YSIS
1 06
the
p
d
n
e
d
rection
but
to
lie
wholly
in
the
la
an from
i
i
-plan e
k
form
upon
the
left
it
is
seen
to
lie
in
the
it s
Hence it must be the l ine of intersec t ion of those two planes
/
T
h
s
gives
the
or
1
i
s
u
magnit
de
Its
m
m
scalar relation s
k
j
j
’
’
.
i
.
— a
magni t ude 1 a l is the square of the sine of
Hence the vector
b etween t h e vectors i and i
z
Th e
th e
angle
’
.
— “ k
2
j
53
( )
is the line of inte rsection of the k and jk-plan e s and
its magni tude is the sine of the angle be t ween the planes
Eight other s imilar vecto rs may b e foun d each of whic h gives
one of the n ine lines of inte rsection Of the t wo sets of m u
t u a ll y ort hogonal planes
Th e magn i tude of the vector s in
each case the sine of the angle b etween the pla nes
Various ex amples in Plane and Solid G eometr ma
be solved by means of products
Exa mp le 1
Th e perpendicul ars from the vertices of a tri a n
gle to the opposite sides meet in a point
et A B C b e the
t rian gle
et the perpendiculars from A to B C and from B
to CA meet in the point 0 To Show 0 0 is perpen di cular
to A B Choose 0 as ori gin and let OA
A O B = B and
00
Then
C
’
’
,
.
,
i
y y
.
.
.
L
.
.
L
.
.
,
,
.
By
y
B C= C— B,
C A = A — C,
AB
z
B — A
.
h po thesis
A
and
Subtra ct ;
(C
B(A
C(B
B)
O
C)
0
A)
0,
.
which proves the theorem
Exa mp le 2 : To fin d the vector equation of a
through the point B parallel to a given vecto r A
.
.
drawn
K W PR O
DIRECT AND
Let
S
E
0 be
D
UC TS OF VE C TOR S
1 07
L
the origin and B the vector O B
et B be the ra
di ns vecto r from O to an point of the required line
Then
B
B is parallel to A
Hence the vector p ro du c t va n is h e s
y
.
.
.
.
q
Thi s
.
AX ( R
—
=0
B
q
.
A
is the desired e uation It is a vector e uati on in the
unkn own vector B Th e e uati on of a plan e wa s seen ( page
8 8) to be a scalar e uation such as
.
q
q
.
3 0
c
0
in the unkno wn vecto r R
Th e point of intersection of a line an
found at once Th e equations are
.
d
a
plan e
ma y
be
.
B
R C
A>
<R
0
)
c
A>
<B
>
<C
ACB
CRA
ACB
c
A
c
Hence
A
i
O In th s cas e h o w
solution evidently fails when A C
ever t h e line is parallel t o the plane and there is no solution ;
or if it lie s in the plane there are an in fi nite number of solu
ti ons
E xa mp le 3
Th e introduction of vectors to represent planes
Heretofore vectors have b e en used to denote plane a re a S o f
de fi nite ex tent Th e direction of the vector was normal to
the plane and the magnitude was equal to the area to be re
presented B u t it is possible to use vectors to denote not a
plane area but the entire plane itself just as a vector represents
a point Th e result is analogous to th e p la n e co ord nates of
analytic geometr
et 0 be an assumed origin
et M N be
a plane in space Th e plan e M N s to be denoted b a vector
Th e
.
,
,
.
.
'
.
.
i
,
y L
.
.
.
i
.
L
y
VECTOR A NAL YSIS
1 08
p
whos e di rection is the direction of the perpendicular drop ed
upon t h e plane from the origin 0 and whose magni t u de is the
Thus the nearer
r e c ip ro c a l of t h e length of t hat perpendicular
a plane is to t h e origin the longer will be the vecto r whi ch
represent s it
If r b e any radius vector dr awn from the o ri gin to a point
in t h e plane and if p be the vector whi ch denot e s the plane
t hen
.
.
,
t
is the equation of the plane
o
1
p
Fo r
.
cos ( r p ) p
N o w p t h e lengt h of p is t h e reciprocal of the perpendicular
distance from O to t h e plane O n the other hand r cos ( r p )
is t hat perpendi cul ar dis t ance Hence rp mus t be unit
Ifr and p b e e x pressed in terms of i
k
t
o
r
p
,
.
,
y
.
.
,
r
=
=
p
Hence
ro p
=
x
u
xu
i + y j+
i +
v
f
+ y
c
z
j
,
.
,
k
j+ w k
+
z
w
l
z
quantities u r w are the reciprocals of the intercepts of
the plane p upon the ax es
Th e relation bet ween r and p is symmetrical
It is a rela
tion of duality If in the equat ion
Th e
,
,
.
.
.
ro p
= 1
be regarded as variable t h e equation represents a plane p
which is t h e loc us of all points given b y r If however p be
regarded as variab le and r as constant t h e equation re p re
sents a point r through which all the planes p pass Th e
development of the idea of duality will not be carried out
It is familiar to all students of geometry
Th e use of vec
to rs to denote planes will scarcely be alluded to again until
Chapter VII
r
,
.
,
.
.
.
.
VE C TOR ANA L YSIS
110
i
vanishes is t hat their vector product van shes
mutative laws do not hold
.
Th e
com
.
AXB =
i>
<i
B XA
>
<j
j
i xj
AXB
( A2
B
S
k xk
jx i
k
xk
j
k>
<j= i
k>
<i
i Xk =
As 8
i
2)
A
B
( s
( A1
AXB
j
A1 B
l
B
3
A2 3
i
j
AI
A2
A
B
3
B
]
0
a
)j
k
1)
k
2
Th e
q l
scalar triple product of three vecto rs [ A B C] is e ua
to t h e volume of the parallelopiped of wh ch A B C are three
edges whi ch meet in a point
i
,
.
[ A B C]
AB xC
B Cx A
CAxB
AxB C
B xCA
CxAB
[ A B C]
o
[A CB ]
.
,
DIRECT
If the
KE W PR ODU
AN D S
T
VE C TOR S
C S OF
component of B perpendicular to
A
be
111
”
B
,
AA
CB
A-
AB C
ACB
CB A
CB D
A-
ADBC
o
C
D]
B
[A C D] B
[
D
D] 0
A
[
[ A B C] D
A
( 26 )
.
q
e uation which subs ists between four vectors
Th e
A, B , C, D
is
[B
C D] A
Th e
( 27)
.
ication of formul of vector analysis to obt i the for
of Plane and pherical rigonometry
system of vectors
is said to be reciprocal to
A ppl
mul a
=
0
A
B
C
1)
[
]
[D A B ] 0
C
[ D A] B
a n
a
T
S
a
’
,
b
’
,
c
.
’
the
sys t em of three non-coplanar vectors
b x
when
[
b
a
c
x a
[a
b
c
c
]
,
r
b,
a,
c
]
C
a
x b
[a
b
2
9
( )
vector r may be expressed in te ms of a set of vectors and
its reciprocal in two similar ways
A
r
r a
.
r
y
’
a
ro a , a ,
b b
r c
bb
r e
r
.
’
’
r
.
’
’
3
( 0)
c
y
( 31 )
necessar and su ffi cient conditions that the two s stems of
non-c oplanar vecto s a b c and a b c be reciprocals is that
Th e
r
’
,
a
a
’o
b
a
’
o
c
y
l
.
,
,
b b
b
1
o
c
2
“
r
Ma
c
form a s stem eciprocal to
form a s stem reciprocal to a b c
Ifa
’
, b
’
,
y
0
’
’
,
[
a
t
b
l
c
’
,
’
a
’o
’
’
]
’
,
’
.
a,
’
o
b
b, C ;
2
3
( )
0
.
then
a,
b, c
il
w l
VECTOR ANAL YSIS
11 2
P A
B
P-
C
P-
A
Q-
AB
a c
BA
AB
B C
o
j
system i k is its own reciprocal and if conversely a
syste m b e its own reciprocal it mus t b e a ri ght or left han ded
syst em of thr ee mutuall y perpendi cular unit vecto rs App l i
cat ion o f t h e theory of reciprocal syst ems to the solution of
scalar and vecto r equati ons of t h e fi rst degree in an un kn own
vector Th e vector equat ion of a plane is
Th e
,
,
.
.
r
A
a
( 36 )
.
Applications
of the methods developed in Chapter IL to the
t reat men t O f a system of forces acting on a rigid body and in
part icular t o the reduction of any system of forces to a single
force and a couple of which the plane is perpendicular to that
force Application of the methods to the treatment of
ins t antaneous motion of a rigid b ody o b taining
,
.
(46)
where
v is the velocity of any point v a translational ve l o c
ity in the direction a and a the vector angular velocity of ro
t a t io n
Further application of t h e methods to Ob tain the
condi t ions for equilib rium b y making use of t h e principle of
virt ual velocities A pplications of the method to ob tain
t h e rela t ions which e x ist b etween the nine di rection cosines
of the angles b e t ween two systems of mutua lly orthogon a l
axes A pplicat ion to special prob lems in geome t r including
t h e form under whi ch plane co o rdinates make their appear
ance in vector analysis and the method by which planes ( a s
,
0
,
.
.
.
y
1 14
VE C TOR
'
y
AIVA L YSIS
b vecto r methods that the formu
ume of a tetrahedron whose four vertices are
12
.
Show
v
(
“
v y
rs
2
1
( x2
)
,
z
312 :
“
H
8
6
a
x
(
3
z)
I
’
I
ll
z
N
3
5
z
y3
z
2
s
“4
z
75
,
l
a
“
)
(
3
p
for the
90
z
l
)
1
,
1
2
1
8
1
,
use of formula ( 34 ) of the te xt Show that
[
where
a
, b,
a
b
c
]
=
a
bc
are the lengths of
c
a
,
n
1
l
m
l
1
i y
respect vel
b, c
an d
where
cos ( b c ) m cos ( c a ) n cos ( a b )
uantit )
I4 Determ ine the perpendi cul ar ( as a vector
which is dropped from the origin upon a plane determined b
the termini of the vecto s a b c Us e the method of solution
given in Art 4 6
1 5 Show that the volum e of a tetrahe dr on is e ual to one
sixth of the product of two opposite edges by the p e rp e n di c u
lar distance between them and the sine of the included angle
1 6 If a line is drawn in each f a ce plane of an t ri e dra l angl e
through the verte x and perpen di cular to the thi rd e ge t h e
t hree lines thus obtained lie n a plane
t
:
,
,
,
,
,
A
r
.
,
,
q
y
y
.
.
q
.
.
y
.
i
.
.
d
,
C
H
AP TER III
TH E DIFFER E N TIA L CA LCULU S OF VE CTOR S
Dife r e n ti a tio n
f
o
Fun c tio ns
f
O
On e Sc a la r Va ri a ble
a vecto r varies and changes from r to r the in cre
ment of r will be the d fference between r and r and will be
denoted as us u al b A r
IF
y
’
i
’
.
Ar =
r
r
y
where A must be a vector quantit
If the variable T b e
unrestri cted the increment A r is of course also unrestri cted
it may have an magnitude and an direction If however
the ve c t o r r be rega rded as a function ( a vector function) of
a Single scalar variable t the value of A r will be completely
determined when the two values t and t of t which give the
two values r and r are known
To o b tain a clearer conception of the quantiti es involved
it will be advantageous to thi nk of the vector r as drawn
from a fi x ed origin 0 ( Fig
When
the independent variable t changes its
value the vector r will change and as t
possesses one degree of freedom r will
var in such a way that its te rminus
describes a cu ve in space r wil l be
the rad us vector of one point P of
the curve ; r of a neighboring point P A
be the
cho d P P of the curve Th e ratio
y
.
y
,
.
’
,
’
.
,
.
,
y
r
i
r
.
’
’
,
’
.
Ar
A t
r
,
VECTOR ANAL YSIS
116
will b e a vector collinear with the chord P P b u t magnifi ed
in t h e ra ti o 1 : A t When A t approaches z ero P will a p
proach P t h e chord P P will approach t h e tan gent at P and
the vector
03r
will approach
’
’
.
’
,
,
dt
which is a vec t or tangent to the curve at P di rected in that
s ense in which t h e varia b le t increases along the curve
Ifr b e e xpressed in terms of i
k as
j
,
the components
(
r
r
1
,
r
Ar =
r
,
r
=
r
2,
r
3
will be functions of the scalar t
’ —
Ar
( r2
j
2
r
3
k
A r2 ) j
A r3 ) k
( r3
= A r i + Ar
1
2 j+ A r3 k
r
A
,
A t
dr
r
.
A rl
At
i +
1
A r1 ) i +
1
.
r
A
2
A t
d rl
d rg
,
d t
r
3
k
A t
,
d t
d
r
3
d t
k
( 2)
'
Hence the components of the fi rst derivative of r with re
spect t o t are the fi rst derivatives with respect to t of the
components of r Th e same is true for the secon d and hi gher
derivati ves
.
.
d
z
dt
d
"
r
‘
i
d r
2
d t
r
d t”
In
d
“
1
2
r
d t
"
s rmrla r
a
manner if r
non-coplanar vecto rs a b
,
'
1
1
,
e xpressed in terms
as
be
c
a a
+ bb +
c c
y
an three
VE C TOR
118
A(a
Ab
b)
o
At
Hence in the limit when
3
7
A
+
0,
At
(
(
a
o
b x
c
)
a
8
(
da
X
-b
x
o
a
Ab
At
At
db
X b)=
Aa
Aa
d
a
NAL YSIS
b x
c
.
x
x[ x
b
c
]
( 6)
.
d
y
last three of these formul a may be emonstra ted exactl
as the fi rst was
Th e formal process of di fferentiation in vector anal s is
differs in no way from that in scalar analysis e x cept in this
one point in whi ch vector analysis always di ffe s from scalar
anal sis namely : Th e order of the factors in a vector product
Th e
y
.
y
r
,
THE DIFFERENTIAL CAL CUL US OF
cannot be cha nged wi thout changing the sign
Hence of the two formul a
ECTOR S
119
V
Of
the product
.
d
d
r
ly
the fi st is evident incorrect but the second correct In
other words scalar di fferentiat ion must take place w thout
altering the order of the factors of a vector product Th e
factors must be fferentiated in s i tu Thi s of cou se was to
be e xpected
In case the vectors depend upon more than one variable
th e results are practicall the same
In place of tota l deriva
t ive s with espect to the scalar vari ables partial derivatives
occur Suppose a and b are two vectors which depend on
three scalar variables x y z
Th e scalar product a h will
depend upon these three variables and it will have th ee
a t al der vati ves of the fi rst order
i
,
,
di
r
.
.
.
.
y
r
.
,
.
,
pri
,
o
.
i
,
.
(
a
o
h)
o
b +
a
o
b +
a
o
b + a
o
o
.
ri
second pa t al de ri vatives are formed in the same
9
2
a wa y
as
9x
r
VECTOR ANAL YSIS
1 20
O ft en
it is more convenient to use not the derivatives but
t h e di f
feren t ials
Thi s is part icularly true when dea ng wi th
rs t differentials
Th e form u l a
(4 ) b ecome
fi
li
.
.
d
(
a
o
b) = da
-b
+
a
o
db ,
and so forth As an illus tration consider the following
If r be a un i t vector
e xample
.
.
r
o
r
l
z
.
Th e
locus of the te rminus of r is a spherical surface of uni t
radius described about the origin r depends upon two vari
ab les Different ate t h e equation
i
.
( d r)
.
.
o
r
Hence
+
= 0
r
.
r
o
dr
z
0
.
Hence the incremen t d r of a u n i t vecto r is perpendicular t o
t h e vecto r
Thi s can b e seen geometrically
If r traces a
sphere t h e variation d r must b e at each point in the tangent
plane and hence perpendi cular to r
V ector methods may b e employed advantageously
d
in the iscuss ion of curvature and torsion of curves
et r
deno t e the radi us vector of a curve
.
.
.
L
.
where f is some vector function of the scalar t In most a p p li
ca t ions in physics and mechanics t represents the time
et
s be t h e
lengt h of a re measured from some de fi nite point of
the curve as origin Th e incremen t A r is the chord of t h e
curve Hence A r A s is appro ximately
equal in magnitude
t o unity and
approaches unity as its limi t when A 3 becomes
in fi nites imal
.
.
.
.
.
L
VE C TOR
1 22
t
d
A
NAL YSIS
t = e -c = n
o
=n
=1
n
o
t
=0
an
Different at ng the fi rst set
dn =0
dc =n t dt =c and the second
d n + dc
c
c
t
d
c
t d +
B u t d t is para e to c and consequentl y pe rpend cula
t
ii
C
o
o
n
o
o
.
o
,
ll l
i r
o
o
n
dt = o
.
to
n
.
.
t = 0
dn Hence
B u t the ncrement
Th e increment of n s pe rpendicular to t
of n is also perpendi cular to n It s therefo e parallel to e
d u /d s it is pa a el t o d n and hence
As the tortuosit is T
to c
Th e tortu osit T is
i
y
y
.
d
dc
d
z
a
s
r ll
X
d
x
a
)
"
z
s
1
h
/
d
x
s
m
d
x
d
s
s
z
+
d
x
s
d
.
1
z
1
3 r
d
dr
2 r
d
dr
+
ds
ds
r
3 r
d
dr
d
r
.
,
r
?
i
.
t
(
i
.
ds
3
Vc
.
c
1
d s x/ c
Th e fi rst
te rm of thi s expression v a n ishes T moreover ha s
2
b een seen to be para llel to C
d r/d
Conse uentl the
magnitude of T is the scalar product of T b the un it vec
tor c in the direction of C It is desirable however to have
the to rtuosit positive when the normal 11 appea s to turn in
the posit ive or counte rclockwise direction if viewed from
that side of the n o-plane upon w h ich t or the positive part
of the curve lies With this convention d 11 appears to move
in the direction — c when the tortuosit is positi ve that is 11
turns away from C Th e scala va ue Of the tortuosit w
therefore be given by — c T
q
.
y
y
y
r
.
.
r l
.
.
y
,
y ill
,
THE DIFFERENTIAL
B
ds
x
d
s
1
d
dr
m
3
is parallel to the vector
ut c
r
>
<
da
An d
AL C UL US OF VE C TOR S
3 r
d
dr
T
— c
C
d
x
ds
al
d
z
d
3
/
s
r
d
z
ds
Henc e
2
r
d
z
1 23
.
z
r
—
2
s
T
(
is a u it vector in the direction H e c
c
C
n
d
C
Hence
n e
.
d
T
2
2
d
dr
d
ds
d
d
2
2
d
3
3
1
r
dr
r
d8
2
s
x
d
3
3
1
C C
r
2
r
z r
d
2
2
d
r iy y
i
r
q
d3 r
3
y
r
d i
to tuos t ma be obta n ed b another metho wh ch
s somewhat sho ter if not uite so st aightforward
u
n
t t = 0
C = c
— d c dt c
t
ence
— dn dc u
c
— dt n
t
dn -
i
H
Th e
o
.
o
.
o
o
o
.
l
H
is parallel to c ; hence perpendicu ar to 11
ence
0 B u t d n s perpendi cular to 11
0 Hence d n t
dt n
Hence d 11 must be parallel to c Th e to rtuosit is the mag
of d n d 3 taken however w th the negative sign
n it u de
because d n appea s clockwise from the positive d rect on of
the cu ve
ence the scalar tortuosit T ma be g ven by
NO W d t
i
.
.
.
r
.
H
r
dn
T=t >
<
i
y
dc
0
.
y
.
i i
y i
1
4
( )
VECTOR A NA L YSIS
1 24
d C
t x
c
d
CC— t x
c
-C
/
C
C
x
.
s
t x
c
-C =
0
d C
t x
c
t
C
dr
d
s
d
2
ds
Cartesian
-C
dr
2
d
2
r
2
s
d
3
d
2
d
d
s
r
s
3
r
2
coordi nates this b ecomes
d
x
ds
d
dy
ds
ds
m
d
s
d
3
d
s
z
a
2
x
3
ol s
2
d
3
d
d y
3
ds
d
s
3
s
2
e
3
Th os e
who would pursue the study of twiste d
surfaces in Space further from the s t andpoint of vecto rs w l
fi n d t h e boo k Ap p lic a tio n de l a M éth o de Ve c to rie lle ole G ra ss
m a n n a l a G eo metri c In n ités i m a l e 1 b
e
xtreme
F
E
H
B
fi
y
il
ly
“
‘
”
1
Pa ri s
,
Ca rré e t Na u d
,
1 89 9
.
VE C TOR A NAL YSIS
1 26
i
d
introduced b y N ewton It w ll als o be conve n ien t to enote
the unit tangent to the curve b y t Th e e uat ons become
.
q i
.
is the rate of change
is a vector uantit
et it be denoted
defi ni tion
Th e
a cc e le ra
q
ti o n
y L
.
Av
LIM
M
"
_
i
Df
ferentiate
dv
G At
i
dt
dv
d
dr
dt
dt
ll t
the expression
dv
—
dt
d
— r
—
v
( v t)
d
dt
dt
dv
dt
z
d
i
s
dt
2
s
dt
dt
d
dt
ds
d t
r
r
where C is the (vector) curvat u e of the cu ve and v is the
speed in the curve Substituting these values in t h e e uat on
the result is
d
“
b
q i
.
“
7
a
a
A=
é
?
Th e
.
s
2
t + v C
i
.
“
i
f
“ t”
i
acceleration of a particle mov ng in a curve has there
fore been broken up into two components of which one is p a ra l
le l to the tan gent t and of which the other is pa allel to the
curvat ure C that is p e rp e n dic u la r to th e tangent That thi s
resoluti on h a s been accomplished would be unimportant were
r
,
,
.
TH E
DIFFERENTIA L CAL C UL US OF VE C TOR S
1 27
it not for the remarkable fact which it brings to light Th e
component of the acceleration parallel to the tan gent is e ual
in magni tude to the rate Of change of sp e e d It is entirel
ha t sort t ll rve the p a rti cle s describing
indep e n de n t p f w
It would b e the same if t h e particle described a right line
with the same speed as it describes th e curve On the other
hand the component of the a cceleration normal to th e tan gent
is equal in magnitude to the product of the s uare of the
Speed of the particle and the curvatu re of the curve
Th e
sharper the curve the g eater this component Th e greater
the Speed of the particle the greater the component B u t the
rate of change of sp e e d in path ha s no e ffect at all on this
normal component of the accelera ti on
k as
If r be e xpres sed in te r m s of i
.
i
.
“
V
q .
[
.
q
,
y
.
.
q
,
r
.
.
,
.
,
x
A = v=
i + y
,
j+
515 1 + y
r
A = v=
j
.
z
k,
j+
z
k,
+ y y +
x x
s
z
z
From
th ese formul a the difi e re n c e between 3 the rate of
change of spe ed and A i the rate of change of velocity
is apparent J
ust when this difi e re n c e fi rst b ecame clearly
recogniz ed woul d be hard to say B u t certa in it is that
N ewton must have h a d it in mi nd when he stated his secon
law of moti on Th e rate of change of velocit is proporti ona
to the impressed force ; but rate of change of speed is not
Th e h o do gra p h was introduced by Hamil ton as an
aid to the s t udy of the curv li near motion of a particle
With any a s sumed origin t h e vector velocit i is laid o ff
Th e locus of its term inus is the hodograph
In other words
the ra d us vector in the hodograph gives the velocity of the
,
i
,
,
,
.
d
l
.
y
.
.
i
i
y
.
.
.
,
VECTOR ANAL YSIS
1 28
y
particle in magnitude and dir ection at an instant It is
pos s ib le to proceed one step further and co n struct the hodo
graph o f the hodograph This is done by la ing OR the
vector acceleration A r from an assumed origin Th e
radius vecto r in the hodograph of the hodograph therefore
gives t h e accelerat ion at each instant
2
F
9
i
et
a
particle
revolve
in
a
circle
e
1
m
l
x
a
E
)
( g
p
of radi us r with a un iform
V
a n g u l a r v e l o c i t a Th e
speed of the particle w ll then
b e equal to
.
‘
y
L
.
.
y
L et
v
a
.
i
r.
the radi us vector
F G 29
drawn to the particle Th e
is perpendi cular to r and to a It is
i
v a >< r
I
he
r
.
.
.
velocity
v
.
.
vector v is always perpen di cul ar and of cons tant magni
a r
t ude
Th e hodograph is t herefore a circle of radius v
Th e radi us ve c t o r i in this circle is just nine t y degrees in
advance of the radius vector r in its circle and it c o n s e
quen t ly desc ri bes t h e circle with the same angular velocity
Th e acceleration A whi ch is the rate of change of v is
a
always perpend cular t o v and equal in mag n itude to
Th e
.
.
'
,
i
A =
Th e
accelera tion
T
:
as
Hence
B
ut
a
A
o
a
ff
v
a
z
r
.
may b e given by the formula
A
xv
a
a ct
a
a oa r
.
is perpendicular to the plane in whi ch
i
z
Th e
A
-
— a a r
—a 2
r
r
lies a r
,
0
.
.
accelerat ion due to the uniform moti on of a particle in
a circ le is directed toward the centre and is equal in magni
tude to t h e square of the angular velocity multiplied b y the
radi us of the circle
.
1 30
VE C TOR
l
A
NAL YSIS
lly
Perhaps it wou d be well to go a li ttle more ca refu into
th s question If r be the ra di us vecto r of the pa rticle in
its pat h at one instant the ad u s vector at the ne x t ins ta nt
Ar
Th e area of the vector of which r an r
is r
A a re
t h e b ounding ra di i is appro ximatel e ual to the are a of t h e
A
a n d th e
triangle enc osed b
chord A r This
a rea is
i
r i
.
,
l
yr r
r
,
1
d
yq
.
,
.
1
1
r
Th e
rate of
consequently
l l
LE I
At r 02
’
-
description of are by
r)
1
a
'
ra
the
Ac
e
s
o
di us
£ 3:
z
A ,
is
B
4
r
.
vecto
L
‘
r
is
t v r
"
5
L et Tand T be two values of the velocity at poin
and which
near together
acce l e rati on at
the limit of
t wo
o
P
x A
”
LIM
At
r
Po
a re
re ak up the vector
r l
Th e
.
A
l
E i
r
r°
A t
into
p i lr
i
io
components
t wo
pa alle and the other per end cu a to the
ts
P0
one
a c c e l e ra t ro n ro
Af
A t
y
_ x
l/ n?
o.
Th e
unit when A t approaches z ero
approaches z ero when A t approaches z ero
r
p oac
he s
q antity
qu tity
u
Th e
.
.
2:
an
a
p
y
DIFFEREN TIAL
TH E
Hence
r
A L C UL US OF VE C TOR S
C
1 31
A
x r — rO X ro
r
i i
ch of the thr ee te ms upon the ri ght-hand s de s an
in fi ni tesimal of the s e c o n d order Hence the rates of de s c rip
ti on of a ea at P and P difi e r b an in fi nitesimal of the
Th s is true for an
s econd order w th respect to the ti me
point of th e curve
ence the rates mu st b e e xactl e ua
at all points This proves the theorem
Th e mo t ion of a rigid bod one po nt of wh ich is
fi x ed is at an instant a rotation about a n i n sta ntaneous a x is
passing through the fi x ed po int
et i k be three axes fi x ed in the bod but mov ng in
spa c e
et the ra di us vector r b e drawn f om the fi x e point
to an point of the bod
Then
B
ut e a
r
y
.
i
o
H
.
y
.
y
L j
L
y
,
y
d
d
Substit uting
r
q
d
.
x
r
r
i + y
o
j+
z
d i + y d j+
k,
dh
z
r
i, d
rj r
,
d
k
z
w
( i
o
di + y i
o
d j+
z
i
+
( j
o
d i + yj
o
d j+
z
j
+
(x k
x
i
i
i
d
= (d r
ut
ce
y
r
the values of d
s econd e u ation
H en
i
.
,
x
B
y
y q l
.
.
th e
i
.
d j+
o
=
j j
j
o
di =0
=
d
k
k
d
0
+
j
j
k di + i dk = 0
r di =
o
o
or j-d i
or k -d j
or i d k
o
=
d
k
dh
j
i
=o
.
obtaine
d k) i
d k) j
d
1 32
VECTOR A NAL YSIS
i i
r
d
x
u
st
t
t
ng
th
s
values
in
the
e
pression
for
e
e
u
S b
—
—
d
i
z k
d r = (z i d k
yj
— x i k
d
k
d
)k
+ (y
j
is
Th
o
is
a vector product
.
~
d i)
i
.
.
d k j+ j
a
=k
dj
di
,
"
o
d t
d t
d
r
d t
k
.
>
<r
a
d t
3
.
Thi s
dy i
li
shows that the instantaneous motion of the b o
s one
a
about t h e ne a
Of rotation with the an gu lar velocit
Th e
Thi s angular velocit chan ges from n s tant to instant
proof of this theorem fi ls the lacuna in the work in Art 5 1
Two in fi nitesimal rotati ons may be added ike vecto s
et a , and n a be t wo angular velocities Th e d spl ac e ments
due t o them are
y
i
l
y
.
L
l
i
.
Ifr
be
d splaced by
i
r
If it
a,
d1
r
d2
r
=
r
y
then be displaced b
a
z
r
,
+
a
x
l
r
d t
.
d i r
( d t ) of or er h ghe tha n the
2
fi
rt
s
,
i
d
r
=
a
l
.
it becomes
Hence
neglected
r
.
.
it becomes
+ d1
If the in fi n ite s im a ls
.
>
<r d t +
r
x
a 2
wh ch proves the theo em
.
Ifboth
a
1
'
1
‘
3
r
d t,
i id d y
sides be d v e b
2)
x
r.
dt
be
1 34
VECTOR A NAL YSIS
r
l
i r
i y
whe e C is some co nstant vector To a ccomp is h th e nte g a
tion in a n y pa rticular cas e ma be a m att er of some d ffi cult
just a s it is in the case of ordi nar integration of sca ars
Integrate the e u at on of mot on of a
Exa mp le 1
rojectil e
Th e equat on of motion is s im l
y
p
y
q i
py
i
.
.
g,
r
r
i
y
a
wa
ve r
.
f= g t + b ,
whe re b is a constant of integration
velocit at the time t 0
y
.
l i i l ys
which exp e s ses th e fact that the acce erat on s
ticall do wnward and due to gravi t
y
l
It
.
is evi dently
th e
.
i
i
is a nother cons tant of integrati on It s the posit on vecto r
of the point at time t = 0 Th e path wh ch is given b th s
as t e uation is a parabola That th is is so ma be seen b
e xpress ing it in terms of x a n d y and eliminating t
Th e rate of description of areas whe n a a
Exa mp le 2
t o le moves under a central a ccelerati on s cons t ant
f ( r)
Since the acce erat on is parallel to the a us
c
.
l q
.
i
y
.
.
i
i
l i
r
H
r
_
—
-
ci
—
a nd
which
>
<
d
d
ence
r
proves
th e
t tement
s a
r di
.
x
.
T
:
C,
0
,
y i
y
pr
TH E
DIFFERENTIA L
L UL US
CA C
q
Integrate
VE C TOR S
OF
1 35
ri
the e u ati on of motion for a pa t cle
moving with an accelerati on to ward the cent e and e ual to
a
constant multiple of the in ve se s ua e Of the stan ce
from the centre
Exa mp le 3
r
r q r
q
di
.
G iven
Then
Henc
1
'
X
rx
e
e
0
r
i
.
C
.
q u tio s together with x
rx( x )
{ r
r
a
n
.
1
r
r
7
r
3
r
-}
r r
r r
.
o
H ence
Ea c h
s
id
e
In te grate
.
r i
i yr
i q ality is a perfect
of th s e
u
difi e re n tia l
.
Then
i
iy
whe e e I s the vector constant of inte grati on e s its ma gni
tude and I a unit vector in its rection Mul t pl the e ua
t on b
di
r x
i
0
2
C
r
.
.
r
+
e r
o
I
.
q
VE C TOR
1 36
A
NA L YSIS
and cos
=
r +
p
y
cit
.
r
,
I)
.
e r c os u .
P
r
This
(
c os
it
:
1 +
q i
e c os u
i
dir i
is the e uat on of the ellips e of wh ch e IS the e c c e n tri
ect on of the major
Th e vector I is drawn in the
Th e length of th s ax is is
i
P
a
1
e
2
i
is p o ssible to carry the integrat on further
the time So far merel the path has been found
It
y
.
Sc a la r Fu n c tio n s
and
obt ain
.
n i n Sp a
P
s
i
t
i
o
o
f
o
ce
Th e Op e ra tor V
.
lr
fun ction V (x y z ) which takes on a de fi nite sca a
value for e a ch set of coordinates x y z in space s called a
scalar function of position in space Such a funct on for e x
A
,
i
i
,
,
,
.
a
m p l e , is
x
2
+ y
2
+
z
2
=
,
r
is function gives the s quare of the d st nce of the point
Th
i
function V will be supposed to
be in general continuous and single -valued In physics scalar
fun ctions of position are of constant occurrence In the
theor of heat the te mperature T at any poin t of a body is a
scalar function of the position of that point In mechani cs
and theories of attraction the potential s the all-importan t
function This too is a scalar function of position
If a scalar fu nction V b e set equal to a constan t t h e e ua
tion
(x
,
y,
2
) from the origin
a
.
Th e
.
y
.
i
.
,
.
,
.
,
V ( x, y ,
z
)
c
.
q
( 2 0)
de fi nes a surface in space such that at every point of it the
function V ha s the same value 0 In case V b e the tem era
.
p
VE C TOR
1 38
A
NAL YSIS
r t i cre s
vector sum which is the re s u l t a irt
o f V is denoted b V V
Th e
y
a e of n
a e
.
VV
i
9
9V
9V
9V
—
9
9 g
x
"
21
o
z
dir cted rate of cha ge of
i
d re cted
or vector derivat ive of V so to speak Fo r th s reason V V
will be c a lled the de riva tive of V; and V the p ri m i tive of
V V Th e te ms gr a die n t and s lop e of V are a so us ed for
V V It is custo mar to egard V as an operator which obta ns
ro m a scala r functi on V of posit on in space
a vecto V V f
V V re p re s e n ts
n
e
a
r
a
l
,
y r
r
.
i
.
,
.
V
i
9
_
i
9
9 31
x
9
i
9
a
V=i
i
is symbolic perator
( 22 )
.
9
9g
x
.
z
y
was introduced b Sir W R
amilton and is now in uni versal emplo ment There
1
seems however to be no uni versally recogniz ed name for it
although owing to t h e frequent occurrence of the s mbol
some name is a pra ctical necessit
It has been found by
experience that the monos ll able de l is so short and eas to
pronounce tha t even in complicated formul a in which V occurs
a num ber of times no inconveni ence to th e speaker or hearer
arises from the repetition V V is read Simpl as del V
Although th s operator V h a s been de fi ned as
H
Th
V
O
y
y
y
i
V
m
p
ha r
i
co ne
th e te r
e us e
Ot h e rs h a ve
the
no ne
t h e wo r
e
tric itcit a vo
ds
‘
na
m e is
y
.
y
—
Ij
TE
C
”
9
ay
a ny s
ea
,
e th n
te
.
o
wn
it s
n o te
ec a
is t o b e
y
so
bla
t o its
e n e ss
e
a nc e
to
an
A tle d
re s e
n ve rt e
a nc e
d Aand
to a n
h a ve
de s
is
re a
b ut fo r
na
t
no t
e c tu r n
to o t h a t
oe s
o n a n d re
d vul
a nd
no t
e
e rs t o
.
In
ur os e s o
c o n us e
the
ee
f
,
s
as
“
r nt n
n s t ru c t o n
th e s e a
er or
ue n t
e t t e rs
h e o rie de r
o
fo r
Ass yr a n
c o ns e
t h e o r e r o ft h e
n ve rt n
Flip p l in h is E infi h rung in die M a xwe ll ’s c h e
.
s
a
t o o e u h o n o us n a
n e c e s sa r
—
q i d
o ft e n re
u re
lta
H o w th
V
”
.
i
,
m Nd liki g f i i d mbl
i
q
l
d
p i
m
by i
i g
d
l
dD
T
E
i
p i l ig i
f
ymb l
O
i
d
i gd
d d
pi i g p i l
l
i g
p p
i
i
m
i g
m
i g
d
f
p k
p d
So
.
y
.
9
.
.
,
,
1
.
so
y
in
le o
die
p e ra t ion
no
a rt c u a r
e
b e a re r
th
n
e ve n
rs re
wh e n
TH E
DIFFERENTIA L
C
AL C UL US OF VECTOR S
1 39
so that it appears to depend upon the choice of the axes it
is in reali t independent of them This would be surmised
from the interpretation of V as the magnitude and direction
of the most rapid increase of V To demons t rate the inde
’
’
k
and
a
new
set
of
n c e tak e another set of a x es i
n
d
e
e
p
’
’
eferred to them Then V referred to th s
variables x y z
s stem is
y
,
.
.
’
y
,
,
r
V
I
,
,
j
’
i
,
.
v
— ‘
9
9
e
9g
'
x
9
+ k
’
9
z
r
( 2 2)
’
i
and
from i j to j
ctually carr ing out the
erentiat ons and
taking into account
identities ( ) an d
act ally be transformed into
B y mak ng use of the formul ae
1 04, for transformation of a x es
a
,
di f
f
y
,
k
i
’
,
’
,
i
the
49
V
u
V
’
page
k and b
b
fi nall
’
V ma
Art 53 ,
’
y
y y
y
.
V
r
.
deta ils of the proof are omitte d he e be caus e
s horter method of demonstration is to be g ven
Consider t wo surfaces ( Fig 3 0)
Th e
,
i
an
othe
r
.
.
V ( x, y , z ) =
c
and
i i y
upon whi ch V is constant and which are more over nfi n tel
near together
et x y z be a given point upon the surface
V
0
et r denote the ra
di n s vector drawn to thi s
d
’
‘
T
poin t from an fi x ed origin
V(f iJ
VW
Then any point near by in
g
the neighbo ring s u rface V
c
d c may b e represente d
by t h e radius vector r d r
Th e a c tua l increase of V fro m
t h e fi rst surface to the second
is a fi x ed uan ti t d o
Th e ra te of in c eas e is a va r a bl e
:
.
L
L
.
,
,
y
.
‘
c
.
q
y
.
r
i
VECTOR A NAL YSIS
1 40
quantity depends upon the direction wh ch is fol
lowed when pa sing from one surface to the other
i
dr
and
s
.
L et be a u it normal to the
segment of that normal intercepted
wi ll then be the least value for
di rection d r
surfaces and d rt the
between the surfaces
Th e quotient
dr
,
n
n
.
d ri
11
.
d
0
i r ll l
will therefore be a maximum when d r s p a a e to
e ual in magn tude of d n Th e e xpress on
q
i
i
.
n
an
d
23
( )
n
di
dr
is therefore a vector of whi ch the rection is the i ection of
most rapid increase of V and of whi ch the m a gn it u de i s t h e
rate of that increase This vector is entirel independent of
the axes X Y Z
et d c be replaced b y its equal d V which
is the increment of V in passing from the fi rst surfa ce to t h e
second Then let V Vb e de n e d again as
y
L
.
.
,
,
fi
.
1
VV
k
w».
From
dV
n
a
24
( )
.
this de fi nition V V is certai nl y the vector whi ch
give s the di rection of most rapid increase of V and the rate
in t hat direc t ion Moreover V V is independent of the axes
,
.
.
V Vdr
fn
dI
—
d
o
2
( 5)
dr
.
n
is a un it normal Hence n d r is the projection of d r o n
n and mus t b e equal to the perpend cular distance
d 77 be t ween
t h e s urfa ces
n
.
.
i
.
1 42
VECTOR ANAL YSIS
q
A?
i
r
Moreover thi s e uati on de fi nes dv dz In a s milar m anner
it is possib le to lay down the following defi nition
function of
Defin itio n : Th e derivative V V of a scal a
po sition in space s h a satis fy the e uation
.
ll
d
q
r
VV
.
dV
for all values of d r
mpo rta n t
This de fi n iti on is cert ainl the most natural an
from theoretical considerations B u t for practical purposes
either of the defi ni ti ons before given seems to b e bett er
Th e real sign ifi canc e of th s last
They are more tangible
de fi ni t ion c a nnot be app e ciate d until the subject of linea
vector functions has been treated Se e Chapte r VII
Th e computation of t h e derivative V of a fun ction is most
frequently carrie on b means of t h e or d nar
dif
ferentiation
y
.
di
.
.
i
r
.
d
.
r
.
y
i y
.
Let
9
Vr =i
W
9 3/
x
x
H
V
W
Oz
y
V
z
Hen
1
(1 x + jy
ce
kz )
Vr
Th e
dir
derivative of r is a unit vecto r in t h e
e ction of r
This is e videntl the di rection o f m o s t ra pid in c re a s e of r
a n d the
ate of tha t in c ease
r
y
r
.
.
TH E D
IFFEREN TIAL
A L C UL US
C
VECTOR S
OF
1 43
V ( x, y, z )
j
a:
1
( 23
2
y
2
z
k
He n c e
V
q
c
s
‘
.
1n
ua re
y
z
(
2
90
”
v
J
w—
— l
— k
y
z
)
1
i
a vector whos e di recti on s tha t
and who se magn i tude is e ua to the recipro c a of th e
of th e length r
l /r is
q l
l
.
VT
“
Th e
)
Z
r
derivative of
,
i
1
V
of - r
2
y
r
r
p oof is left to the ea der
Exa mp le
A:
Let
r
n r
.
log m
V( co, y, z )
+
y
J2 +
33
y
”
0k
1
x
If
(
cc ,
y,
r denote the vector
+ y
2
from the origin
z
) of spa c e the fun cti on V m a y be wr t te n
dra wn
i
,
i
H
2
x
+
=
jy r
r —
e n 00
-
r
r
(
r—
o
kk
r
3
r
(k )
r—
— k kd
r)
k k-
o
-( r — k k r)
o
t o th e
as
pit
o n
VE C TOR
1 44
There
A N AL YSIS
is another method of computing
upon the identit
VV= dV
Ci r -
i
wh ch
V
y
.
—
dr
Hence
L et
V:
dr VV
o
.
v
r
where a is
a,
d V= dr
o
Exa mp le 3
:
r
constant vecto s
L et
o
2 dr
o
Henc e
(
rx s
a
V=
da
a
r a
o
.
) (
r xb
o
a
(
o
h
.
dr b
ro b
a
a ro b
o
ra
) , W here
-
V V= 2 r a h
V V=
.
ro r a o h — r a ro'b
(
’
w
v
a
r Ar a de
PN
dr
h
.
.
.
Ar m
dV
V:
r
= dr V V
a
V V=
Hence
consta n t ve c to
a
a ro h
)
(
a
x
b
ra
(
o
A
fi
r “
-
A
d
b
"
dr V V
r a
-
r a
b
h
rxb
;
an
a
-)
r a
)
.
Which of these two methods for computing
l
shal b e
applied in a particular case depends enti rely upon their
relative ease of execution in that case Th e latter method is
independent of t h e co ordinate ax es and ma therefore be
preferred It is als o shorter in case th e funct on V e an be
expressed easily in te rms of
B u t when V can not be s o
e xpressed the former method has to be resorted to
Th e great importance of the ope rator V in mathe
me ti cal physics ma be seen from a few illustrations Su p
P08 9 T ( w 31 z ) be the tempe atur e at the point a y z of a
.
r
.
y
V
i
.
.
y
,
,
r
.
,
,
1 46
VE C TOR A NAL YSIS
il
y
l
y
Pote nt a in el e c tricit or magnetism is the pote nti a energ
per un it charge or pole ; and pote ntial in attracti on problems
is potent ia l energ per unit m ass ta ken however w th t h e
negative sign
It is ofte n convenient to t eat a n op era to r a s a
uan tit provided it obe s t h e same formal laws a s tha t
f
fe e n t a to s
ua n tit
Cons ider for e x ample the a t a
y
,
y
y
y
.
9
9
i
r
p r i l di r i r
.
q
q
,
9
9
9 3/
x
9
z
d r l
d di r
far a s combinati ons of these a re c oncern e t h e fo mal a ws
a re precisely what the
woul b e f in s tea of ffe entiators
three true scalars
As
y
d i
a
r i
b,
,
,
c
we e g ven Fo r insta nce
t h e commutativ e l a w
.
9
th e
as s
9
i
oc ati ve
x
9
9
9v
a
9g
b =ba,
la w
9
—
a nd
i
9
3:
9y
9
9
9
z
(b c )
the d s tributive law
9
9
92
a
9
9y
x
+
9
9
x
9
9 y
r
+
9
x
9
z
l
hold for the differentia to s just as for sca ars
formu a a s
l
.
Of cou
rs
e
such
DIFFERENTIA L
TH E
the same
oo k ng upon
In
l i
v e c tor
It is
.
S
VECTOR
OF
y
1 47
i d y
a great advantage ma be obta n e b
wa y
a
_
i
V
as a
CALC UL us
9
cc
a
a
9y
9
r
z
i
not a true vector fo the c o e ffi c e n ts
,
9
l
9z
9 3/
3:
i
are not true sca ars It is a vector d fferentiator and of
cou se an operand is alwa s implied w th it As far as form a l
opera t ions a re concerned it be haves like a vector Fo r
ns tan c e
r
y
.
i
i
.
.
=
) V n + V v,
V (n
V
(
a
v
( V ie ) v
)
v
e
V
V
n
(
( V v)
n
e n
)
,
,
if and are any scalar f nctions of the scalar variables
and if
a scalar independent of the var ables with
regard to which the i erentiations p rformed
rep resent any vector the formal comb nati o
x,
y
t wo
v
n
,
z
e
u
i
be
df
f
e
a re
.
i
If A
A V is
o
A V
°
r i
his operator
—
Al
9
am
+ A2
9
9 3/
A = Al i + Ag j
p o v ded
Aa k
.
is a scalar di fferentiator When
to a scala r function V ( a y z ) it gives a scalar
T
A V
.
,
.
,
9 V
=
A1
V
V
A
(
)
.
Suppos e
9
x
for convenience that
( V)
a
V=
A
7
1
9
-
da
1
+ Az
9 V
ay
+4
9 V
,
.
9
z
is a unit vector
9 V
——
;
ag
a
9 V
+
a
3
az
.
n
VE C TOR A N ALYSIS
1 48
li
a
h
t
e
the
rection
c
sine
s
of
ne
referred
o
d
i
where l
to the axes X Y Z Cons equentl ( a V) V appe a rs as the
well -known direc tio n a l derivative of V in the d rection a
Th s is ofte n written
a
,
y
“ r a s a re
.
,
,
i
i
WM
9V
7
1
9
S
.
Z
e xpresses the magnitude of the rate of increase of V in
the direction a In the particular case where this direct on 1 8
the normal 11 to a surface of constant value of V thi s relation
becomes the normal derivative
i
It
.
,
.
(
”
W
V—
7
1
9
9V
my
”
197 :
i
be
the
d
rection cosines of the normal
a
Th e operator a V a p p lie d to a scalar function of position
V yields the same res ul t as the direct pr o duct of a and t h e
vecto r V V
if n ,
,
”
2a
n
.
.
Fo r
(
i
a
o
V) V =
a
( V V)
( 3 0)
.
yy
th s re a son either operation may be denoted simpl b
a
VV
wi t hout parentheses and no amb igui ty can result from the
o m ission Th e two different forms ( a V ) V and a ( V V)
may however b e interpreted in an important theorem
t h e dire c tio n a Ld e riva t i va e f V in t h e di rect io n
a
0 n th a o t h e r ha nd a ( V V) is t ha n o mp o ne nt a f V V_in
t h e di re c t io n a
Hence : Th e directional derivative of V in
any direction is equal to the component of the derivative
V V in that direction If V de n o t e gravitational potential the
t h e orem b ecomes : Th e directional derivative of t h e po t ential
in any direc ti on gives the component of the force p e r uni t
mass in that direction In case V b e electri c or magnet ic
o tentia l a difi e re n c e of si n must b e observed
g
.
.
-
.
~
-
.
.
.
.
p
.
.
VE CTOR
1 50
This m a y
NAL YSIS
A
be writt e n in the form
(
a
.
7
1
9 1
V) V
9
972
.
9
3
9 17 3
.
9
s
k
( 31 )
.
s
’
Hence ( a V) V is the directional derivative of the vector
function V in the irection a It is possib le to write
d
o
.
(
wi thout parentheses
V) V
a
Fo r
.
a .
V V
a
y
the mean ing of -the vector s mbol
vs
de fi n ed Hence from the present sta ndpoint the e xpression
V V can have but the one in terpretation given to it b
a
y
;
V) V
(a
.
Although
the operation V V has not been de fi n ed and
1
cannot be at present two formal combin ations of the vector
O perator V and a vector functi on V ma
be treated These
are the ( formal) scalar product and the ( formal ) vecto r p od
u c t of V in to V
The a re
y
,
y
.
9
VV
9
o
V is
re a
9
d de l do t V;
Th e di fferenti a to m
9
9
x
by the dot and the cros s
.
i
9
V x V, de l
9
9 3/
9
a
That
z
9
9y
a nd
9
9
3
56
z
,
z
c ro ss
V XV=
I
X
9V
a:
V
.
b e m g s c a l a r o p e ra t o ra p a ss
is
3
9m
r
9
9 3/
x
i
v
V
.
9 y
9V
9 y
2
9
+ k x
z
9V
9
z
DIFFERENTIA L
L UL US OF VE C TOR S
9 73
9
9
x
9
ar
9a
ae
9V
9 VI
9 V2
a
2
7
1
z
1
.
a
9
n
z
9
x
av
9
H
9
x
J+
.
k,
x
,
W
,
k’
9v
7
1
9 3
9
k‘
z
e nc e
More o ve
r
H
VxV
en
ce
1
9
9
z
9 V2
Th
is
m a y be
ri i
1 51
CA C
9
x
i
x
9 Vl
9 3;
w tte n n the form of a determ nan t
y
1 52
VE C TOR A NAL YSIS
r
is to be unde stood that the operators are to be applied to
t h e func t ions V1 V2 V8 when e xpanding the determin ant
From some standpoints o b jections may b e brought fo rward
a gains t treating V as a sym b olic vector and intr oducing V
v
and V x V respectively as t h e symbolic s ga g r a n d ve gtg;
These objections ma be avoided by
L
OL M
Q LQQM
simply laying down the de fi nition that the s mbols V and
V
whi ch may be looked upon as entirel new operators
quite d s tinct from V shall be
It
,
,
x
y
y
y
i
,
.
,
VV—
x
e
9V
gx
V x V= i >
<
B
<
+ k >
for practical purposes and for remembe ring fo
seems by all means advisable to regard
ut
e
z
i
a
9
r ul i t
m
a
a
96
9g
i
l
a symbolic vector differentiator Th s symbo obeys the
same laws as a vector just in so far as the difi e re n t ia t o rs
as
.
9
9
w
9
y
y
q
obe
the
same
laws
as
ordinar
scalar
uantities
7
3
97
That the two functions V V and V
V ha ve ver
important physical meanings in connection with the vector
fun ction V may be easily recogni z ed B the straight
y
forward proof indicated in Art 6 3 it wa s seen that the
,
,
.
.
x
y
.
S
VE C TOR A NA L Y IS
1 54
9V
m
dx
d y dz
V
to tal flu x outward from the cube through thes e
faces is therefore t h e algebraic sum of these quan t ties
is simpl
i
Th e
y
9 VI
9 V
9
a:
d a: d g d z
.
.
i
like man ner the flux es through t h e other pa rs of
the cube are
gV
v
d x d g d z and k — d cc d g d z
In
o
g
e
3!
Th e
to tal
flu x
9
i
fa c e s
of
.
z
out from the cube s therefore
9V
9V
9
9 g
cc
q
This
d cc d g dz
.
y fl
is the net uantit of uid whi ch leaves the cube per
unit time Th e quotient of t his by the volume d x dg d z of
the cub e g ves the rate of di minution of de n sit This is
.
i
V v= i
°
y
av
av
o
v
s
_
__
o
9
9 g
3:
9
W
7
1
9 1
9
z
x
.
e V,
,
9
9g
i
'
z
y
ecau se V V thus represents t h e di minut on of densit
or the rate at which matter is leav ng a point per u n it volume
per unit time it is called the dive rge n c e M ax well emplo ed
t h e term c o n ve rge n c e t o denote the rate at which
uid a p
This is the
p ro a c h e s a point per unit volume per unit time
negative of the d vergence In case the uid s in c o mp res s it le
as much matter must leave the cube as enters it Th e total
change of contents must therefore be z ero Fo r this reason
the characte ristic differential equation wh ch an in c o m p re s
s ib e fluid must sat sf
is
B
i
,
fl
.
i
fl i
.
.
,
.
l
i
iy
V V
0
0
.
y
y
TH E
DIFFERENTIAL
i
C
AL CUL US OF VE C TOR S
fl
1 55
where V s the flu x of the uid This equation is often
kn own as the h ydro dyn a m ic e qu a tio n It is satis fi ed b any
ow of wate r since water is practical l incompressible Th e
great importance of the equation for work in e l e c t ric ity is due
to the fact that according to Max well s h pothesis electric d s
placement obe s the same laws as an incompressible ui
If
then D be the electric d splacement
fl
.
,
y
i
i
fl d
.
,
the operator V x Max well gave t h e na me
nomenclat ure has be come widel accepted
To
Thi s
.
y
’
y
y
.
y
V x V=
cu
e
c u rl .
.
.
i i
y
i
curl of a vector fu nct ion V s t self a vecto r funct on
of pos ition in spa ce As the nam e in diga tgs it is closel
connecte d with the an gular velocit or spin of the flu x at
each point B u t the interpretat on of the curl is neither s o
easily obtained nor so simple as that of the di vergence
Consider as before that V represents the flu x of a u d
Take at a de fi ni te insta nt an in fi nitesimal sphere about an
point ( a g z ) At the ne xt instant what has b ecome of the
In the fi rst place i t may have moved o f
f as a whole
Sphere ?
in a certain di rection b an amount d r In other words it
ma have a translational veloc ity of d r/d t In addi tion to
this it ma have un dergone such a deformation that it is n o
onger a sphere It may have been subjected to a s tra in b
virtue of which it becomes slightly ellipsoidal in shape
Finally it ma
have been rotated as a whole about some
ax is through an angle d w That is to say it ma ha ve an
angular velocit the magni tude of whi ch is d w/d t An
infi nitesimal sphere therefore ma have an one of three
di sti nct t pes of moti on or all of them combined
Er}r a
translation with efi nite velocit
Se c o ri i a strai n w th thr e e i
ij
de fi nite rates of e ongation a ong the ax es of an elli s oid
Th e
.
i
.
L
g
fli
.
,
y
l
,
y
.
.
y
.
y
.
y
.
y
.
y
y
y
d
y
l
y
,
.
y
l
y
.
.
.
,
i
,
p
.
VE C TOR A NAL YSIS
1 56
Tha d,
i i
an angular velocity about a de fi nite axis It s th s
third t p e of motion whi ch is given b the c u rl In fac t
t h e c u rl o f the flu x V is a vector whi ch has at each point of
space the direction of the instantaneous ax is of rotation at
t ha t point and a magni tude equal to twice the instantaneous
angu lar velocit about that ax is
Th e analytic discussion of the motion of a uid presents
more diffi culties than it is necessar to introduce in treating
the curl Th e motion of a rigid body is su ffi cientl complex
to give an adequate idea of the operation It wa s seen ( Art
5 1 ) that the velocity of the pa ticles of a igid bo
at an
instant s given by the formula
y
y
y
.
fl
.
y
y
dy
.
r
”
i
v
v(,
.
x
a
vo
a
a
i +
1
a
r
y
.
r
.
<
+ V >
2 j+
a
8
(
a
)
x
r
z
k
.
k
+
x x
x
,
.
e xpand V ( a r ) formally as if it were the vector tri ple
product of V a and r Then
V
r a —
v V
V
a) r
V
(
)
(
,
,
.
o
is a con stant vector
v0
V
As
a
Hence the term
.
9m
'
r
55
-
lV
x
v0
9
r
Hmm
Therefo
“
1
re in
9 75
+
,
i
van shes
.
9y
9z
+
+
é7l
5
is a constant vector it may be place d upon
of the differential operator
"
V
.
V
a
a
V
th e
other side
.
9
a
2
7j
9
=3
v
a — a
= 2a
r
q l
i r
case of the motion -o f a igid body the curl
of the line a r vel o c it at an point is e ua to tw ce the
Ia n gu la r velocity in magn tude and n di ec t io n
th e
y
i
y
i
.
VE C TOR
1 58
A
NAL YSIS
( V ie )
a
v
Vu
Vu x
v
v
=
n) x
V
(
v
.
ll
is to be applied to more th an the one te rm whi ch fo ows
it the terms to which it is appli ed are enclosed in a paren
thesis as upon the left-ha nd side of the above e uat ons
Th e pr o ofs of the formul a ma be given most natu all
by e x an di n g the expressions in terms of three assumed unit
vecto s 1 k Th e sign 2 of summation will be found con
B y mean s of it the opera t ors V v A
ve n ie n t
tak e th e
fo m
IfV
q i
,
p
r j
,
r
,
d
summation e xten s over cc
demonst rate
V x
(
u
)
v
=
,
g,
z
.
r y
x
.
,
.
Th e
To
y
.
,
.
V x
z
i x
v
a
v
9
11.
9;
Xv +
+
2
2
u 1
1
x
+
u
it
9v
>
< —
9w
Hence
To
demonstrate
V (n
o
v)
v
Vu + u
V
( X
)
V
.
THE DIFFERENTIAL
L UL US
VE C TOR S
CA C
o
1
In k ke m a n n e r
Henc
v
)
(
V
e
E
u
=
9
u
=
—
-
a
v
+ v
r
v
+
u
<
v >
v
.
r
z
x
u
Vv
z
Vn
x(
+
(V
u
V x
u
)
+
x v)
u >
<
+ r
V
X
V)
(
l
v
.
.
other formul a a e demonstrated in a simi ar manne
1
4
Th
not
a
tion
e
7 ]
Th e
V
ill
0
1
)
V
.
( 44)
n
yi
r
w be us ed to denote that in appl ng th e Op e ra t o r V to the
product (11 v) the quantity u is to b e egarded as c on s ta n t
That is the operation V is car
ri ed out o nl y
p a rtia l ly upon
the product ( u In general if V is to be carried out
v)
pa rtially upon an numbe r of functi ons which occur afte r
it in a a enthesis those functions which are con stant for the
different ati ons are writte n a fter the parenthes is as subscripts
,
r
.
,
y
pr
i
,
.
m
ide a
m
km
m m
m
m
p a rt ia l V s o t o s pe a m a y be a vo ide d by e a n s
o f t h e fo r u la 4 1
B u t a c e rt a i n a o u n t o f c o p a c t n e s s a n d s i pli c i t y is
lo s t t h e re by Th e id e a o fV (n v) is s ure ly n o o re c o pli c a te d t h a n u V v o r
1
Th is
.
a n d n o ta
X (V X
u
o fa
.
,
.
v
ti o n
)
.
VECTOR
1 60
V
NAL YSIS
A
(u
i
This
v)
4
5
( )
v.
formula correspon ds to the follow n g one
tion of difle re n t ia l s
in
the nota
y
form ul a ( 35)
given above ( Art 7 3 ) ma be
written in the following manner as is obvious from analog
with the correspondin g formula in di fferential s
Th e
.
,
V
V >
<
V
( 11
4
y
1 62
VE C TOR ANAL YSIS
l
form a l y a s if V
,
11
n,
r
l
r
v we e all rea vecto s
x ( x v)
=
V
V
r
.
Then
V
v
:
— u
z
.
d
i
second term is capable of interpretation as it sta n s
Th e operator V ha s noth ng
Th e fi rst term however is not
upon which to operate It the efo re must be trans posed so
that it shall have n v as an operand B u t u being ou t side
of the parenthesis in u x ( V v) is cons tant for the d fferen
Hence
ti a t io ns
Th e
.
,
,
.
r
x
o
.
i
.
.
u
o
vV
=V
(
u
o
)
v
u
u
V v
be a unit vector say a the formul a
v
and
—
Ifu
,
,
a
o
=V
(
.
/
.
v
a
4
( 7)
a
presses the fact that the directional derivative a V v of a
vector f u nction v in the direction a is equal to the derivative
of the projection of the vector v in that d ection p us th e
vector product of the curl of v int o the direction a
Consider the values of v at two neighboring po ints
ex
ir
l
.
.
v
B
ut
y
1
i +
v2
j+
v3
d v1 = d r o
1
d v2 = d r o
2
d v8 = d r o
3
dv=dr
Hence
By
v
k
d v = d v1 i + d v2 j+ d v3 k
b
Hence
=
dv = dro
dv=V
(dr
.
.
.
THE DIFFERENTIAL CAL C UL US OF VE C TOR S
if v denote the value of v at the point ( a
value at a neighboring point
Or
o
v
=
vo
+ V
d
(
g,
z
)
the
a nd v
( 49)
r
is e pression of v in ter
Th
,
1 63
ms
of its value v at a given po int
t h e de ls and the di splacement d r is analogous t o the e xp a n
sion of a scalar functor of one variable by Ta lor s theorem
x
y
,
Th e
f CC)
i
+1
”
(
)
500
dx
derivat ve of ( r v) when v is cons tant is e
Th a t is
V
V (r v) v
Fo r
(
v
v~
=
Vr
5c
v
1
V
q
ual
to v
.
9
7
2
5 1,
i + y j+
i +
V x
e
,
Vr
0
r
’
r
9
Henc
,
0
e
v
r
( r v)
z
2 j+
k,
v
3
k = v,
= 0
.
v
v
.
In like
manner if in stead of the fi nite vector r an infi nitesimal
vector d r be substi t ute d the result s t ill is
,
,
V (d r
By
( 47)
v
=
vo +
V
(d r
V (d r
v)
1
Th
c
1
is gives another form of (
onvenien t
It
- v
.
1
which is sometime s
is also sl ightly more s mmetric a l
4 9)
y
.
m ore
VE C TOR ANAL YSIS
1 64
Cons ider
L et
a mo ving fluid
velocity of the uid a t the point (x
round a poin t (T g z ) with a sma
fl
o,
At
p
.
ll
,
o,
o
r r
d
d
i
c
phere
S
r
i
the inc ement of t me
moved the distance
In
(
8t
r
d
v,
.
y
V
v
z
,
the
t ) be
t
.
Su r
.
is
.
i p r ll
the points of t h s s he e wi
St
)
vo + d r o
i
point at the center w ll ha ve move
Th e
g
,
2
each oint of th s sphere the velocit
v
(
x,
) at t h e time
z
g,
v
v
,
8t
.
d dista ce
th e
n
.
distance between the center and the points th at were
upon the sphere of radius d r at the comm encement of the
interval 8 t has become at the end of that interval 8 t
Th e
d r = d r + d rVv8t
’
locus of the e xtremity of
e m nate d r from the e uati ons
To fi n d t h e
li i
q
c
.
Th e
dr
’
i
it s necessar
y
to
= dr dr
o
.
y
Th e fi rst
Art
2
.
y
equation ma be solved for d r b t h e method of
4 7 page 9 0 a n d the solution substitute d into t h e second
result w show that the infi nitesimal sphe e
,
ill
r
,
d rdr
.
2
y
has been transformed into an ellipsoid b the motion of the
fluid during the time 8 t
A more de fi nite account of the change that has taken plac e
m a y be obta in ed b making use of equation 50
( )
.
y
VECTOR ANAL YSIS
1 66
to
them
s e c tive l y parallel
p
b ecomes sim m
y
Then
.
i
the e xpress on
a
bo ve
i
point whose coord nates referred to the c ente r of the
infi n itesima sphere are
Th e
l
d oc ,
dz
d g,
is therefore endowed with thi s velocity
will have moved to a new position
i
t me
In t h e
.
,
Th e
8t
—
dz
dy
+
5
li y
9
71
2
9y
8t
it
1 +
dz
,
8 t
tota t of the points upon the sphere
dr
li y
dg
2
dz
2
goes over into the tota t of points upon the e
which the equation is
to
3
z
9
"
e
x
llipsoi d
of
z
v3
55
8
statements made before ( Art 7 2) concerning the three
types of motion which an in fi n itesimal sphere of fluid ma
possess have therefore now b een demonstrated
Th e symbolic operator V may be applied several times
in succession This will correspond in a general wa y to
forming derivatives of an order higher than the fi rst Th e
e xpressions found b y thus repeating V w ll all be in de p e n d
e n t of t h e a x es because V itself is
There are s ix of these
dels of the second order
Le t V ( re
z ) be a scalar f u nction of position in space
Th e derivative V V is a vector f u nction and hence h a s a cu l
and a divergence Therefore
Th e
y
.
.
.
i
.
.
.
r
,
.
V
°
V V,
V x VV
.
THE DIFFERENTIA L
AL CUL US OF VE C TOR S
1 67
C
y
are the two derivatives of the second order whi ch ma be
obtained from V
.
V x V V=
Th e
V
e
cu
5
2
( )
.
second e xpression V x V Vva n is h e s ide n tic a lly
the de ri va tive
fa ny
o
sc a
la
r
f
V p o ss e ss e s
tio n
x
unc
That is ,
.
n o c u rl
j
rl
This
.
may be seen by e x pandin g V V V in terms of i k All
the terms cancel out
ater ( Art 8 3) it will be shown con
ve sel that if a vector functi on W poss esses no cu i e f
r y
L
.
,
.
.
.
,
W is the derivative of some scalar functi on V
Th e fi rst e x pression V V V when e xpande
.
o
i, j
k be c o m e s
,
9 V
2
V
o
VV
9
x
9
V V
Sym b o h c a lly ,
2
+
2
x
2
+
9y
9
2
9 y
L
.
9 V
i
i
9
x
9 V
2
2
+
2
9 y
2
r
te ms of
+
.
+
9
z
9
2
9
z
2
2
q
’
i
9 V
2
2
in
.
2
2
o
9
9 V
d
operator V V is therefore the well-known o
ap ac e
aplace s E uation
Th e
L l
,
p rator of
e
9 V
2
+
y
9
z
2
bec omes n the notat on here emplo ed
V VV
i
When applied to a scalar funct on
0
y d
.
operator V V iel s
a scalar function which is more over the divergence of t h e
derivative
et T be the te mperat u e in a bod
et c be t h e con
a n d k the s e c fi c
heat Th e
du c t ivity p the densit
V th e
,
,
L
r
.
,
y
y L
p i
.
,
flo w f is
f=
— c
VT
.
.
VECTOR ANAL YSIS
1 68
i
i
rate at which heat is leav n g a point per un t volume
unit time is V f Th e increment of temperature s
Th e
.
V fdt
o
d T
c
k
p
dt
is
Let
Th
is Four
.
VT
.
ier s e quation for the rate
of
’
ch a nge of tempera
a vector function and VI V2 V3 its three com
Th e operato r V V of aplace ma be applied to V
V be
ponente
-V
i
r
e
p
.
L
,
,
L
If a
y
,
.
vector functi on V satis fi es aplace s Equati on each of
O ther de ls of the second
its three scalar comp onents does
order may be obtained by considering the divergence a n d curl
of V. Th e di vergence V V has a derivative
’
,
.
Th e
cur
l x
V
V VV = V div V
V ha s
V
V
V
O f these
in
i
turn a d vergence and a cur
v
V
x V
expressions
.
,
x
x
.
cu
e
curl cu
V
x
,
V x V x V
V = di v
V V
l
V
rl
5
6
( )
5
( 7)
V
.
ide n tic a lly That is ,
Thi s ma b e
is z e ro
va n is h e s
.
y
the diverge n c e of th e c u rl of a n y ve c to r
seen b y e xpanding V V V in terms of i k
ate r ( Art
8 3 ) it will b e shown conversely that f the d vergence of a
vec tor function W vanishes identicall y i e f
x
j L
i i
i
.
,
.
,
V
W is the curl of some vecto r function V
.
.
,
.
.
VE C TOR
1 70
A
NAL YSIS
i
i
V
V u is nte re st ng
geometric
interpret
a
tion
of
Th e
It depends upon a geometric interpretation of the second
derivative of a scala r function u of the one scalar variable cc
et u , be the value of u at the point as ,
et it be required
to fi n d the second derivative of u with respect to x at the
point
et re , and a , be two points e uid s tant from :c
Th at is let
L
L
q i
.
L
,
=
“2
“1
a
.
.
o
.
o
“a
2
?
I
“
r
is the ratio of t h e difference between the ave age of u at the
point s x, and and the value of u at d to the s u a e of t h e
distance of the points xl a , from ec Th at
o
,
o
i d oc 2
y
q r
.
—O
is e as il proved by Taylor s theorem
et u be a scalar function of positi on i n space Choose
k and eva uate the
three mutually orthogonal lines 1
e xpression s
L
’
.
,
9
2
9
u
2
,
Let
mo ;
(c o
;
9
x
j
a
a
l
.
,
z
u
,
2
di
a
1
and
be
two
points
on
the
l
ne
at
a
stance
a
from
i
,
,
x and a
at the same di stance a from
4
s two points on
cc
an a s two oints on k at the same d stance a from w
a
,
d
,
,
p
j
i
“1
+
u
2
_
u
o
e
.
TH E
DIFFERENTIAL
9
1
2
9
9
2
9
2
2
7“
Lm
2
a i
u
x
2
L UL US OF VE C TOR S
CA C
+
9
2
O
VVu
a
2
9y
1 71
+
6
a
r
p
s
ri
and V a e inde endent of the pa t c ul ar ax es chosen
this expression m a y be evaluate for a different set of ax es
then for st a difle re n t one etc B y add ng together all
thes e results
As V
d
ill
,
u
V
L et
-V
1
+
i
.
u
2
+
,
o
,
6 n t e rm s
u
u
o
u
become infi nite and at the same time let the
sets of axes point in ever direction ss uing from
frac tion
u
u
6
n te rms
+
1
2
y
n
i
difle re n t
$0.
Th e
then approaches the a ve ra ge va lue of it upon the surfa ce of a
D enote th s
sphere o frad us a s u ound ng the point
by n
i
a
rr i
i
.
v Vu
.
LIM
-0
a
.
a
2
y
is equ a l to s ix ti mes the limit approached b the ratio
of the e xc es s of u on the surface of a sphere above t h e value
at the cente r to the square of the ra di us of the sphere Th e
same reas oning held in case n is a ve c to r function
VV u ( e x cep t for a
If u be the tempe rature of a bo
co n stant factor which depends upon the mate ri al of t h e
V Vu
.
dy
.
VE C TOR
1 72
A
NAL YSIS
q al to the rate of increase of t mperature (
- is p sitive the average temperature upon a
small sphere
greater than
tem perat re at the center
center of the sphere is growi g warmer
the case
of a
flow the temperatu e at the center m st remain
constant
vi dently therefore the condi tion for a steady
body ) is e
Vu
If V
Art
e
u
o
th e
is
u
Th e
.
n
s te a
.
dy
In
.
u
r
E
.
flo w is
V
That
O
V
‘
tt
o
z
.
L l q i
q y
y
lr
l
ddi i i
is t h e temperature is a solut ion of ap a ce s E uat on
V V it whether
M axwell gave the name c on c e n t ra tio n to
Conse uentl V V u ma
u b e a scalar or vector function
be called the disp e rs io n of the function i t whether it b e sca a
or vector Th e dis persion is proportional to the excess Of
the average value of the function on an in fi nite sima surface
above the va ue at the center In case it is a vecto r f u nction
the average is a vector ave age Th e a t ons n it a re
vector add tions
’
.
,
.
.
l
r
.
i
.
.
SUMM ARY
i
If a
OF
CHAPTE R III
i i
vector r s a function of a scalar t the der vat ve of
r w th respect to t is a vector
uantity who se direction is
that of the tangent to the cu rve described b y the terminus
of and whose magnit ude is e ual to the rate of advance of
that terminus along the curve p e r un it change of t Th e
derivatives of the com onents of a vecto are the components
of the derivat ve s
i
r
q
q
i
p
r
.
r
A
.
r
lr y
combination of vecto s or of vecto s and sca a s ma b e
dif
ferentiated just as in ordinar scalar anal s s e xcept tha t
th e difle re n tia t io n s must be performed i n si tu
y
yi
'
.
VE CTOR
1 74
A NAL YSIS
i
denote the position of a mov n g
v the velocit A the acceleration
If r
y
rtic l e
a
p
,
t the
tim e
,
,
,
p
whi ch one is parallel t o the ta ngent and depen ds u on the
v of the parti cle in its
rate of change of the scalar velocit
path and of which the other is perpendi cular to the tan gent
and depends upon the velocit of the a t c e a n d the cur va
t ure of the path
y
y
,
p ri l
.
A=
s
t +
Applicat ions
v
2
0
.
i l
to the hodograph in part cu ar motion in a
circle parabola or under a central acceleration Application
to the proof of the theorem that the moti on of a rig d body
one point of which is fi xed s an ins tantaneous rotation abo ut
axis through the fi x ed point
Inte gration with respect to a scalar is merel the inve se
of difle re n tia t io n Application to fi n di ng the paths due to
given accelerati ons
Th e ope ator V applied to a scalar fun ction of pos t on in
Space gives a vector whose
rection s that of m o st rapid
in crease of that function and whose magni tude is e ual to
the rate of tha t inc eas e per un t chan ge of po sit on in tha t
direction
,
,
,
i
.
i
y
.
r
.
r
.
di
r
i
9
v=i
i
90
a
9
a:
i
9 g
5
A
1
9 3/
9
’
,
z
ii
q
THE DIFFERENTIA L
C
AL C UL US OF VE C TOR S
ope rator V is invari ant of the ax es
de fi ned b the e uation
Th e
y
j
i,
q
k
,
ma
It
.
1 75
y
be
( 24 )
VV d
o
r
d
( 2 5y
.
ion of the derivative by two me ho ds depend
ing upon e q uat ons ( ) and
llu tration of the
currence of in mathe at cal physics
may be looked upon as a ctitious vect r a vector
di erentiator
obe y s the formal laws of vect rs just in
so far the scalar
erentiato rs of /
bey
the fo rmal la s of scalar q uantities
Computat
VV
t
21
i
V
I
m
i
s
oc
.
V
fi
f
f
o
It
.
o
di f
f
as
,
9 9
9
cc ,
9 y, 9 9
z
O
w
9V
9V
be a un it vector a
n the rect on a
Ifa
i
di i
9V
i
i
is the d re ctional deri va t ve of
VV
.
a
c
V V=
If V is
i
(
a
o
V) V =
a
( V V)
.
l
a vector funct on a V V is the di rectiona
o f th a t vector f u nct on in the d rection a
i
i
.
av
o
9
9 g
0:
9v
9
2:
9 Vl
9
v
=i
x
9 g
9 V2
9 y
<
+ k >
9 V3
9
9
’
z
9
9 V3
V
VE C TOR A NA L YSIS
1 76
P roof tha t V
V
is the
o fV.
dive rge n c e
of
V
an
d x
V
V,
V V = d1 V V,
°
V xV=
V
(
+
u
cu
V
(
n
o
)
-u
(
u
v)
o
.
v
V
V
e
)
v
=V
u
+V
-v +
u
-( u
V x (u
x v)
v,
)
q
— v
V
o
—
u — u
l
d
x(
11
u
v
)
v
,
1
4
( )
4
2
( )
,
v ( 43 )
Vv + uV .
i
of the partia de l V (u
in wh ch the dif
a e performe upon the h othesis that u is
fe re n t ia tio ns
r
x(
.
u
If a be
-
v
Introducti on
constan t
o
(q
=v Vu
x v)
V
v
+
V
v,
o
i
=
V
)
v
r
yp
,
(
u
i
)
v
u
-u
o
.
a un t vecto the direct onal derivative
a
V v = V (a
4
7
( )
i
expan sion of any vector funct on v in the neighborho o
of a oint ( x y z ) at wh ch it takes on the va ue of v s
Th e
p
0,
o,
i
o
l
0
i
49
( )
x dr
.
Appl
ication to hydrod n mics
Th e de ls
y
a
.
of the second order are
s ix
in numbe
r
.
50
( )
d
VECTO R ANAL YSIS
1 78
r
Obtain
p
the accelerations of a moving pa ti cle ara llel
and perpendicular to the tangent to t h e path and re duce the
resul t s to the usual form
3
.
y
.
d
a s stem of polar coor inates in Spac e
where r is the di sta nce of a point from the origin (1) the
m e ridia n a l angle and 0the polar angle obtain the e x pressions
for the components of the velocity and acceleration along the
radius vector a meridi an and a parallel of latitude Reduce
these e xpressions to the ordinar form in terms of x y z
4
If
.
0 be
r,
,
,
,
5
Show
.
operator
V
y
,
,
.
,
the direct method suggested in
is independent of the axes
,
.
that the
Art 63
.
.
i
the second method g ven for computing V fi n d
the derivative V of a triple product [a b 0] each te m of which
is a function of x y z in case
a = (r
r) r
b = (r a) e
c = r x f
where d e f are constant vectors
6
By
.
r
,
,
o
,
,
,
‘
,
7
,
.
Compute V V t
.
Vis
en
r
z
1
1
r
2
7
,
and V x V w h e n V is
equal to r and when V is e ua to 5 and show that n these
£
cases the formula ( 58) holds
8
9
Compute V V V, V V
o
.
Expand V
.
x
q l
o
V,
.
V V a nd V
o
i
,
V
x
V
in te rm s
show that they van ish ( Art
1 0 Show by e xpandin g in terms of i
.
.
,
V>
<V >
<V = V V
11
.
Prove
A V (V
o
W)
o
V— V
j
,
i,
, k
j
and
k th at
VV
= V A V W+
o
of
.
WA -V V
.
and
(W
W)
—
V (v W) ,
H
C A P TER IV
TH E IN TEG R AL CALCULU S OF VECTORS
L et W( ) be a vector function of position
L et be any curve in space and the rad us Vector
ac,
y
,
in
z
i
space
r
C
dra wn from some fi xed origin to the points of the curve
Divide the curve into in fi nitesimal elemen t s d r From t h e
sum of the scalar product of these elements d r and the value
o fthe function W at some point of the element
.
,
.
.
thus
2
W
dr
.
i
r
i
limit of th s sum when the elemen t s d be come n fi nite
in number each approac hi ng z ero is called the lin e in tegra l of
W along the curve C and s wri tte n
Th e
i
,
,
i d x + jd y + k d z ,
f
w
1
( )
.
e
de nition of the line integral therefore coincides ith
the de nition usual y given
is however necessary
spec fy which direction the radius vector r is supposed to
describe the curve during the integration
the elements
ave opposite igns when the curve is described in opp
Th e
w
fi
fi
i
l
.
It
to
in
.
dr h
s
Fo r
o
1 80
SS
VE C TOR A NAL Y I
r
site di ection s If one method of descripti on
0
C and t h e other b
y
.
be
d y
denote b
,
W -d r
.
r
case the curve C is a closed curve bo undi ng a po t i on of
surface the curve will alwa s be regarded as described in
such a direct on that the enclosed area appears pos i t ive
In
y
i
( Art
y
.
If f denote t h e
y
force which ma be supposed to var from
point to point along the curve 0 the work done b the force
when its point of application is moved from the initial po nt
is the line inte g a
r of the curve 0 to it s fi nal point
y
,
r
,
f
dr
f-
rl
J
r
fd
:
i
xo
r
.
'
0
r
line integral of the de ivative V V of a
scalar function V (z y z ) along any curve from the point
r to the point r is equal to the di f
ference between the values
of the function V ( w y z ) at the point r and at the point t
That is
Th e o re m
:
Th e
,
,
0
,
,
o
.
,
VV dr
By
V (r)
z
V ( r y, z )
,
V
(
900,
yo ,
z
o
)
.
dr V V= d V
de fi niti on
V = V G)
The o re m :
V ( ro
V Oo )
'
17
z
.
6
0 y )
V( xo r yo , z o )
‘
( 2)
line integral of the derivative V V of a
single valued scalar function of position V taken around a
c lo s e d cu ve vanishes
r
Th e
.
VE C TOR ANA L YSIS
1 82
value of the integral is therefore a scalar fun ction of
positi on of t h e point r whos e coordinates are x y z
Th e
th e
o
a
together
ut
.
W
.
dr
Let the integr l be taken between two points
B
,
,
d V ( x, y , z )
dr
it ly near
in fi n
e
.
V V dr = dV
by defi n ition
o
W
Hence
VV
.
theorem is therefore demonstrated
et f b e the force whi ch ac t s upon a unit mass near
the surface of the earth under th e influence of gravity
et
a system of axes i k be chosen so that k s vertical Then
Th e
L
.
,
j
i
,
—
f
.
L
.
k
g
.
li i
work done by the force when its point of app cat on
moves from the position r to the position r is
Th e
o
r
r
r
H
enc e
i
Th e
1?
w=
force f s said to be derivable from a fo rc e -flen e tio n V
when there e xists a scalar func t ion of posit on V such that
the force is equal at each point of the derivative V V
Evidently if V is one force -function another may be obtained
by addi ng to V any arbitrar constant In the abov e e x
ample the force -function is
i
.
y
more simply
Th e fo c e is
Or
r
V
f
VV
,
.
gz
.
TH E
INTEGRAL
C
y
i
AL CUL US
i
OF
VECTOR S
1 83
necessar and suffi cient cond ti on that a force-fun cti on
V ( T y z ) e xist is that the work done by the force when its
point of applicat on moves around a closed circuit be z ero
Th e work done by the force is
Th e
,
,
,
.
20
i i
:
l i
r
y
th s ntegra van shes when taken a ound ever clos e
contour
f= V V = V w
If
r y
inten vanishes
er o y by a consta t
An d
th e
.
conve sel if
di f
f
nl
n
VV
f
Th e
.
d
force-functi on and the work done
.
const
V= w
.
case there is fric tio n no force-function can e xist Fo r t h e
work done by fri ction when a particle is moved around in a
closed c rcuit is never z ero
Th e force of attraction e x erted by a fi xed mass M upon
a uni t mass is di rected to ward the fi x ed mass and is propor
nve se square of the distance between the
t i o n a l to t h e
mass es
In
.
i
.
i r
.
i
Th s
i
is the law of universal gravita t on as stated b y N ewt o n
It is eas to see that this force is derivable from a force
function V Cho ose the orig n of co ordinate s at the center
of the attractin g mass M Then the work done s
y
.
i
.
i
.
w:
r
r
w=— oM
o
dr
z
r
—
dr
d r,
0
M
.
1 84
VE C TOR
r
A
NA L YSIS
y
d
a p oper choice of units the constant 0 ma be ma e
equal to unity Th e force -function V ma the efo e be
chosen as
By
y r r
.
di
there h a d been several attracti ng bo es
the force -fun ction would have been
If
M 1 , M 2, M 3 ,
d i
where r 1 r 2 r 3
are the distances of the attra cte un t
mass from the attracting masses M 1 M2 M 3
Th e law of the conservat ion of mechanical energ requires
that t h e work done b y the forces when a point is moved
around a cl o sed cu rve shall be z ero This is on the a s s u m p
tion that none of the mechanical energ has been converted
into other forms of energy during the motion Th e law of
conservation of energy therefore requires the fo ces to be
derivable from a force-function
Conversely if a force
function e x ists the work done by the forces when a point is
carried around a closed curve is z ero and conse uently there
is no loss of energy A mechanical system for which a force
function e xists is called a c o n s e rva tive system From the
e xample just cited ab ove it is clear that bodies mov ng under
the la w of universal gravitation form a conservative s stem
at leas t so long as they do n o t collide
et W ( as y z ) be any vec t or function of position in
sp ace
et S b e any surface Divide this surface into in
fi n it e s im a l elements
These elements may be regarded as
plane and may be represented b y in fi n itesimal vecto s of
whi ch t h e di rection is at each point the direction of the
normal to the surface at that point and of which the magn i
tude is equal to the magnitude of the area of the in fi nitesimal
,
,
,
,
y
,
y
.
r
.
.
q
.
.
.
L
L
i
y
.
,
,
.
.
r
VECTOR A NAL YSIS
1 86
gives the amount of that substance which is passing through
the surface per unit t ime It was seen before ( Art 7 1 ) that
the rate at wh ich matter was leaving a point p e r unit
volume per unit time was V f Th e total amount of mat
ter which leaves a closed space bounded b a surfac e S er
unit time is the ord nar triple inte gra l
.
.
y
.
i y
V
p
fd v
f
o
.
y
Hence the ver important relation conn ecting a surfac e in
tegral o f a flu x taken over a c lo s e d surface and the volum e
integra of the divergence of the flu x taken over the spa c e
encl osed b the surface
l
'
y
V
fd v
.
i
7
a
f
a
r
i y l
Wr tten out in the notation of the ord nar c a c ul u s this
becomes
X
d
d
z
y
[
d
or
e
x
.
dx
d
9 17 + o z
9 3,
oz
dy ]
dx d y d z
( 8)
where X Y Z are the three components of the flu x f Th e
theorem is perhaps still more familiar when each of the th e e
components s treated separatel
,
.
,
i
y
ff
d
This
dy =
r
.
fff
d x d y dz
.
is known as G a us s s Th e o re m It states that the surfac e
integral ( taken over a closed surface ) of the product of a
f u nction X and the cosine of the angle wh ch the e x te rior
normal to that surface makes with the X-axis is e u a l to
the volume integral of the pa tia derivative of that funct on
’
.
i
r l
q
i
THE INTEGRAL
CA
L C UL US OF VE C TOR S
1 87
y
l
with respect to x taken throughout the volume enclosed b
that surface
If the surfa ce S be the surface boun di ng an nfi n tesima
sphere or cube
i i
.
f da = V
o
W here d o is
o
fd v
l
the vo ume of that sphere or cube
1
fda
do
H ence
.
.
is e quation may be taken as a fi of the vergence
f
divergence of a vector function is equal to the
limit approached by the surface integral of f taken over a sur
Th
V
de
.
di
n i tio n
Th e
f
face boun ding a n in fi nitesimal body divided by that volum e
when the volume approaches z er o as its limit That is
.
f a
0
a
( 1 0)
.
y
From
thi s de fi nition which is evidentl indepe ndent of the
a x es a ll the properties of the divergence may be deduced In
order to make use of this de fi nition it is necessary to develop
at least the elements of the integral calculus of vecto s before
the differentiati ng operato s can be tr eated This de fi nition
of V f consequently is interesting more from a theoretical
than from a practical standpoint
Th e o re m
Th e surface integral of the curl of a vector
f u nct ion is equal to the line integral of that vector function
taken around the closed curve b oundi ng that surface
.
r
r
.
.
.
x
W-d a
W
e
a
r.
1
( 1)
is is the celebrated theorem of
account of
great importance in all branches of mathematical phys ics
n mber of di erent pr ofs will be given
Th
u
Sto ke s
f
f
o
.
.
On
it s
9.
'
VECTOR A N ALYSIS
1 88
First
Proof :
a small tri angle 1 2 3 upon the su rface
et the value of W at the verte x 1 be W
Chap III the value at an neighboring point is
F
i
( g
S
y
.
Then
b
Consider
L
.
.
y
,
0.
w
Sr
whe re the s
y
di
m bol 8
r has been introduced for the sake of s
t i n guis hin g it from at r which is t o be used as the element of
inte gration Th e inte gral of W taken aroun d the
1 9 3 is
.
FIG 32
.
l
.
1
x 8
r
dr
.
A
Th e fi rs t t e rm
f
w -d r
o
A
i
l
r i
van s hes be cause the integral of d r around a c osed fi gu e n
th s case a small tri angle is z ero Th e second te rm
i
,
m
L
v
i
y
va n sh e s b virtue of ( 3 ) p a ge
.
.
8 r)
1 80
.
.
dr
Hen
ce
,
Th e
x
\
VE C TOR A NAL YSIS
1 90
2
second member
W
7
8
r
da
rl
is the su fa ce inte n of the cu of W
E
.
da
V x W-
V x
8
W
o
da
.
i
r
addi ng together the line integrals wh ch occur in the fi s t
member it is necessary to notice that all the sides of the ele
mentary triangles e xcept those which lie along the boun di ng
curve of the surface are traced twice in Op p o s ite di rect ons
Hence all the terms in the sum
In
i
.
y
which arise from th ose sides of the triangles l in g wi thin the
surface S cancel out leaving in the sum onl y the terms
whi ch arise fr om those sides whi ch make up the boundi ng
curve of the surface Hence the sum reduces to the lin e in
t e gra l of W along the curve whi ch bounds the surface S
,
.
.
Henc
W
e
.
da
L
Second
l
.
L
.
.
Fro
.
33
.
y
Proof : et C be an closed
contour drawn upon the surface S
F
It wil be assumed that C
( ig
is continuous and does not cut itself
’
et C be another such contour near
to 0 Consider the variation 8 which
takes place in the line integral of W
in passing from the contour C to th e
’
contour C
.
NTEGRAL
TH E I
s
f
w
.
f
d
A L C UL US
C
W
dr =
o
OF
VECTOR S
1 91
dr
01
e
w
8
r
W
(
)
e
r) fW-
Sd r +
= d W8r +
f
8 Wdr
.
d 8r
W-
8dr = d 8r
.
H ence
o
8 dr
f
f
W-d 8 r =
d(
8 r)
W-
Sr
d W-
.
r
expression at ( W 8 ) is by its form a perfect di fferential
Th e value of t h e integral of that ex press ion wi ll therefore be
the difference be tween t h e values of W d r at the end and at
the beginning of the path of integration In this case t h e
integra is taken around the closed contour C
ence
Th e
.
l
Hence
.
.
f
W -Sd r
d
W-8 r
,
H
VE CTOR
1 92
Subs tituting
A
NAL YSIS
these va lues
a
w
dr i
o
Br
8r
il r
sim a terms in y and z
B
ut
y
b ( 25) page
.
111
CW
3
io d
9r
dr i-
a
r
i
o
dr
.
x
f
f
Henc e
9
W-d r =
S
‘
Z
o
a
e
dt
s imilar terms in
f
W
S
o
f
dr =
y
an
d
z
W
i
ri
r
it w ll b e seen that d s the element of a c
along the curve C and 8 r is the distance from the curve C to
the curve C Hence 8 r d r is equal to the area of an ele
mentar parallelogram included between C and C upon the
surface S That is
In Fig
.
33
x
’
y
.
’
.
B
r
f
S
Let the curve
x
W -d r
dr = da,
s
x
da
W-
.
il
l
starting at a point 0 in S ex pan d unt it
coincides with the contour boundin g S Th e line integra
C
.
f
w-d r
y
will var from
0
at the point
f
ol r
W-
O
valu e
VE C TOR
1 94
AN AL YSIS
defi nition of V >< W which is independent of the axe s
k may be o b t ained b y applying St okes s theorem to an in
i
Pass a plane
C onsider a point P
fi n ite s im a l plane area
t hrough P and draw in it concent ic with P a small circ e of
area d a
W dr
V x W da
( 1 3)
A
,
j
’
,
r
.
,
l
.
,
.
o
o
When d a has the same
direction as
.
W the value of the
line integral w ll be a max imum for the cosine of the angle
Fo r th is
between V x W and d a will be e ual to unit
value of d a
i
V x
q
,
y
,
w
Hence the curl
d
.
r
of a vector function W has at each
point of space the direction of t h e normal to that plane in
which the line integral of W t aken about a small circle con
centric wit h t he point in ques t ion is a max imum Th e mag
n it u de of the curl a t the point is equa l to the magnitude of
that line integral of max imum value di vided by the area of
the c ircle about whi ch it is taken This de fi nition lik e the
one given in Art 8 1 for the divergence is interesti n g more
from theoretical than from practical cons ideration s
Stokes s theorem or rather its converse ma be used to de
uce Maxwell s e quations of the electro-magnetic fi eld in a
simple manner
et E be the electric force B the magnetic
induction H the magnetic force and C t h e flu x of electri cit
p e r unit a ea p e r unit time ( i e the current densit )
It is a fact lea rned fro m e xperiment that the tota el e ctro
mot ive fo ce a ound a closed c ircuit is e ual to the negati ve
of t h e ate of chan ge of to tal magnetic induction through
th e ci cuit
Th e tota electromot ve force is the l ne ntegra
of the e ectr c fo c e ta ken aroun the circuit That is
V >
<W
.
.
.
d
y
’
’
.
r
r r
L
.
y
,
,
r
r
l
l i r
.
y
,
.
.
q
d
i
l
.
i i
.
Edr
.
l
NTEGRAL
TH E I
AL CUL US
C
i
VE C TOR S
OF
1 95
total magnetic induct on through the circuit is the sur
face inte gral o f the magnetic induction B taken over a surfa c e
bounded b the circuit That is
Th e
y
.
B
ri
Ex pe m ent
-d a
r
the efore shows that
Eu h
t
a
r
m
o
H ence
by t h e
fiL
da
B
r
conve se of Stokes s theorem
’
V x E
a
—
l
w flE
y
is a so a fact of e xperiment that t h e work done in carr
ing a unit positive magneti c pole aroun d a closed circuit is
equal to 4 7: times th e to ta elec t ric flu x through the cir cuit
Th e work done in carr ing a unit pole around a circuit is
the lin e integra of H aroun d the circuit That is
It
l
l
y
.
.
H
l
o
dn
y
i
i
to ta flu x of elec tri cit through th e c rcui t s the
surfa ce in tegra of 0 taken over a s u face bo u n e d b t h e
circ ui t That s
Th e
.
l
i
r
C da
'
Ex
p rim nt the efore tea hes th t
e
e
c
r
H
o
a
dr = 4
7r
Co da
.
d y
1 96
con ve
B y th e
r
V ECTOR
se
A NAL YSIS
of Stokes s theorem
’
V X H =4
p
C
7r
With a pro er inte rpretation of
.
r
c u rent 0 a s t h e dis
placement current in addit on to the conduction current
an interpretation dependi ng upo n one of Maxwe s pri mar
h pothes e s this elation and the preced ng one are the fun da
mental equati ons of Maxwell s theor in the form u sed b
Heaviside a n d H e rtz
Th e theorems of Stokes an G a uss ma be use to
strate the ide n tities
y
,
i
th e
r
y
’
VV x
W= 0
di v
,
rl
ing to
G auss
i
d
=O
.
.
V x
Wda
’
.
r
to Sto kes s theo em
Hence
Wdr
W -d a
VV d
i
d
V V= 0
v
’
V x
Apply
’
s theorem
VV
A ccord n g
y
y
,
,
e
cu
cu
-
Accord
i
y
V x V V = O,
’
ll
d
.
,
.
W
'
o
o
dr
.
'
th s to an in fi n ite s im a l s p h e re Th e su rfa c e bounding
the sphere s cl o sed
ence its boun di ng cu ve e uces to a
o nt ; a n d t h e integra around t to z e o
pi
i
.
H
l
r rd
.
i
V V X Wd v
'
O
V V X
O
r
,
W
o
W= 9
.
.
d r = 0,
1 98
i
VE C TOR ANAL YSIS
y
i
y
those connected w th integra t ion b parts
n o rdinar
calculus The are obtained b y inte grating both s des of the
formul a page 1 61 for di fi e re n tia t in g
y
.
V
(
Hence I
p res ion
u
ex
d
r
)
u v
'
V
i
.
,
,
Th e
“
o
u
dr
[
:
V
u u
v
'
dr
v
r
]
v
u
u
dr
dr
o
.
.
1
4
( )
r
[
s
u n
]
represent s the difle re n c e between the value of ( u v) at r the
end of the path and the va lue at t the beginning of the path
Ifthe path be closed
'
,
o,
,
u
V
vo
'
dr
.
v
V
u
o
dr
.
Second
v
x
(
u
d
a
)
v
ff
=
u
V
-
Xv da +
Vu
-
xv da
.
8
V
u
xv
da
f
n
O
v
dr—
fq
S
da,
X v-
1
( 5)
THE INTEGRAL CAL C UL US OF VECTOR S
V
x
(q
d
a
)
V
v
V
X
u
u
1 99
da
.
Hence
V
u
x
V
v
'
d
o
u
a
V vdr
vV u
'
,
v
Fourth
V
d r ( 1 6)
.
(
o
u
v) d o
u
V vd v +
V
'
u
vd v
.
Hence
u
vd v =
V-
ff
u
v
da
—
S
V
u
vdv
'
s
u v
-d a
—
S
fff
V
fff
u
v
u
o
vdv
,
vdo,
v
—
ff
H
ence
v(V
ff
v
u
u
x v) d
x v
da
v
f
—
s
V
u
fff
—
V
fff
V
u
-v
dv
f
.
( 1 8)
u
these formul a which contain a triple integral the
surface S is the closed surface b oun di ng the bod thr oughout
whi ch the integration is performed
Ex amples of integra tion b y parts like those ab ove can be
m ul tiplied almost without limit O nly one more will b e
given here It is known as G re e n s The or e m and is perhaps
the most importa nt of all If u and r; are any t wo scalar
func t ions of position
In
a
ll
y
.
.
’
.
.
,
VE C TOR A NAL YSIS
200
v (u V o)
=
u
v
V
V ( v )
-
VV
u
v=
VVv
'
,
JV V
V(v
u
q
O
)
a
.
V ( vV u )
—
—
c
ffq
v vu
d v;
o
f
Vva d o
dv
f
,
.
e
u
o
v
vd v
'
'
fv
s
«
u
V
,
u Vv+
V V e dv
ff
By
u
'
'
°
Henc
Vv+
=Vu
.
:
u
da
—
q i
da
ffv
u
o
V
u
d
V V v d o,
v
.
ubtractin g these e ualit es the formula
— v
v
v
( V V
u
o
v )
u
( V
u
i
( 1 9)
( 20)
ff
dv
'
'
o
r— v
v ) da
u
o
.
obta ined B y expand ng the e xpression in terms of i i k
t h e ordi nar form of G reen s theorem ma
be obta ined A
further gene aliz ati on due to Thoms on ( ord Kelvin) is the
following
is
.
y
V
u
r
v
where w is
u
i
q
d
a —
vo
,
.
ff
V
fff
V v dv=
jf
y
L
’
,
da
u
V
2
1
( )
v
'
a th rd scalar function of po sition
Th e element of volume d v has nothin g to do w th the s ca a
fu nction e in these equations or in those th at go before Th e
us e of o in these two di f
ferent senses can ha dl g ve se t o
an
y misunderstanding
In the precedi ng articles the scalar an vector fun c
ti ons which h ave been subject to treatment h ave been su
i
lr
r y i ri
d
p
.
.
.
VE C TOR A NAL YSIS
2 02
the value along one path being the negative of the value
along t h e other Th e integral around the circle whi ch is a
closed curve does not vanish but is e ual to i 2
It might seem therefore the results of Ar t 7 9 were false
and that consequently the enti re bottom of the work whi ch
follo ws fell out Th s however s not so Th e di fii c u lt y is
that the function
q
.
,
z z
7 r.
.
i
.
i
V
.
ta n
:
is not single-valued At the point
funct on V takes on not o nl the va ue
i
y
.
V
tan
:
l
fo r
i
ns
tan ce
,
th e
1
1
but a whole series of values
I
+ k 7r,
where la is any positive or negative integer Furthermore at
the origin which was included bet ween the t wo semicircul ar
paths of integrat ion the function V becomes wholly inde
terminate and fails to possess a derivative It will b e seen
therefore that the origin is a peculiar or s i n gu la r point of the
V
If t h e two paths of integration f om
function
r
1 O) to
had not included the origin the values of the integral
would not have di ffered In other words the value of the
integral around a closed curve which do es n o t n
i c lu de th e
o ri i n
vanishes as it sho ul d
g
.
,
,
.
.
,
.
.
vitiates the results obtained let it be considered as marked
by an impassable barri er An closed curve
C which does
y
,
.
NTEGRAL CALCUL US
TH E I
y
OF
VE CTOR S
2 03
ma shrink up to no t hin g without a break in its continui t y ;
b u t C can only shri nk do wn and fi t closer and closer ab out
the origin It canno t b e shrunk down to not hing It mus t
always remain encircling the origin Th e c urve C is said to
be re du c i ble C i rre du c i ble In case of the func t ion V then
it is true that the integral taken around any re du c i ble circuit
C vanishes ; b u t the integ ra l around an i rre du c i ble circuit C
does n o t vanish
Suppose ne x t that V is any function whatsoever
et all
the points at whi ch V fa ils t o b e con t inuous or to have con
t in u o u s fi rst partial derivatives be marked as impassab le
Then an circuit C whi ch contains within it no
b arriers
such point may be shrunk up to nothing and is said to b e
re du c i ble ;
but a circ ui t whi ch contains one or more such
points cannot be so shrunk up wi t hout b reaking it s continuity
and it is said to b e i rre du c i ble Th e theorem may then b e
stated : Th e li n e in tegra l of th e de riva t ive V V Of a n y fu n c tio n
V va n is h e s a r o u n d a n y re du c i ble c i rc u i t C
It may or ma not
vanish around an irreducib le circuit In case one irreducible
circuit C may b e dis tort ed so as to coincide with another
irreducible circuit C without passing throug h an of the
singular points of V and wi t hout break in g its continuity
the two circuits are said to b e re c o n c ila ble and t h e values of
t h e line in t egral of V V about them are the same
A region such that a n y closed curve C within it may b e
shrunk up to nothing without passing through any sin gu ar
oint of V and without b reaki ng its cont inuity that is a
region every closed curve in which is reducib le is said to be
a c y c lic
Al l other regions are c yc lic
B y means of a simple device any cyclic region may b e ren
dered acyclic Consider for instance t h e region ( Fig 3 4 ) e u
closed between the surface o f a cylinder and the surface of a
cube which contains the cylinder and whose bases coincide
with those of the cylinder Such a region is reali z ed in a room
.
.
.
,
.
,
,
y
L
.
.
y
.
.
y
.
.
y
,
.
l
p
,
,
,
.
.
.
,
,
.
.
2 04
VE C TOR
i
A
NAL YSIS
fl
wh ch a column reaches from the oor to the ceili ng It
is evident that this region is cyclic A circuit which pass es
It cannot be contra cted to
a ound the column is irreduci b le
nothing without breaking its continuit If
now a diaphragm be inserted reachin g from
the surface of t h e cylinder or column to the
surface of the cube the region thus forme
bounded b y the surface of the c linder the
s urface of the cube and the t wo side s of the
O wing to the ins a r
dia p h ra gm is ac clic
F G 84
ti on of the di aphragm it s no longer poss ible
to draw a circ ui t which shall pass completel around the cyl
inder — the diaphragm revents it Hence ever cl o sed cir
cuit which may be drawn in the region is reducible a n d t h e
region is acyclic
In like manner a n y region ma be rendered acyclic b
inse ting a suffi cient number of di aphragms Th e bounding
surfaces of the new region consist of t h e b ounding surfaces of
the given cyclic region a n d the two faces of each di aph ragm
In acyclic regions or regions rendered acyclic by the fo e
going device all the res ults contained in A rts 7 9 e t se g
hold tru e Fo r cyclic regions they ma or ma not hol
true To enter further into these questions at this point is
unnecessary Indeed even as much di scussion as has b een
given them already may b e super uou s Fo r the are ques
tions which do not concern vector methods an more th an the
correspondi ng Cartesian ones They belong properly to the
subject of integration itself rather tha n to the particular
notation which may be employed in connec t ion with it and
which is the primar o b ject of e x position here
In th s
respect thes e uest ons are s mi ar to uestions of gor
in
r
.
.
y
.
d
y
I
y
p
,
,
i
.
.
.
.
y
y
.
y
.
r
y
.
r
.
y
.
’
y
d
.
.
.
.
fl
,
y
.
y
.
,
q i
y
i l
q
.
ri
i
.
2 06
Th e
VECTOR A NAL YSIS
length of
positive
7
r
and
12
wi
ll
be
assumed to be
.
Vr
’
is then
r1 2
Consider
“3
2
( 31 2
0
(2 2
90
z
1)
2
'
the triple integral
0
1
( x2
r
2
5
z
1,
V05 2 ,
)
z
2
)
r
d x z d l/s d z
r
i
integra t ion is performed w th respect to the variables
i
x
z
that
is
with
respect
to
the
b
ody
of
wh
ch
V
y
2
2
2
D uring
represents the densit ( Fig
‘
t h e integration the point ( x 1 I
l v z l ) re
main s fi x ed Th e integral I has a defi ni te
value at each de fi nite point (x 1 y 1 z l )
It is a function of that point
Th e in
F G 35
if
t e rp re t a t io n of this integ a I s eas
the fu nction V be regarded as the density of matte r n space
Th e element of m a ss d m at ( x 2 y2 z z ) is
Th e
,
,
y
,
.
,
.
,
,
rl i y
i
.
.
r
.
.
,
V ( n a , y2 ,
dm
r
z
2)
,
.
,
d wa d yz d z
2
Vdv
.
integra l I is therefo e the sum of the elements of m a ss
in a body each di vided b it s s tance from a fi xed point
Th e
,
(
33
1,
$
1
1,
z
1
)
y di
°
This is
what is term e d the p o te n tia l at
due to the bod whos e ens it is
y
d y
17
02
5
z
,
a
th e
po i t (
n
x1,
yl ,
z
)
l
)
limit of i t grat on in the inte ral
l oked at
in either of
ways
the st place the y may be
reg ded coincident with the li its of the body of w ch
the densi y
hi i deed ight seem the most nat ral
set
limit
other h d the integral
be
Th e
s
i
n e
t wo
ar
In
.
m
t
s
.
.
T
s
On t h e
n
o
fir
as
V is
of
g
I ma y b e
m
an
hi
u
I ma y
THE INTEGRAL
regar ed as taken over
AL C UL US OF VECTOR S
2 07
C
value
of
the
int
gr
e
al
p
is t h e same in both cases Fo r when t h e limits are in fi n i te
the fun ction V vanishes identi call y at every point ( x 2 y2 z z )
situated outside of the body and hence does not augment
the value of the integral a t all It is found most convenient
to consider the limits as in fi nite and the inte gra l as e x tended
over all space Thi s saves the trouble of writing in special
limits for each particular case Th e function V o f i ts elf then
practically determ ines the limits owing t o its vanis hing iden
ti cally at all points unoccupied b matter
Th e O peration of fi nding the pote ntial is of such
fre uent oc currence that a s p ecial symb ol P o t s used for it
d
a
ll
a ce
s
Th e
.
.
,
,
.
.
.
y
.
q
,
Pot
y
z ?
y”
V
r
)
p
e
x
,
,
i
.
d y, d z
z
.
p
s mbol is e ad the otential of V Th e otential
P o t V is a functi on not of the variab les x 2 y2 z 2 with
re gard t o which the inte grati on is performed but of the point
which
is
fi
x
ed
during
the
int
e grati on
hese
22
T
z
( 1 yl l )
Th e function V
variables enter in the e xpression for
and P o t V t herefore have di fferent sets of variab les
It may be necessary t o note that although V h a s hitherto
be en regarded as the densit of matte r in space such an
interp etation for V is entirel to o restri cted for convenience
Whenever it becomes necessary to form the inte gral
Th e
“
”
.
,
,
,
,
,
.
,
.
y
y
r
”
x
”
if”
Z
2
)
dz
l
cry
,
,
,
.
dz
5
,
se
,
my
.
of any sca a r function V no matter what V represents that
inte gral is call ed the potential of V Th e reas on for c a lling
such an inte gral the pote ntial even in cases in wh ch it h a s
no connection with ph s ical potent a is that it is formed
la was the true po tent al and
a ccord ng to the same form a
,
,
.
i
y
l
il
i
i
VECTOR ANA L YSIS
2 08
y
b vi rtue of tha t formation ha s certa in simple rul es of Ope
tion which other t pes of in t egrals do not possess
Pursu a nt to th s idea the potential of a ve c to r f un ct o n
i
y
9
,
2,
z
a
a
i
.
W ( 372
r
)
q
this case the i ntegral is the sum of vector uantiti es
and is consequently itself a vecto r Thus the potential of a
vector function W is a vector function just as the po tent al
of a Scalar function V wa s seen to be a scalar func tion of posi
tion in space IfW be reso ve into its thr ee com
ponent s
In
i
.
,
l d
.
W ( x2
,
9
3”
Z
2)
i
X ( 002 ,
z
a)
+
jY ( wa
( x2
kZ
Pot
,
W = i P o t X + jPot Y + k Pot Z
.
z
92,
r
l ,
I
la
)
z )
z
z
( 2 4)
.
r
potential of a vector function W is equal to t h e vecto
s u m of the potentia s of its three components X Y Z
Th e potent al of a scalar function V e x sts a t a
o nt
r
e
‘
z
( l Il v 1 ) when a n d o nl when the nte gra
Th e
l
i
,
y
,
i
l
i
,
pi
.
,
.
Pot V
l
taken over all spa c e converges to a de fi nite va ue If
for instance V were ever where constan t in Space t h e in
t e gra l would become greater and greater w t hout
m t as
the limits of integration were ex tended farther and farther
out into spa ce Evidentl therefore if the pote nt a is to e x s t
V mus t approach z e o as its
m t a s t h e po int (212 y, z g)
ecede s inde fi ni te y A fe w mportant s u ffi c ent condi t ons
fo r t h e converge nce of the potent al ma b e o bt a n e
by
t an s forming t o pola c o b rdin a t e s
Le t
y
,
r
.
r
r
l
.
r
i
y
i
,
.
li i
il
li i
i
.
‘
y
i
i
'
i
i d
,
.
VECTOR A NAL YSIS
21 0
f
u nc tio n
If the
V
re m a
we a kly th a t the p ro du c t
re ma
a s
in s
fi
wh e n
n i te
r
f
fin ite
in
or i
th e n the in tegra l
o rigin a re c o n c e rn e d.
to th e
a r a s regio n s n e a r
f
so
Vr
a c h e s z e ro ,
ro
pp
a
i t be c o me inf
in ite
c o n ve r es
g
Fo r
let
Vr < K
=R
r
r :
Vr
s rn
ddr a s a
s
<
jfd
r
r
a s a
s
=0
=R
Kdr
r
r
a s a s
n
:
KR
.
=0
Hence the t iple ntegral taken over all space inside a spher
of ra us ( where is now supposed to be a small quant ty )
is less than
and consequently converges as far as
K
i
r
di
e
R
R
4
7r
i
R
regions near to the origin which is the poin t ( sol
concerned
,
z
yl ,
are
)
l
.
If
i e
.
a
y p o in t ( x 2 , y 2 , 2 2 )
p o in t ( x 1 , y l , z l ) , th e
t
a n
t he
.
we a kly th a t th e p ro du c t
x
( 2 , y2 , Z 2 ) by th e s q u a
(x2
f
o
re
n o t c o in c i de n
u n c tio n
f
t h e va lue
f th e
O
fi
V be c o m e s i n
fV
o
dis ta
t wi th th e
nce
a
t
a
p o in
f th a t
O
t
o rigin
,
n i te
so
nea r
to
p o in t
fro m
a in s
r
e
m
n
i
a
a
h
e
n
i
h
a
i
a
t
e
s
d
a
e
ro
c
h
e
s
z
e
r
o
t
t
t
s
t
n
c
)
f
y
pp
2
th e in te gr a l c o n ve rge s a s fa r a s re gio n s n e a r to th e p o in t ( x 2 y 2 Z 2 )
,
2,
Z
,
,
,
proof of this statement is like those given
before These three con di t ions for the convergence of the
integral P o t V are suffi cient They are b y no means n e c e s
sary Th e integral may converge when they do not hold
It is however indispensable to know whether or not an integra
under discussion converges Unless the tests given ab ove
show the conver gence more stringent ones must b e resorted
to Such however wi ll not be discussed here They b elong
t o t h e theory of integration in gene ral rather than to the
a re c o n c e rn e d
.
Th e
.
.
l
.
.
.
,
.
,
,
.
y
THE INTEGRAL CAL CUL US OF VECTOR S
21 1
theor of the in t egrating operator P o t Th e dis cussion of
the convergence of the potential of a vector function W re
duces at once to that of its th re e components whi ch a re sca ar
functions and may be treated as above
Th e potential is a function of the variables x1 y l a 1
which a re con stant with respect to the integration
et the
value of the potential at the point ( x1 y 1 z l ) b e denoted by
.
l
.
L
,
.
,
P
Ot
[
i
Th e fi rst
is
m
a
.
,
,
l
311 , a
partial derivat ve of the potentia with respect to
therefore
a P Ot V
9
33
P
O
t
V
[
]z
LIM
A
1
x
1
1
O
i
A
value of this li mit ma
device ( Fig
C onsider
the potential at the point
Th e
21
,
P
O
t
V
[
l
y r, 3 1
’
yl ’
A 93 1
y
be
determ ined
by
21
a
,
( 2 5)
a simple
.
A 901 ,
z
y
)
r
i
due to a certain bod T Th s
is the same as the potent a at
the po int
(
x
1
,
y1
,
z
l
.
il
:
M
)
Fro
.
36
.
y
y
due to the same body T di splaced in the negative di rection b
the amount A ee l Fo r in fi nding the potential at a point P
due to a bod T the absolute positions in space of the bod
T and the point P are immaterial
It is only their positions
re l a ti ve to e a c h o th e r which determines the value of the poten
tial If both bod and point be tra nslated by the same
amount in the same d rection the value of the potent al is un
changed B u t now if T be displaced in the negative di rect on
b y the amount A ac the v a l ue of V at each point of s a c e s
chan ged from
7
7 05 2 yr z z ) to V s
A 502 9 2
where A x 2 A re ]
y
.
.
y
.
i
i
,
,
.
i
p i
.
,
,
,
VE C TOR A NA L YSIS
21 2
Henc
e
P
ot V
[
H
A ‘n S
in
J
P
O
t
V
[
e nc e
Aa l
i
P
015 V( x
[
2
31
”:
A 3 1 9 Vi a
3;
A x 2 ’ ga
P
L
O
Vs
[
,
z
e
”
21
.
h
o
31
31
y”
A 33 1
O
d
will be found convenient to intro uce th e li mits o f
inte gra t ion
et the portion of space originall fi ll ed by t h e
body T be denote d by M ; and let the portion fil led b t h e
body after its translation in the negative d rection through
’
’
Th e regions M and M
the dista nce A re ] b e denote d by M
overlap
et the eg on common to both be M ; and let t h e
’
’
Then
em a nder of M be m ; the remainder of M m
It
L
.
y
y
i
r i
L
.
r i
.
,
M=M + m
Pot
A
7
1
6
73 1 ”
2
,
V( a
29 2 ,
A x 29 y 2 9
s
r
2
2
)
d v2
12
Axe , Se i z e )
"
Pot
.
12
V ( x2 , y a ,
e
V ac
a)
:
2
y”
2
)
d v2
12
V ( 932 ya
Hence (
i As
in di c e s
a
25)
2
)
do
V 03 2 9 2
i
,
g
.
2
2
)
g
be comes when A x , s replaced b y
fo llo wi n po t e n t ia ls
be e n dro ppe d
ll t h e
ha ve
2
a re
fo r t h e
po in t
x1 ,
y
1,
it s
z
de
q
g
e ual A a
i the
bra c
k
n,
e t a nd
21 4
Then
if it be
VECTOR
A NA L YSIS
ssumed that the region
upon the surface b oun di ng T
vanishe s
T is fi n
a
LIM
é l)
A xe —
r
V( x 2 y2
’
r
12
’
e and
th a t
V
d v2 = 0
s
12
it
z
?
)
d
s
,
”
2
O
.
Cons equently
the expression for the derivati ve of
tial reduces to merely
poten
Pot
d vz
Th e p a rti a l de riva tive of the p o te n ti a l
th e
f
o
a
s c a la r
f
u n c tio n
V
l to th e p o te n tia l of the p a rtia l de ri va tive of V
Th e de ri va tive V of the p o t e n t ia l of V is e q u a l to th e p o te n tia l
of th e de riva tive V V
is
e qu a
.
.
V
is statement follo
Pot V Pot V V
2
7
( )
:
immediately from the former As
the V upon the left-hand side applies to the set of vari
ables xl Z‘l v z l it may b e written V I In like manner t h e
V upon the right-hand side ma be written V2 to call atten
ti on to the fact that it applies to the variables wz yz z z of V
Th
,
ws
.
y
,
.
,
To
i,
j
i
VI
Pot V = Pot V2 17
y
demonstrate th s identi t
k
,
V
.
y
d
ma be e xpa nde in
.
9
Pot V
9 I
f
!1
—
i Pot
+
,
jPot
+k
Pot
.
j
THE IN TEGRAL
CA L C UL US
y
OF VECTOR
S
21 5
are consta nt vectors the may be pla ced under
the sign of integration and the terms ma be coll e cted Then
by means of ( 2 6)
As i,
,
k
y
.
V 1 P 0t V = P Ot V z V
The
Vx
c u rl
f
it/no t io n
W
dive rge n c e
di ve rge n c e V
a nd
a re e u a
q
l
r e sp e c
h
a t
u n c t io n
t
f
f
o
the p o te n tia l
ti ve ly to th e p o te n tia l
f
o
f the
o
vec to r
a
c u rl a n d
.
W P ot c u e
V l P o t W Pot V2 W
di v Pot w P o t d v w
curl
Pot
:
:
i
These
f
o
i
y
,
9
.
relat ons ma be proved in a mann er analogous to the
It is even possible to go further and fo rm the de l s
a bove
o f h gher order
i
.
V
0
V Pot V
V Pot
V-
:
Pot V
V V,
( 3 0)
W = P o t V -V W
Pot Wfi Pot V V VVW
V x V x P o t W= P o t
V xW
(31 )
,
32
( )
,
( 33)
.
pon the left might have a sub script 1 attached to
show that the differentiations a re performed with res pect to
the va riables x1 gl z l and for a simil ar reas on the de ls upon
the right might have been written wi t h a sub script 2 Th e
esul ts of this article may be summed up as follo ws
Theorem : Th e dife re n ti a tin g op e ra to r V a n d th e i n tegr a tin g
Op e ra t o r Pot a re c o m m u t a ti ve
In the foregoing w ork it has been assumed that the
egion T wa s fi n ite and t ha t the function V wa s everywhere
fi nite and continuous inside of the region T and moreover
decreased so as to approach z e ro continuously at t h e surface
bounding that region These restrictions are inconvenient
Th e de ls
U
,
,
,
r
.
.
r
.
21 6
VE CTOR
y r
y
A NA L YSIS
i
r
ma be e moved by ma k n g use of a su fa c e
Th e deriva tive of the po te nti al wa s o bta n e d ( a ge
essenti all th e form
and
i
p
it l
n e
g
ra
.
2 1 3) in
9 Pot V
9
x
1
V032
1
LIM
A
A xe — O
A x2 , ya , 5 2 )
s
r
12
VG” ? y 2 ’
i
l
q l
2
2
)dU
‘
2
i
a d re cted element of t h e surface S bo u nd ng t h e
'
egion M Th e e ement of volume d v z n t h e reg on m is
there fore e ua to
Le t d a b e
r
.
i
d v2 = A x 2 i
He c
V023,
n e
A
7
VO5 2
A
7
Th e
da
lmn
e e
e
t
l
i
x
2,
'
f
lz a
z
2
)
23
2
2
r i
i
o
V( x2 ’
7
V03 2 » ya ,
z
a
)
s
)
d v,
32
of vo ume d v2 n th e eg on
A
Conse
.
93 »
2
12
i
i
da
z
m
i
da
0
.
is q l
e
ua
to
.
2
)
d ?) a
12
day
quently
9 Pot V
i
o
dS
.
34
( )
VECTOR
21 8
A NA L YSIS
L
thi s point with a small Sphere of radi us R
et S den ote t h e
s u rface of this Sphere and M all the region T not included
within the sphere Then
.
.
9
Pot V
9
x1
.
1
ii i
the cond t ons mposed upon
By
.
da
.
V
Vr < K
h
E de d¢ = 2
da
C onsequently
7r
K
.
ll
when the sphere of radi us R becomes sma er
and smaller the surface integra l may or ma not be come z ero
Moreover the volume integral
y
y
y
.
ma or ma not approach a li mit when R becomes
and smaller
ence the equation
.
H
sm
ll
a er
9 Pot V
9
58
y
r
1
r i
not alwa s a de fi nite meaning at a point of the eg on
T at which V becomes in fi nite in such a manner th at t h e
product V r e m ain s fi n ite
If however V remains fi nite at the point in uesti on so
that the product Vr approaches z ero the constant K is z e o
and the surface in tegra bec omes sma ller a n d sma er a s R
appro a che s z e o Moreove t h e volume inte g a
ha s
,
q
.
,
r
l
.
,
r
rl
1
ll
r
THE INTEGRAL
OF VECTOR
CA L C UL US
appro a ches a defi nite li mit as R becomes infi ni tesim
sequently the equati on
OP o t V
Pot 67V
9
x
S
al
21 9
Co n
.
9 wz
1
holds in the neighborhood of all isolated poin ts at which V
remains fi nite even though it be dis continuous
Suppose that V becomes in fi nite at some single po int
z
x
( 2 y 2 z ) not coincident with ( 1 1 gl z l ) A ccording to the
condi ti ons laid upon V
2
W K
.
,
,
,
,
i
.
where l s the distance of the point ( x2
near to it Then the surface inte gra
l
.
i
d
nee not be come z ero
and
o
c
y
oo
z
rom a po int
f
)
z
da
ly the e quation
OV
9
l
need not hold for an point
f V becomes in fi nite at
gz
i
y2 ,
consequent
9 PotV
A
,
,
Vl
gl ,
z
2
<
x
z
2
r i
of
the
eg
on
)
l
.
B
ut
in such a mann er th at
K
,
then the surface integral will approac h z ero a s its limit a n d
the equat on will hold
Fin all y suppose the fu ncti on V remai n s fi nite upon the
surfa ce S boundin g the region T but does not van ish there
In th is ca se there exists a surface of dis cont nuiti es of VI
Within th s su face V is fi nite ; w thout it s z ero Th e
i
.
,
i r
i
i
da
,
i
i
.
.
VE CTOR
2 20
doe s not vanish
general
in
9
A NA L YSIS
H ence the e q atio
Pot V
9V
9
9
x
1
a
c a nnot hold
Similar reasonin g ma be applied
pa rtial derivatives w th espect to co ]
t h e resu ts it is seen that in general
y
i r
.
l
Vl
L et
n
u
.
s
to
,
g] ,
Pot V = Pot V, V
i
( 35 )
fun ction n spa c e and let it be gra n t e d th a t
Pot V e xists Surround each point of spa ce at whi ch V
ceases to be fi ni te by a sma ll sphere
et the surface of the
sphere be denote d by S Draw in spac e a thos e s u rfac e s
which a re surfa ces of di sconti nuit of V Le t thes e su
faces also be denote d b S Then the formu a ( 8 5 ) hol s
whe e the surfac e integ al s taken over all the s u rfaces
which have been e signated b S If the inte n t aken
over all these surfaces van shes when the adi of the sphe es
a b ove mentioned becom e nfi n i tesi m al then
V hs
any
,
L
.
.
.
y
r i
d
y
i
i
r
This
Pot V
Pot
:
n i te
su re ly
or
o
d
r
uc
p
t
ho ld
r
pp li ed
a
t
fi
be c o me s in
VI
re m a
dis c o n t in u it y,
as
r
d
l
.
r i
.
r
,
V2 V
.
fo rmul a
Pot V, V:
VI P o t V :
fi
y
.
VI
wil l
ll
be c o me s
a
p o in t
n it e
ins
a
t
(
x
1
,
Yb
2
V
f
i
)
1
p o i n t ( x) , Ya ,
n d if V
o
s
p se s ses
a
re m a in s
2
2)
i
to
n o su rfa c es of
fi n i te a
3
a n d f u r th e rm o re t he
/n i te
u
c
a
i
ns
d
o
t
V
r
re
m
r
i f
fi
p
infl a te }
In other cas e s special tes ts m u st b e
,
be
as
x
lwa ys
th a t th e
so
certain whether the formul a
can
or the more c omplicate d one ( 35) mus t be re s o rte d to
a
a
Fo r e t e ns i o ns
a nd
m difi
o
ca
i
i
.
t o ns o f t h s t h e o re m , se e e xe rc iws
.
us e d
VEC TOR
222
A NA L YSIS
irregul i ies which may arise are th own into the inter
not into the analy tic appearance of the formul e
his is the essence of Professor
meth d of treatment
Th e
re
p
T
ar
t
r
t a ti o n ,
a
G ib b s
’
s
o
.
.
Th e fi rst
partial derivatives of the potential may also
be obtained by differentiating under the sign of in te gration 1
.
d x2 d y a d z z
( 36 )
w.
V[<
a
like mann er for a vector function
In
Pot W
9
—
9
x
W
W
x
( 2
1)
517
,
y2 , z z )
1
V P ot V= i
9 Pot V
9 Pot V
9
a 1
wi l l,
k
(z 2
3
s
B
ut
m
1
r]
doe 2 dy“dz “
If a n
a n n e r,
i
(
a
tt e
:c
2
mldp
it wo u
—
x1
)
t we re
+
md
be se e n
j( y g
—
y 1)
e
to
o
k (z
mi g
bta i n
t ha t t h e vo l u
a
+
the
e
g
g
pa rt i a l d e ri va t i ve s in t h e
ra ls n o l o n e r c o n ve r e d
se c o nd
n te
z
.
sa
m
e
S
IN TEGRAL CAL C UL U OF VE CTOR
TH E
Henc
lik
In
e
22 3
III
V Pot V
e
S
mann e r
W
V x Pot
V
o
P ot
III
W=
fff
il
These
th ree integrals obtained from the potent a
differentiating O perators are of great importance in mathe
m a t ic a l physics
E ach has its own interpretation
Co n s e
quent y although obtained so simply from the potential each
M oreover inasmuch as these
is given a separate name
integrals may ex ist even when the pote ntial is d vergent
they must be considered independent of it The are to
be looked upon as three new integrati ng operators defi ned
each upon its own merits as the potential was de fi ned
et therefore
l
.
.
i
y
.
.
L
,
.
,
,
r
1
12
'
12
X
°
W ( 39 2 g a
z
,
w(
(13
2,
y 2a
z
3
If the p o te n ti a l
e xis ts ,
e
2
)
d xg d y g d z
>d x
2
dy2 dz
z
2
L ap
43
( )
W
= Ma x w
.
then
V P o t V = Ne w V
W La p W
P o t W = Ma x W
VV X P ot
Th e fi
rst is writ en
t
.
N e w V and
read
“
Th e N ewto ni an
of
( 44 )
VE CTOR
224
A NA L YSIS
reason for calling this integral the Newt n is t t
repre ent the density a body the integral give the f
due
to
the
bod
y
at
the
point
)
(
f
wi be proved later
second is written
r ad Lap acian of W his integral was used to a
c n iderable tent by L aplace
is of fre quent occurrence
in electricity and magnetism
represent the current
in space the L aplacian of gives the
at the
f
oint (
due
to
the
c
rrent
th
i
rd
is
w
tten
)
and read the Ma wellian of W
his integral
used by Ma we l
too occurs freq ently in electricity
magnetism
instance if
represent the inten ity
o n ia
Th e
V
a
o
o
s
x
1
s
l
m a gn e tic
0
p
pl
,
z
u
l
Ma x W
l
and
.
It ,
.
orc e
Th e
.
ri
wa s
T
”
x
x
d
If W
.
x ,
l
an
It
.
0
Thi s
.
T
.
ex
s
,
if
o rc e
La p W
”
l
th e
pl
,
Th e
.
“
o
s
ttra c tio n
ll
e
f
ha
.
u
,
Fo r
W
s
of magnetiz ation I the M axwellian of I gives the m a gne tic
p o te n ti a l at the point ( 501 p l s l ) due to the magnetiz ati on
To show tha t the N ewtonian gives the force of attr actio n
acc ording to th e la w of t h e inverse s ua e of th e distan ce
et d m z be an e ement of mass situa ted at the point
x
T
h
force
at
e
( 2 y,
(x1 y1 z l ) due to d m s e ua t o
,
,
L
,
.
q r
y l
,
,
,
.
i q l
,
r
in magnitude and h as t h e di e cti on of the vector rm from the
po nt ( x1 9 1 2 1 ) to the point (mg 31 2 z z )
enc e th e fo ce is
i
,
,
,
Integrating
,
.
H
.
ll
r
over t h e entire body or over a spac e a cco
to the conventi on here adopted the total force is
,
,
r
12
V
31 ,
4
7
r denot s the den ity of matter
whe e
V
e
s
.
d v2 = Ne w V
r
din g
VEC TOR
226
Integrating
the point
A NA L YSI S
over all space the total magnetic force acting
z ) upon a unit positive pole is
gl l
,
,
tl v
L ap W
j L et
’
This
integr a l may be ex panded in terms of i
W ( x2
z
,
a
i X Q’ s i
)
z
z)
j1
+
k
x
Th e i,
j
2
.
,
k
1
)
L
components of ap
(ya
9
1
z
,
a
)
Z
(
“
z
_
z
2
,
z
z
z
'
—
z
.
19 2 ,
2,
are respectivel
W
—
3
(
k
,
0
7
Z
(
( ya
i +
,
k
)
1
a
)
z
)
.
y
Y
)
1
r
j La p W
k
t
— x
1
La p W
)
Y — (y 2
—
g1 )
X
3
y
erms
the potential (if one ex ists ) thi s ma be
i
La p W
” La p
k
.
La p
W
9 Pot V
9
a yl
a
9
1
9 Pot V
9
50
,
1
9 F ot z
9 Po t X
9
a
x
1
9 Pot X
y
i
be the intensit of magnet z ation at the
point ( x 2 v2 z 2 ) that is if I be a vector whose magnitude is
equal to the magnet ic moment per un t vo ume an whos e
To
show that if
W
9 Fot z
,
,
,
I
,
i l
d
IN TEGRAL CAL CUL US
TH E
VECTOR S
OF
2 27
direction is the direction of magnetiz ati on of the element d v2
fr om south pole to nort h pole then the Max wellian of I is th e
magnetic potential due to the di stribut ion of magneti z ati on
Th e magnetic moment of the element of volume d v2 is I d v2
Th e potential at (513 1 p l s l ) due to thi s element is equal to it s
magnetic moment di vided b y the square of the distance r 1 2
and multi plied b y the cosine of the angle b etween the dire c
tion of magnetiz ati on I and the vector rm Th e potential is
therefore
,
.
.
,
,
.
Inte g
rati g the total magneti c potential is seen
n
,
Ma x I
d vz
l
Let
0
0 ,
2
.
integral may a so be writt n out in term
Thi s
1
to b e
V
2,
z
a
11 4
)
(
as
,
02
0
— x
e
z
,
1
)
( V2
“
,
k 0 002
B
z) +j
A
of x
s
V1 )
(
B
z
— z
1
z
)
.
y,
9 2a
z
z
.
z
)
C
.
inste ad of se , Z/v z 1 the variables x g z ; and inste ad of
1
xpression takes
be
used
the
e
x
z
the
variables
C
n
y
5
2
2
2
Maxwell
o n the form g ven b
If
,
,
i
,
Ma x I
,
y
,
,
,
.
A (g
’‘
ing to the notation employed for the Laplacian
Accord
Ma x
W
1
Ma xwe ll
:
El e c t ri c it y
and
Ma gn e t is
m
,
Vo l II
.
.
p
.
9
.
VECTOR
2 28
q
y
Maxwell ian of a vector fun ction is a scalar uanti t
ma be wr tten in te rms of the pote ntial ( if it e xists ) a s
Th e
It
A NA L YSIS
y
i
Ma x
_
W
9 Pot X
9
x
9 Pot Y
.
9 Fo t Z
+
9
1
z
r
Thi s
1
y
form of expression is much used in o dinar t reatises
upon m at hematical physics
aplacian and M axwellian however should
Th e N ewtonian
not b e associated indissolub ly with the particular physical
interpretations given to them above They should be looked
upon as integrating operators which may b e applied as the
potent ial is to any functions of posi t ion in sp a ce Th e N e w
t o n ia n is applied to a scalar function and
elds a vector
function Th e aplacian is applied to a vector fun ction
and ields a fun ction of the same sort Th e Max wellian
is applied to a vector function and ields a scalar functi on
Moreover these integrals should not be look ed upon as the
derivatives of the potential If the potential e xists the
are its derivatives B u t they frequently e xist when the
potential fai ls to converge
et V and W b e such f u nctions that their potentials
9 L]
e xist and ha ve m general de fi nite values Then by ( 2 7) and
,
L
.
,
,
,
.
,
yi
,
y
L
.
y
.
.
.
y
,
.
.
L
.
.
( 29 )
V V Pot V
:
o
B
ut
by ( 45)
V Pot V V=
o
V Pot V
and
:
Pot V
N e w V,
o
V
o
V Pot V= V
o
Ne w V = Ma x V V
Pot
B y ( 2 7 ) and ( 2 9) V V
ut
VV
V P o t V V = Ma x V V
Hence
B
o
b y ( 4 5)
'
and by (45)
V
V Pot
VVV
W= V P o t V W
Pot W
Ma x W
V W Ne w V W
Pot
:
,
z
.
4
6
( )
P ot V V
W
.
VE CTOR
230
P o is s o n
92
]
Le t V be
h a s in ge ne ra l
a
a ny
fi
de
9
P 0t V
9
33
E qu a tio n
s
fun c tio n
i n sp a
t he p o te n tia l
c e su ch
P ot V
n ite va
V
2
’
A NAL YSIS
-V
lu e
Th e n
.
Pot V
92 P ot V
—
9
2
Pot V
9
2
1
Thi s
4 vr V,
z
4
2
qr
V
1
q
equation is known as Poisson s E uation
Th e integral which has been de fi ned as the p o te n tia l is
solution of P oisson s Equation Th e proof is as follows
’
.
a
’
.
.
Pot V
V
r
V
V
12
2
3
subscripts I and 2 have been attached t o designate
clearly what are variables with respect to which the di fi e re n
t ia t io n s are performed
Th e
~
.
Vl
and v,
o
P o t Vl
V v,
o
Ne w V
Vs
v
z
.
TH E
IN TEGRA L
A LCUL US OF VECTOR
C
v,
v1
or
V2 V = V V 2
-V2
+ v2
S
231
V v,
VV
.
Integrate
V2 V d v 2
That
He e
h
is to say
VI
s
i L
sat s fi es aplace s
’
Vl P o t V
E
quation
.
An d
y
b ( 8)
V2 Vd v2
( 5 3)
l
surface integra is taken over the surfac e which boun ds
the region of integration of the volume integral This is
taken over all space
Hence the surfac e integral must b e
taken over a sphere of ra di us R a large quantity and R m us t
be allowed to increase without limi t At the point (x1 y 1 z l )
however the integrand of the su rface integral becomes in
fi n i te owin g to the presence of the term
Th e
.
”
.
,
,
,
,
1
,
,
VEC TOR
23 2
A NA L YSIS
Hence the surface S must include not only the surfa ce of the
Sphere of radi us R but also the surface of a Sphere of ra dius
’
R a small quantity surrounding the point ( x1 p 1 z l ) and R
must be allowed to approach z ero as its limit
As it has been assumed that the potential of V e x ists it is
assumed that the conditions given (Art 8 7) for the e x istence
of the potential hold That is
,
’
,
,
,
,
.
,
.
.
Vr
K when
a
X,
Vr
when
is large
r
,
r
is small
Introduce
( x1
,
y1 ,
.
i
polar co ordinates wi t h the orig n at the point
c )
T
hen
r
becomes
simpl
r
l
y
.
1
r
Then
r
3
r
12
for the large sphere of radius R
VI
1
r
z
s in
O d O d cp
.
Hence the surface integral over that sphere approaches z ero
as its limit Fo r
.
2 7r
da
Ff
l
Hence when R becomes infi nite the surface integra over the
large sphere approaches z ero as its limit
Fo r the small Sphere
.
Vl
1
o
d
a
r
z
s in
’
d db dd
)
.
Hence the integral over that sphere becomes
V s in O d d d d
)
.
VECTOR A NA L YSIS
2 34
Theorem
:
If V a n d W a
th a t the ir p o t e n ti a l s
V
W
a nd
P o is s o n
’
s
fi
a re
re s u c h
f
unc
ti on s
fp o si ti o n
i n sp a c e
ll p o i n ts a t wh ic h
o
in ge n e ra l, th e n fo r a
a n d c o n t in u o us th o s e p o te n t ia ls
e xis t
n i te
sa
tisfy
E qu a tio n ,
V
V
Pot
V
V
Pot W
V
2
5
( )
modifi cations in thi s theorem w h ich are to be made at
points at which V and W become dis c o n tin u o u s will not be
taken up here
It was seen (4 6 ) Art 9 1 that
Th e
.
.
V
V P ot V
V
V
:
o
-Ne w V
—
Ma x V V
In
Ma x V V
Ne s
4
4
vr
vr
V
V
.
a similar man ner it was seen ( 51 ) Art 9 1 tha t
VV P o t W = V Ma x W— V >
< La p W
—
Ne w V W
.
~
V Ma x
Ne o
By
W — V x La p W
W — La p V x W
li
vi rtue of this equa ty
W
Lap x
V
W
where
and
W1
W2
is
WI
Lap x W
V
N ew V
W
—
di vided
W
4
—
7r
4
W
54
( )
,
7r
W
.
into two parts
Ne w V W
5
5
( )
L ap curl
( 56 )
.
W2
.
,
N ew
W
div W ( 57)
.
THE INTEGRA L
E quation
AL C UL US OF VEC TOR
C
S
2 35
( 55) states that any vector fun ction W multiplied
L
by 4 71 is equal to the difference of the aplacian of its c ur l
and the N ewtonian of its divergence Furt hermore
.
-W1
V
B
ut
the
o
V
-W1
div W1
1
V >
< W2
ut
WI
ivergence of the curl of a vector function is
d
Hence
B
VV x La p
V La p V x W
= 0
ero
4
”
V X V Ma s
i
the curl of the derivative of a scalar funct on is z ero
Hence
Conse
V
xW
c u rl W2
2
quently any vect r fu ction
.
5
8
( )
1
-W2
z
.
O
.
.
( 59)
.
whi ch has a potential
may b e di vided into two parts of which one has no divergence
and of which the other has no curl This division of W into
t wo such parts is un ique
In case a vector fun ction has no potential but b oth it s curl
and divergence possess potentials the vector function may be
divided int o th ree parts of which the fi rst has no divergence
the second no curl ; the third neither di vergence n o r curl
o
W
n
.
.
,
,
L et
W
La p V x
W
V
.
,
Ne w V
-La p V x W
V
x
Ne o
W
Vo
—
—
1
4W
i
—
W l—W3
.
W= 0
Pot
V X V Pot V
l
W= O
.
d vergence of the fi rst part and the cur of the secon
part of W are the refore z ero
Th e
.
d
VECTOR A NAL YSIS
23 6
x La p
V
V x
W
W
P ot
VV -
VV
V
W
P ot
o
V
x Pot
W
V Pot
W
V x V
.
W= O
V Pot V o
-V
x
,
W= O
.
Hence
Hence
1
4
'
71
l
q
cur of W is e ual to the curl of the
Th e
1
4
7r
fi r st
part
La
PV XW
into whi ch W is divided Hence as t h e second
c u rl the third part can have none Moreover
.
,
p rt has
a
.
1
4
W1
W = V -W= V -
V Ne o
7r
the divergence of W is equal to the
the second part
Thus
di vergence
of
W
N ew V
i
.
.
into which W is d vided Hence as the fi rst part has no
divergence the third can have none Conse uently the thi rd
pa rt W8 has neither curl nor divergence This proves the
statement
B y means of Art 9 6 it may be seen that an function W3
which po ssesses neither curl nor divergence m u st either
q
.
.
.
.
.
y
,
VECTOR A NA L YSIS
238
With
re sp e c t
to
scalar fu n c tio n
a
V
o
i
dv
or
a re
a nd
Ma x
a n d a lso
in ve rs e Op e r a to rs
V th e op e ra to rs
1
4
a nd
7
V
.
Ne w V :
V
Ma x V V = V
.
Wi th re sp e c t to a solenoidal fu n c tio n W1
1
4
a re
Pot
a nd
7r
in ve rse op e ra to rs
.
That
o
th e
r c url c url
is
Pot
Wi th re sp ec t to
a n
a re
in ve rs e op e ra to rs
1
—
4W
Wi th
O e ra
p
to rs
.
Pot V V
res e c t
p
to
W1
l
irrotationa fu n c tio n W2
Pot
That
w,
a n
y
Op e ra to rs
a nd
—
th e
=
W1 ( 64)
.
op e ra
to rs
V V
is
—
vv
Pot
w2 = W2
scalar or vector fu n c tio n
V,
W the
NTEGRA L
TH E I
1
4
Op e r a
I7r
P ot V
re sp e c t
t
h
e
f
to r s
V V
o
to
a
VW
V
-V
239
Pot V= V
v v
—
Pot
o
solenoidal fu n c tio n W1
W= W
( 66)
.
t he defe re n ti a ting
s e c o n d o rde r
V
a re e qu i va
—
S
"
‘
With
AL CUL US OF VECTOR
71
1
—
Pot V
C
V
V x V X
a nd
le n t
( 67 )
Wi th
t to
r esp ec
i n g Op e ra to rs
an
f th e
irrotational fu n c tio n W2
se c o n d o rde r
o
V
a re e u iva
q
By
le n t
V
a nd
is
VV W2
V V
That
.
i
q i
= V V
4
= V x La p
7r
i
7r
4
Pot V
7r
—
7r
L
lr
1
is
a n op e ra
fu n c tio n s
rr
op e ra
to r
W
Pot
—
Ma x N e w V
9
6
( )
W — P o t V Ma x W
W — N e w M a x W ( 7 0)
La p
irrotationa
l
.
ve c to r
u n c tio ns
f
N ew Ma x
to r wh ic h is e q u i va le n t to
th e
V Ma x
La p
a nd
4
—
Ne w V
W= P ot
P o t W = ap
sca
a
u
n
c
t
i
o
n
s
f
f
or
al
n
Pot V
P ot
4
enc e
.
V N ew V
d
W
W
by means of the potent al i tegr
an
H
-WZ
integra t n g the e uat ons
4 WV
4
the dife re n ti a t
Pot
L ap Lap
.
Fo r
solenoi
dl
a
vec to r
VECTOR A NAL YSIS
2 40
i
rs t Op e r a to r
F
or any ve c t o r fu n c ti o n th e f
h
e p o te n ti a l
t
i
v
es
g
a rt ;
the se c o n d, the
irrotational
h
e p o te n t i a l of th e
i
t
v
es
p
g
o te n ti a l of th e solenoidal p a r t
p
.
.
There
are a numb er of do u ble volume integrals whi ch
are of such frequent occurrence in mathematical physics as
to merit a passing mention although the theor of them will
not b e developed to any considerable e x te nt These double
integrals are all scalar quantities They are not scalar func
tions of position in Space They have but a Single value
Th e integrations in the e x pressions may be considered for
convenience as e x tended over all Space Th e fu nctions b y
vanishing identi cally outside of cert ain fi nite limi ts deter
mine for all practical purposes the limits of integration in
case t hey are fi n i te
G iven two scalar fun ctions U V of positi on in Space
Th e m u tu a l p o te n ti a l or p o te n ti a l p ro du c t as it ma be called
of t h e t wo functions is the sex tuple integral
y
,
.
.
.
.
.
.
y
,
,
”
Pot ( U V)
,
On e
y
fff
.
,
Wd
vl
d v2
.
71
of the integrations ma be performed
ff
P o t V d v,
V ( :c 2 , y, ,
z
Pot
v
U
d
)
,
z
2
7
( )
.
l
In
a Similar manner t h e mutual potential or potentia product
of two vector functions W’ W” is
,
z
l)
°
W
I!
d vl d e g
.
Th is
l
q
y
is a so a scalar uantit
ried out
.
On e
y
integrati on ma be car
VECTOR ANAL YSIS
24 2
yields
O n e integration
M a x Wat v l
M a x (W, V)
8
7
( )
A
rt 9 3
3
5
( )
By
.
4
v
.
.
UPot V=
7r
P
o t V] = ( V
N
w
U
e
[
(V
Ne w U) P o t V
.
V P ot V
Ne w U) P o t V + ( Ne w U) -
o
.
—
=
V
t
V
P
o
N
e w U)
V
(
P
o t V ] + N ew U N e w V
N
w
U
e
[
o
o
Inte grate
U P ot Vdv
+
47
ff
—
fff
V
Ne w U
o
d
v
P
N
t
e
w
o
V
U
]
[
Ne d
v
.
fff
Pot ( U V) =
,
Ne w U Ne d
ff
v
P o t V Ne w U d
_
a
9
7
( )
.
S
surface integral is to be taken over the entire surface S
bo undi ng the region of integration of the volume integral
As t his region of integration is all Space t h e surface S may
be looked upon as the surface of a large sphere of radius R
If the functions U and V va n is h identicall y for all points out
side of certain fi nite limits the surface integral must vanish
Hence
Th e
.
”
,
.
.
,
( my
( 54) Art 9 3
.
4
n
w
”
o
,
Pot
W
’
V x
Lap W
V Ma x
”
W
”
W
Pot W
’
Pot
’
.
TH E I
B
[ L ap W
V
ut
and
NTEGRAL CA L CUL US
”
M
a
x
[
V
W]
’
X Pot
W
”
Pot
W
ap W
W]
W
Ma x W
’
V
”
H ence
V x
L ap W
’
V
”
V Ma x
and
W
Pot W
’
”
M a x W Ma x
H ence substitut ng
W]
’
’
[ Ma x W
V
.
Pot
”
”
,
”
[ L ap W x
Lap W Lap W
Pot W
’
”
W
’
W
Pot W
V Ma x
’
P ot
”
V x Pot
”
24 3
x Lap W
V
’
Pot
L
VECTOR S
OF
,
W]
’
Pot
W
”
’
i
4
7r
Lap W L ap W
[ L ap W x
W Pot W
’
”
’
Ma x
'
”
V
V
M
a
x
[
Pot (W W )
ff
ff
P ot
W
”
W]
’
Pot
Integrating
4
7r
’
”
,
Pot W x La p W
”
’
da —
La p
W
’o
La p
W
M a x W Ma x
’
ff
”
W
dv
”
dv
( 8 0)
Ma x W P o t W d a
’
”
o
.
.
W and W e x ist only in fi ni te s p ace these surface
integrals taken over a large sphere of radius R must vanish
a n d then
If now
4
7:
’
Pot
There
”
fff
W
L ap W
dv
ff
W Ma x W
dv
=
La p
Ma x
’
”
’
”
.
are a number of useful theorems of a function
theo re tic nature whi ch may perhaps be mentioned here owing
VECTOR A NAL YSIS
244
i
l
to their intimate conn ection w t h the integra calcul us of
vecto rs Th e proofs of them will in some instances be given
and in some not Th e theorems are often useful in prac t ical
applications of vector analysis to physics as well as in purel
mathematical work
Th e o re m : If V ( x y z ) be a scalar function of position
in space which possesses in general a de fi nite derivative V V
and if in an portion of space fi n ite or infi nite but necessarily
continuous that derivative vanishes then the fun ct on V s
constant throughout that portion of Space
.
y
.
.
,
,
y
i
,
,
,
i
.
V V= O
.
V
Choose
a
fi x ed
point ( 901
:
cons t
.
) in the region
, y ,, z ,
By
.
2
( )
p ge
a
1 80
VV
y
f
f
dr =
V V-
Hence
.
If V
(x
,
y,
z
z
V a l , ?/p
)
z
1
)
dr = 0
0-
V ( x1 , y 1 ,
V ( v, y , z )
Th e o re m
,
s
const
l)
.
) be a scalar function of position
in sp a ce which possesses in general a de fi ni te derivative V V;
if the divergence of that derivative e xists and is z ero through
out any region of Space } fi nite or infi nite but necessaril
continuous ; and if furthermore the derivati ve V V vanishes
at every point of any fi ni te volume or of any fi nite port ion of
surface in that region or bounding it then the derivative
vanishes throughout all that region and the function V re
duces to a constan t b y the preceding theorem
y
,
m g
.
m
Th e t e r
thro ugho ut a ny re io n of s a c e
us t b e
g
p
bo un da rie s o f t h e re i o n a s we ll a s t h e re gi o n i ts e lf
1
.
re
g
d d as
ar e
in c l u din
g
t he
VE CTOR ANAL YSIS
24 6
an e x tension of the reasoning V is seen to be constant
throughout the enti re region T
The o re m If V ( x y z ) b e a scalar function of position in
space possessing in general a derivative V V and if through
1
out a certain region T of Space fi nite or in fi nite continuous
or di scontinuous the divergence V V V of that derivative
e xists and is z ero and if furthermore the function V p o ss e s s e s
a constant value 0 in a ll the surfaces b ounding the re gion
and V ( x y z ) approaches 0 as a limit when the point ( cc y z )
recedes to infi nit then throughout the entire region T the
func t ion V has the same constant value c and the derivative
V V vanishes
Th e proof does not di ffer essentially from the one given
in the case of the last theorem Th e theorem ma be gen
e ra liz e d as follows
Th e o rem : If V ( x y z ) be any scalar function of position
in space possessin g in general a derivative V V; if U ( x y z )
b e any other scalar function of position which is either posi
tive or negative throughout and upo n the b oundaries of a
region T fi nite or in fi nite continuous or discontinuous ; if
t h e divergence V [ U V V of the product of U and V V
]
ex ists and is z ero throughout and upon the b oundaries of T
and at in fi nity ; and if fur t hermore V b e constant and equal
to c upon a ll the b oundaries of T and a t infi ni t y ; then the
function V is constant throughout the entire reg on T and
is equal to c
Th e o re m : If V ( x y z ) be any scalar function of position
in Space possessing in general a derivative V V; if through
out any region T o f space fi nite or infi nite continuous or
dis con t inuous the di vergence V V V of this derivati ve e x ists
and is z ero ; and if in a l l the b ounding surfaces of the region
T t h e normal component of the derivative V V va n is h e s and
at in fin ite distances in T ( if such there be ) the p roduct
.
,
,
,
,
o
,
,
,
y
,
,
,
,
.
y
.
,
,
,
,
,
i
.
,
,
,
,
,
1
Th e
re
gi
on
i n c l u de s
it s
bo un da ri e s
.
,
NTEGRAL CAL C UL US
TH E I
OF
VECTOR S
247
vanishes where r denotes the distance measured
from any fi x ed o ri gin ; then throughout the entire region T
the derivative V V va n is h e s and in each continuous portion
of T V is constant although for difi e re n t continuous portio n s
this constant may not b e the same
This theorem may b e generali z ed as the preceding one
wa s b the substitution of the relation V
U V V)
O for
r
2
9 V/ 9
r
,
,
.
y
VV V=
Oa nd
Ur
2
9 V/9
r
= 0 fo r
r
2
9 V/9
r
= 0
.
corollaries of the foregoing th eorems t h e following
statements ma be made Th e language is not so precise
as in the theorems themselves but will pe rh aps be under
stood when they are borne in mind
If V U = V V then U and V di fi e r at most b
a
constant
f V U = V V in any fi n ite
If V V U = V V V and
port ion of surface S then V U V V at all points and U
difi e rs from V only b y a constant at most
V V and if U
V in a ll the bounding
If V V U= V surfaces of the region and at in fi n ity (if the region e x tend
thereto ) then at all points U and V a rc equal
V V V and if in all the bounding surfaces
If V V U
of the region the normal components of V U and V V are
2 9
equal and if at infi nite d stances r ( U 9 r — 9 V/9 r) is
z ero then V U and V V are equal at all points of the region
and U difi e rs from V only b y a constant
’
”
Th e o re m If W and W are two vector functions of position
in Space which in general possess curls and divergences ; if
for any region T fi nite or in fi n i te but necessarily conti nuous
’
the curl of W is e ual to the curl of W and the di vergence
’
of W is equal to the divergence of W ; and f moreover
”
the two functions W and W are equal to each other at
every point of any fi nite volume in T or of any fi n i te surface
’
”
in T or bounding it ; then W s e ual to W at ever point
o f the reg on T
As
y
.
,
y
.
,
i
.
o
o
,
.
:
.
,
i
,
.
,
q
,
”
”
i
’
i
i q
.
y
24 8
xW
VECTOR A NAL YSIS
A vec
W V x (W
1
h
a
t
e
tor function whose curl vanishes is equ l to
derivative
—
=
V
V
W W
of a scalar function V ( page
et
Then V V V O owing to the equalit of the divergences
Th e theorem therefore b ecomes a corollar of a precedi ng one
Th eo re m IfW and W are two vector functions of pos i
ti on which in general possess de fin ite curls and divergences
2
if throughout any p e rip h ra c tic region T fi nite b u t not
of
n ecessarily continuous the curl of W is equal to the c u r
W and the divergence of W is equal to the divergence of
W ; and if furthermore in a ll t h e b oundin g surfaces of the
region T the t a nge n ti l components W and W are e ual
then W is equal to W throughout the a p e rip h ra c t ic region T
Th e o re m : If W and W are two vector functions of posi
tion in space which in general possess de fi nite cur s and
divergences ; f throughout any a c yc lic region T fi nite bu t not
necessarily continuous the curl of W is equal to the curl
’
W and the di vergence of W is e ual to the divergence of
W ; and if in a ll the b ounding surfaces of the region T the
d l components of W and W are equal ;
then the fu nc
tions W and W are equal throughout the region acyclic T
Th e proofs of these two theorems are carried out by me a ns
of the device suggested before
The o re m : If W and W are two vector functions such
that V V W and V V W h ave in general de fi nite values
in a certain region T fi nite or in fi ni te contin uous or discon
ti n u o us ; and if in
ll the bo unding s u rfaces of the region
and at infi nity the functions W and W are e u al ; then W
is equal to W throughout the en t r e region T
Th e proof is given by t e a tin g separatel t h e th e e c o m
p o n e n t s of W and W
Since V
’
= V x
’
”
,
L
y
y
’
”
.
.
.
”
’
a
l
,
’
,
’
”
”
’
a
’
q
”
”
.
’
”
l
i
,
’
q
,
”
”
’
”
”
’
.
.
’
”
’
”
,
,
q
a
’
”
r
’
”
i
y
’
r
.
”
.
1
Th e
A
ti c
.
re
If it
ggi i
re
on
on
l
Tm
wh i c h
ay
h a ve t o b e
gi
e n c l o se s
e n c o se s n o re
on
wi t h
it is
mi d f y li by gi i
a
e ac
n i ts e
l
a p e rip hra c
c
c
the
a n o t h e r re
t ic
.
i
n se rt o n o f
on
is
sa
id t o
dia ph ra
gm
s
.
b e p e rip h ra c
VECTOR A NAL YSIS
250
fff
—
ff
8
d c + Z d wde l.
d
e + d
d
X
y
[
if X Y Z be the three components of the vector f u nction W
Stokes s Theorem : Th e surface integral of the curl of a
vec t or func t ion taken over any surface is equ al to the line
integral of the function taken around the line boun d ng the
surface An d convers ely if the surface inte gra l of a vector
func t ion U taken over any surface is equal to the lin e integral
of a function W taken around the boundar then U is the
curl of W
.
,
,
’
i
.
y
,
.
W
V X
a nd
if
ff
U
-d a
S
s
‘
da
W
t
dl
'
,
( 1 2)
0
Application
of the theorem of Stokes to deducing the
equations of the electro -magnetic fi eld from two e xperimental
facts due to Faraday Application of the theorems of Stokes
and G auss to the proof that the divergence of the curl of
a vector func t io n is z ero and the curl of the derivative of
a scalar function is z ero
Formul a analogous to integration b parts
.
y
.
q
ff
V
S
ff
s
u
X V
Vu v
v
v
o
da
u
v
-d r —
ff
u
-d r
q
,
v
S
da
q
-d r
—
;
j
vV u
d r ( 1 6)
,
VECTOR A NA L YSIS
25 2
V
-V P o t
P ot V
-P o t W = P o t
V V
-V W
31
( )
-W
3
2
( )
VV
,
,
3
3
( )
Th e integrating
operator Pot and the differentiating operator
V are commutative
Th e thr ee additional integrating operators known as the
N e wto n i a n the La p la c i a n and the M a xwe lli a n
.
.
,
,
Ne w V
Ma x W
If the
i
rl
i
potent al e xist s these integ a s are related to t as fol
lows :
V
V
Pot
V:
N e w V,
W = La p W;
P o t W = Ma x W
x Pot
V
.
interpretation of the physical mean ing of the N ewtonian
on the assumption that V is the density of an attra cting
b ody of the aplacian on the assumption that W is electric
flu x of the Max welli an on the assumption that W s the
intens it of magnetiz at on Th e e x pression of these integra s
or their components in terms of ac y z formul a
and
Th e
L
,
,
y
i
i
.
,
V
o
,
N e w V = M a x V V,
V Ma x
W = Ne w V -W
,
l
VE C TOR A NAL YSIS
2 54
1
4W
Ne w V = V
Ma x V V = V
.
4
6
( )
P ot V
VWZ
P ot V
P ot V
—
-V V
V
—
VW
—
V
Pot
1
-V
V
V
4
W2
=
W2 ( 65)
.
P ot V
7r
1
-V -— - P o t W = W
;r
1
£
.
( 6 7)
VV
4
4
7r
7:
W2
Pot V
Pot W
M u tu a l pote ntials N
-W2
( 6 8)
M a x Ne w V
( 6 9)
= VV
L ap L ap W
W ( )
ewtonians L aplacians and Ma well an
Ne w M a x
x
,
,
70
.
i
a
may b e formed They are sex tuple integrals Th e integra
tions cann ot all b e performed immedi ately ; b u t t h e fi rst three
may be Formul a ( 7 1 ) to ( 8 0) inclusive deal wi th these inte
grals Th e chapter closes with the enunciation of a number
of t heorems of a function -theoretic nature B y mean s of
these theorems certain facts concernin g functi ons may be
inferred from the conditions that satis f aplace s equati on
and have certain b oundary conditions
A mong the e x ercises numb er 6 is worthy of especial atten
tion Th e wo rk do n e in the te xt h a s fo r th e m o s t p a rt a s s u m e d
.
.
.
.
.
yL
’
.
.
t h a t th e p o te n tia l
N e wto n ia
ti a l do e s
n s,
e xis ts
La p l a c i a
n o t e xis t
.
.
B
ut
n s, a nd
Th ese
ma
f
ny o
M a xwe lli a
th e
ns
o r m u l ae c o n n e c tin g
f
h o ld whe n th e p o te n
are taken up in Ex ercise 6 referred to
.
NTE GRA L CAL CUL US
TH E I
EX ER CISE S
OF
VE C TOR S
55
CHA PTER IV
ON
is a Scalar function of position in space the line
If V
integral
q
is a vector uantity
Vd
Show
.
that
Vd
r
d
a
a
>
<VV
.
That
is the line integral of a scalar fun ction around a
closed curve is equal to the s ke w surface integral of the deriv
ative of t h e function taken over any surface spanned into
the contour of the curve Show further tha t if V is constant
the in t egra l around any closed curve is z ero and conversely
if the integral around any closed curve is z ero the function V
is constant
Hint : Instead of treating the integral as it sta nds m u ltiply
it ( with a dot) by an arb itrary constant un it vector and thus
reduce it to the line integral of a vector function
2 If W s a vector function the lin e integral
.
.
i
.
.
d
q
r
is a vector uantity It may be called the s ke w line integra l
of the function W If e is any constant vector Sho w that if
the inte gral b e taken aro u nd a closed curve
.
,
.
( V
c
Th e
1
we ll
’
s c he
fi
W—
fo u r e xe rc is e s a re
Th e o ri e de r El e c t ric ita t
rs t
ta
c
-V W) -d a
k f ym
en
ro
wh e re t h e
P o p pl
a re
’
d
kd
wo r
r,
Ein fu h run g in die M a x
s
e
o ut
.
VECTOR A N ALYSIS
2 56
and
H
o
=
c
c
V
o
Wd a
W [c
e
(W
V
o
d
a
)
o
case the integral is taken over a plane curve and the
surface S is t h e portion of p ane included b y the curve
l
In
[
VWd a
—
V
(W
i
Show
that the integral taken over a plane curve van shes
when W is constant and conversely if the integral over an
plane curve vanishes W must be constant
3 Th e s u rface integral of a scalar function V is
y
.
.
Vd a
is is a vector quantity
.
that the surface integral
of V taken over any closed surface is e ual to the volume
integral of V V taken throughout the volume boun ed b
that surface That is
Th
.
Show
q
d y
.
V Vdv
.
r
Hence conclude that the surface integral over a closed su
fac e vanishes if V b e constant and conversely if the surfac e
integral over any closed surface vanishes the function V must
be constant
4 IfW be a vector function the surface integral
.
.
,
da >
<w
may be called the s ke w surface integral It is a vector
quantity Show tha t the Skew surface integral of a vector
.
.
VECTOR A NAL YSIS
2 58
By
e xercis e ( 3)
V2
( p 1 2 V)
d v2
p
Vda
12
.
can be Shown that if V is such a function that N e w V
e xis t s then this surface integral taken over a large Sphere of
ra di us R and a small sphere of radius R approaches z ero
when R b ecomes inde fi n itel great ; and R inde fi nitel
small Hence
It
,
’
y
y
’
,
.
P
Pot
Ne w V :
12
V2 V d vz r
VV
Prove in a Sim ilar manner that
W= P o t
Ma x
By
means of
-N ew
V Ma x
fff
=
p
V Ma s
enc
e
Hence
7
.
An
W
.
r
( 8 7) it is possible to p ove
V
H
V
La p
W—
V >
< La p
12
V = Ma x V V,
W=
VV
o
Ne w V
W d vz
fff
p
V Ma x
W—
integral us ed by
—
.
fff
pl
Vd
l
W
—
V Ma x
H elmholt
r
W
fff
p
W= 4
z
12
z
7r
is
V d vz ,
12
W
.
o
Vd
z
.
V V W d vz
°
’
S
THE INTE GRAL CAL CUL U OF VECTOR
S
2 59
or if W be a vector function
7
Show
l
that the integra converges if
Vr
5
y
VH
H
( W)
V XH
( W)
=H
(
o
V) = H ( V
VV H
o
(V
V
(
.
W)
=E
(
= La p
Z
T
(
Z
7
,
W)
2
,
W)
,
V V) = 2 P o t V
(V
Pot Pot
H
V)
Ma x ( r
W)
V V) = M a x
( W)
so rapi dl y
V dim in is h e s
=
=
N
H
e
w
V
V
V
)
(
( )
= H
V
V VE
'
9
0
( )
.
K
when r becomes inde fi n i tel great
o
12
W d vz
V
Pot Pot W
( W)
9
( 7)
.
9
8
( )
a proof of G auss s Theorem which does not depe nd
upon the physical interpreta tion of a func t ion as the flu x of a
fluid Th e reasoning is similar to that emplo ed in Art 51
and in the fi rst proof of Stokes s Theorem
Show tha t t h e division of W into two parts page 2 3 5
9
s unique
1 0 Treat in a manner analogous to that upo n page 2 2 0
the case in which V has curves of s continuit es
8
G ive
.
’
y
.
.
’
.
i
,
.
,
.
.
,
di
i
,
.
H
C
AP TER V
LINE AR VE CTOR FU N CTION S
AFTER the
de fi ni tions of products h a d be en laid down
and applie d two paths of advance were open O n e was
differential and integral calcul us ; the other higher algebra
in t h e sens e of the theory of linear homogeneous substitutions
Th e treatment of the fi rst of these topics led to new ideas
and new symb ols to the derivative dive rgence c u rl scalar
and vector potential that is to V V V x and P o t w th the
au xiliaries the Newt onian t h e aplacian and the Maxwell ian
Th e treatment of the second topic will likewise introduce
novel t y b oth in concept and in notation the linear vector
function the dyad and the d adic with their appropriate
symboliz ation
Th e simplest e x ample of a linear vector function is the
’
product of a scal ar constant and a vector Th e vector r
.
,
,
.
,
,
,
L
,
,
,
,
,
,
i
,
.
,
y
,
.
.
1
( )
is a linear function of
more g neral linear funct on
may be obtained by consideri g the components of r indi vi d
Let jk be a system of a es
components of
A
r
.
i
e
n
ua
ll y
.
i,
x
,
r a re
i
-r
j
o
,
k
r,
Th e
.
-r
.
L et each of these be mul iplied by a scalar constant which
t
may be different for the di fferent components
0
1
i
0
r,
C
z
j
'
r’
08
k
.
ro
.
2 62
fi
VE C TOR A NA L YSIS
vector r is said to be a linear vector fun c
tion of another vector r when the components of 1 along
three non-c oplanar vectors are e xpressible lin early with scalar
coeffi cients in terms of the components of r along those same
vecto rs
De
'
A
n i tio n :
.
"
.
yb
se a
a
’
z
z c
w
l
where [ a b c ]
,
bl y
ba y
e
x
ba y
c
a
g
z ,
l
e
“2 x
0,
z ,
z
a
z
,
then r is a linear function of r ( Th e constants a l bl c l
etc have no connec t ion with the components of a b 0 par
allel to i k ) A nother de fi nition however is found to b e
more convenient and from it the foregoing may be deduced
De n i ti o n : A continuous vector func t ion of a vector is
said to b e a lin e a r vector function when the function of the
sum of any two vectors is the sum of the functions of thos e
vec t ors That is the function f is linear if
’
.
.
,
,
j
,
,
,
,
,
,
.
fi
.
,
.
f(
r
r
1
2
)
fOi )
4
( )
'
:
be any positive or negative scalar and if f
be a linear function then the function of a times r is a times
the fun c t ion of r
Th e o re m
If a
,
.
f Ca r)
An d
a
hence
f< 1 r l
a
a
a
z
r
m)
“
z
r
f
( 2)+
z
a
r
a
e
3
.
‘
l
‘
f( r3 )
Th e
proof of this theorem which appears more or less
obvious is a t rifle long It depends upon making repeate d
use of rela t ion
.
f<
r
)
r
=f
( )
r
+
f ( r)
me
.
L INEA R VE C TOR FUNC TI ONS
Hence
In like
manner
f ( 2 r)
2 f ( r)
f ( n r)
n
.
f ( r)
where n is any positive integer
et m be any other positive integer
jus t obtained
L
.
f( )
r
Hence
=f
2 63
Then
.
y
b
relation
m
n
n
r
f( )
.
That is ,
equation ( 5) has b een proved in case the constant
is a rational positive num b er
To show the relation for negative numbe s note that
r
.
=
0
f( ) f( 0
0) = 2 f
‘
i d»
—
=
0
r
f( ) f <
)
r
o
.
—
=
r
>
> f ( ) + f(
= f<
r
f <>
—
r
a
— r
—
fc e
)
r
.
prove ( 5 ) for incommensurab le values of the cons tant
it becomes necessary to make use of the continuity of the
a
function f That is
To
,
.
f(
L et
=
) f
(
a? r
)
90 r
approach the incommensurable n umb e r a
through a suite of commensurab le values Then
a:
.
f(
H ence
)
x r
r
w
f( )
a
f( )
r
passing
VECTOR A NAL YSIS
264
(
a: r
r
f( )
Hence
)
a
a
f ( r)
which proves the theorem
Th e ore m : A linear vector function f ( r ) s e ntire
mined when its value for three non -coplan ar vec t o s a
known
i
.
Let
Since
ly det r
r are
e
,
.
b, c
.
l = f ( a ),
r
m
f
I
=f
.
b
( )’
is any vector whats oever it ma
,
=
r
x a
+ yb +
y
presse
be
ex
.
e xp re s s e
d
i
a particular case of a l inear funct on
as
i
o
l
r c
r
i jz j
fk os k
i
as
z a
Hence
In Art 9 7
d
o
wa s
r
.
r
the sake of brevity and to save repeati ng the vector
whi ch occurs in each of these terms in the same wa th s
may b e written in the symbo l ic form
Fo r
In
b
y i
like manner if a 2
be any given vectors and b v b 2
another set equal in numb er the e xpre s sion
,
,
,
,
r
'
a
l
b
l
o
r
+
a
2
bé
o
r
+
a
3
bg
o
r
+
u
( 6)
o
is a linear vector function of r ; for owing to the distrib ut ive
c haracter of the scalar product this function of
satis fi es
Fo r the sake of brevity r may be written s m
relation
b o lic a lly in the for m
r
’
r
'
y
VE C TOR A NAL YSIS
266
multipli ed into r by direct or scalar multiplication Th e
order of t h e factors O and r is important Th e direct
product of r into O is
.
.
r
O=
r
(
o
a
b1 +
1
_r a
b1 +
1
a
r
y the vectors r
2
b2
b
aa
-a 2 b 2
+
r
o
a
a
b
3
9
( )
+
3
are in general difle re n t
M
De n i ti o n : When the dyadi c O is multiplied in to r a s O r
When r is multiplied in O as
O is said to b e a p refa c t o r t o r
r
O O is said to be a p o s tfa c to r to r
Evidentl
fi
Q;
Q
Q
.
,
.
,
.
A dy a dic O u s e d e i th e r a s a p refa c to r o r a s
ve c to r r de t e rm in e s a lin e a r ve c to r fu n c ti o n of r
a
p
t
”
a
/ c to r
os
to
a
two li near
vector functions thus ob t ained are in general different from
one another They are called c o nju ga te linear vector func
tions Th e t wo dyadics
.
Th e
.
.
¢ =
a
b1 +
1
a
z
bz
'
l
a
'
g
bg
each of which may b e ob tained from the other by inter
changing the an t ecedents and consequents are called c o nju
Th e fac t tha t one dyadi c is the conjugate of
ga te dy a dic s
another is denoted by affix ing a sub script C to either
,
.
’
.
r
dyadic used as a postfactor gives the sam e
esult as its conjugate used as a prefactor That is
Th e o re m : A
.
fi
De
be
e
n i tio n :
wh e n
O
r
s
o
O
r
T
r
O
r
T
r
z
s
o
$0
o
An y
qu a l
when
or when
or
t
T
y
0
( 9)
12
two d a di cs
O
and
for all values of
for all values of
o
r
fo r a ll
T
are said to
r
r
,
,
va l u e s o f s a n d r
.
L INEA R VECTOR FUNCTI ONS
2 67
third relation is equivalent to the fi rst Fo r if the
vectors O -r and T r are equal the scalar products of any
vector 3 in t o t hem must b e equal A n d conve rs ely if the
scalar product of any and e ve ry vector 8 into the vectors O r
and T r are equal then those vectors must be equal In
like manner it may be shown that the thi rd relation is e q u iva
lent to the second Hence all three are equivalent
Th e o re m : A d adic O is completely determined when t h e
values
Th e
,
.
,
.
o
,
.
y
.
.
O
O
a,
y
b,
$4
,
where a b c are an three non-c oplanar vectors are known
This follows immediately fr om the f a ct that a dyadic de fi n es
a linear vector function If
r
a: a
zc
yb
Or =
Oc
,
,
,
.
.
,
,
y
consequently two d adi cs O and T are equ a l provided equa
tions ( 1 0) hold for th re e non -coplanar vectors and th re e
non-coplanar vectors 8
Th e o re m A n y l inear vector function f may b e represented
by a dya dic O to be used as a prefactor and b y a dyadic T
w hi ch is the conjugate of O to b e used as a postfactor
Th e linear vector function is completely determin ed when
its values for thr ee non-coplanar vectors ( sa i k) are
known ( page
et
r
.
,
.
,
y j
L
= b.
f O)
'
f ( i)
Then
,
a,
k
f( )
c
,
.
the l inear function f is equivalent to the dyadic
O
a
i
b
j
c
k,
to be used as a postfactor and to the dyadi c
T:
to be us ed as a prefactor
fo )
r
.
r
r
T
O
2 68
VECTOR A NAL YSIS
y
i
stud of linear vector functions therefore s identical
with the study of dyadics
De n i tio n A dyad a b is said to be mu ti plied b y a scalar
a
when the antecedent or the consequent is multipli ed b y
that scalar or when a is distrib uted in any manner between
the antecedent and the consequent If a a a
Th e
fi
l
.
,
’
"
.
(
a
a
(
b)
a a
)b
a
(a
b)
(a
'
a
) (a
”
b)
.
dyadic O is said to be multiplied b y the scalar
each of its dyads is multiplied b that scalar Th e
is written
a O
or O a
A
y
.
.
dyadic a O appl ied to a vector r either as a prefactor or
as a postfactor yields a vec t or equal to a times the vector
obtained b appl ing O to r th at is
Th e
y y
(
Th e o re m
utive
.
:
Th e
a
o
r
a
(
Or)
.
y i
combination of vecto rs in a d ad s
dis t rib
Th a t is
(a
and
This
O)
+ b)
c
a c
+ b
y
c
d
q y
follows immediatel from the e fi nition of e ualit of
dyadi cs
Fo r
(
a c
+ b
c
)
o
r
and
a
b
o
r
+
a c
o
r
Hence it follows that a dyad whi ch consists of two factors
each of which is the sum of a numb er of vecto s may be
multiplied out accordi n g to the la w of ordi nar algeb ra
-e x cept t hat th e o rde r o
f th e fa c to rs i n the dy a ds m u s t be
r
m a i n t a in e d
.
,
y
,
3
form the scalar coeffi cients of the correspondin g
e ual
If the coeffi cients be equal then obviously
O r = T r
q
,
da
ds be
.
,
for any value of r and the dyadi cs b y ( 1 0) must b e aq u a l
Conversely if t h e dyadics O and T are e ual t hen b ( 0)
q
,
s
-O
o
L
s
r
T
o
y
,
r
o
for all va lues of s and r
et s and r each take on the
Then
i jk
=
O j i TO
ii
k
i
j
.
,
,
v lu e s
.
z
o
O
i
j
k
,
z
T
i,
j
z
k
-O i
o
o
o
o
,
o
O
k
j
.
o
z
T k
j T
k
Ti,
'
these quantities are precisely the nine coe ffi cients in the
e xpansion of the dyadi cs O and T Hence the correspon din g
1
coeffi cients are equal and the theorem is proved
Th u
anal tic s t at ement of the equality of t wo dyadics can some
t imes b e used to greater advantage than the more fundamental
de fi nition ( 1 0) b ased upon the conception of the dyadic as
de fi ning a linear vector function
The o re m A dyadi c O may be e x pressed as the sum of nine
dya ds o f which the antecedents are an three given non
coplanar vectors a b c and the conse uents any three given
non -00planar vectors 1 m 11
E very antecedent may be e xpressed in te rms of a b c ;
and every consequent in terms of 1 m n Th e dyadic ma
then be reduced to the form
B
ut
.
y
.
.
,
q
,
,
,
.
,
,
O=
a
a
a
1
As
a c o ro
de pe n de n t
.
No n e
o ft h e
11
21
,
1 +
a
bl +
a
31 +
a
a
m
m xp
lla ry o f t h e
y
31
C
t h e o re
ma y be
a
b m +
a
0m
a
82
+
ev
,
12)
a re
.
y
n
a
la
23
38
b
n
o n
.
m
id e n t t h a t t h e n i n e dya ds
re ss e d li n e a rly in t e r s o f t h e
it is
e
m +
e
lz
gz
,
,
o t h e rs .
ih
L INEAR VECTOR FUNCTI ONS
2 71
This
e xpressi on of O is more general than that given in
It reduces to t ha t e x pression when each set of vectors
a b c and l m n coincides with i k
Th e o re m : An y dyadic O may b e reduced to the sum of
th re e dyads of which either t h e an t ecedents or the consequents
but not b oth may b e arb i t rarily chosen provided they be non
coplanar
et it be required to e xpress O as the sum of three dyads
of which a b c are the consequents
et 1 m n be any o t her
three non -coplanar vectors O ma then be expressed as in
Hence
,
,
,
,
,
j
.
,
,
,
L
.
,
L
y
.
,
.
c
(
a
,
31
1 +
,
a
az
m +
“
32
0
1
,
( 1 6)
like manner if i t be required to e xpress O as the sum of
three dyads of whi ch the three non-COp l a n a r vectors 1 m n are
the conse uent s
In
q
,
O
where
L
Mm
Ll
a
M=
a
N =
a
ll
12
ls
a
a
a
+
a
a
+
a
N
n,
b
a
c
,
81
gg
b +
a
0
,
32
23
b +
a
m
21
,
33
e x pression s
for O are u n iqu e Two equal
dyadics which have the same three non -coplanar ante
cedent s a b 0 have the same consequents A B C
these
however need not b e non-coplanar A n d two equal dyadics
which have the same three non -00planar conse uents 1 m n
have t h e same three antecedents
1 02 ] De n i tio n : Th e symb olic product fo rmed by the jux ta
position o i two vectors a b without the intervention of a dot
or a cross is calle the in de te rmin a te p ro du c t of the two vecto s
9 and b
Th e
.
,
,
,
,
,
.
fi
.
q
,
,
,
.
d
.
,
,
r
27 2
VECTOR A NA L YSIS
reason for the term indeterminate is this
two
products a h and a x b have defi nite meanings O n e is a
cer t ain scalar the other a certain vector O n the other hand
the product a b is neither vector nor scalar — it is purely
sym b olic and acquires a determinate physical meanin g onl
Th e product a b does not ob ey
when used as an operator
the commutative law It does however obey the distrib utive
law ( 1 1 ) and the associative law as far as scalar mul tiplication
is concerned (Art
T he o re m Th e indeterminate product a b of two vectors is
the most general product in whi ch scalar multiplication is
associat ive
Th e most general product conceivable ought to have the
property that when the product is known the two factors a re
also known Certa inl y no product could be more general
Inasmuch as scalar multiplicati on is to be associative that s
Th e
o
.
Th e
.
.
,
y
.
.
.
i
.
,
a
(
a
b)
(
a
a
)
b
(
a
a
b)
(
d
’
a
.
)
it will be impossible to completely determine the vectors a
and b when their product a b is given An y scalar factor
may be transferred from one vector to the other Apart from
thi s possib le transference of a scalar factor the vectors com
posing the product are kn own when the product is known In
other words
Th e o re m : If the two indetermi nate products a b and a b
are equal the vectors a and a b and b must be collinear and
the product of the lengths of a and b ( taking into account the
positive or negative sign accordi ng as a and b have re s p e c
t ive l y equal or opposite directions to a and b
s
e
ua
to
t
h
e
)
product of the lengths of a and b
.
.
,
.
’
’
,
’
,
’
’
’
.
’
i q l
’
VECTOR A NAL YSIS
27 4
indetermin ate product a b imposes five conditions upo n
the vectors a and b Th e directions of a and b are fi x ed and
lik ewise the product of their lengths Th e scalar product
a
b b eing a scalar quan t i t y imposes only one condition upon
b b eing a vector quantity
Th e vector product a
a and b
imposes three conditions Th e normal to the plane of a and
b is fi x ed and also the area of the parallelogram of which they
are the side Th e nine indeterminate products ( 1 2 ) of i 1:
into themselves are independent Th e nine scalar products
are not independent O nly two of them are different
Th e
.
.
x
,
,
.
,
,
.
,
.
j
,
.
.
.
i
-i
z
k
jj
o
z
o
k
z
l,
and
Th e
n ine vector products are not in dependent either ; for
and
jx k
i xj
k xi
two products a b and a x b ob tained respectively from
the inde t erm inate product b y insert ing a dot and a cross b e
tween the factors are functions of the indeterminate product
That is t o say when a b is given a b and a x b are determined
Fo r t hese products depend solely upon the directions of a and b
and upon the produ ct of the length of a and b all of which
are known when a b is known That is
Th e
.
,
,
.
,
.
if
a
b
z
a
’
b
’
,
a
o
h
1
( 7)
x
does not hold conversely that if a b and a b are known
a b is fi xed ; for taken toge t her a b and a x b impose upon t h e
vectors only four condi t ions whereas a b impos es fi ve Hence
a b appears not only as t h e mo s t general product but as the
mos t fundamental product Th e others are merely functions
of it Their functional nature is brought out clearl b y the
notation of the dot and the cross
It
,
.
y
.
.
.
fi
L INEAR VE CTOR F UNCTI ONS
2 75
scalar known as t he s c a la r of O may be o b
t a in e d b y inser t ing a dot b et ween the antecedent and c o n s e
quent of each dyad in a dya di c Thi s scalar will be denoted
I
b y a sub script S attached to O
De
A
n itio n :
.
.
If
( 1 8)
like manner a vector known as th e ve c to r of O may be
o b tain ed b y inserting a cross bet ween t h e antecedent and con
sequent of eac h dyad in O This vector will be denoted b y
attac hing a sub script cross to O
In
.
.
¢x
If O
3
1
X b
+
1
3
2
X b2 + 3
X b3 + 0
8
be e xpan ded in nonion form in term s of i
,
j
k,
,
2
0
( )
“
)
23
i +
(
4
a
a
)
j
(
13
-O j+
k
o
Ox =
(j
o
O k
+ (i
q
O
k
o
j
)
i +
k
(
o
a
o
—
-O O
i)
j j
( 21 )
Ok,
O i
o
)
k
k
—
i
c
O k)
j
.
e uations ( 2 0) and ( 2 1 ) the scalar and vector of O are
e x pressed in terms of the coe ffi cients of O wh en ex panded
in the nonion form Hence if O and T are two equal
dyadi cs the scalar of O is equal to the scalar of T and the
vector of O is equal to the vector of T
In
.
,
.
O
If
T,
From
OE
:
and
T5
Ox
TX
( 2 2)
.
thi s it appears that O and O are functions of O
uniquely determined when O is given They may s ometimes
be ob t ained more conveniently from ( 2 0) and (2 1 ) than from
and sometimes n o t
( 1 8 ) and
,
)(
.
1
a nd
A
su
fre e
m
g
m
m
.
b s c ript do t i h t b e us e d fo r t h e s c a l a r o fQ if it we re
fro li a bilit y t o i s i n t e rp re ta t io n
.
i
s uf
fi c e nt
ly dis t i n c t
VE C TOR A NAL YSIS
276
a dic s
D
f y
P ro du c ts
o
giving the de fi ni t ions and proving the theore ms
concerning products of dyadics the dyad is made the unde r
lying p rinciple What is true for the dyad is true for the
dyadic in general owing to _the fact that dya s a n d dyadics
ob ey the distrib utive law of mul t iplication
Th e dire c t p ro duc t of the dy a d a b into the
De n i ti o n
dy a d c d is written
( a b ) ( c d)
and is by de fi nition equal to the dyad ( b c ) a ll
In
,
d
.
fi
.
:
(
That
a
b)
a
t
c
l
2
( 3)
is the antecedent of the fi rst a n d the consequent of the
second dyad are taken for the antecedent and consequent
respectively of the product and the whole is multiplied b y
the scalar product of the consequent of the fi rst and the
antecedent of the second
Thus the two vectors which stand together in the product
,
.
( a b ) (c d)
are multiplied as they stand Th e other two are left to form
a new dyad Th e direct product of two dya dic s may be
de fi ned as the formal e x pansion ( according to the dis trib utive
law) of the product into a sum of produc t s of dyads Thus
1f
.
.
.
u
T=
(
0
1
d1 +
c
2
d2 +
c a
d3 +
o
u
)
)
O
(o l
1
_
a
+
a
+
a
Th e
1
b1
2
b2
a
b3
d1
c
l
d1 +
a
°1
d1 +
a
-c l
d1 +
a
°
p a re n t h e s e s
ma y be
b
1
l
b
2
2
3
o
o
'
c
c
z
d2 +
2
d2
a
1
b
1
-c s d3
+ u
b3
mi
tte
d in
e a c h o ft h e s e
t h re e e
xp
i
re s s o n s .
VECTOR ANAL YSIS
278
an
O
d
T
( +
o
T+ O
T
’
l
Hence in genera the product
W4
9
4
( +
a
.
rr
f
r
q
+
m
+ q +
i
may be e xpa nded formally according to the d stributive la w
Th e o re m Th e product of th ree dyad cs O T Q is associa
tive Th a t is
i
.
,
,
.
q ly
y
and conse uent
parentheses as
( 26)
i
either product ma be wr tten
(D
,
‘
g
p
g
.
without
,
proof consis t s in the demonstration of the theorem for
three dya ds a b e d e f taken respective from the th e e
d adi cs O T Q
=
=
a b
d
c
e
f
d
d
i
b
a
c
a
e
f
e
b
o
)
(
(
)
)
(
)
(
a h c
d
a
f
d
b
e
e
b
t
d
e
c) a f
e
(
)
)
(
(
(
)
Th e
y
,
,
r
.
:
o
Th e
ly
,
,
o
:
y
o
o
z
o
o
o
.
r
proof ma als o be given by conside ing
as o p e ra t o rs
r
{ < T>
O, T,
-r >Q
< o
Q
o
r
r
’
T)
r = (OT
{(
O
o
r
’
r
”
,
T) -Q §
.
-r =
(
T
o
Q) r =
T
OT
[
d
(
{
.
values of r
.
Conse
(O
.
quently
r]
T
o
r
O
o
’
,
=
r
”
r
.
.
an
d
Q
L INEA R VECTOR FUN C TI ONS
27 9
y
theorem ma be e x tended b y mathematical induction
to the case of any numb er of dyadics Th e direct product
of any num b er of dyadics is associative P arentheses may
b e inserted or omitted at pleasure without altering the resul t
It wa s shown above ( 2 4 ) t hat
Th e
.
.
.
Hence the product of two dyadi cs and a vector is associative
Th e theorem is true in c a se the vector precedes the dyadics
and als o when the number of dyadics is greater than two
B u t the theorem is u n tru e when the vector occurs b e tween
the dyadics Th e produc t of a dya d c a vector and another
dya di c is not associative
.
.
i
.
,
,
.
L et
a
b
y
be a d ad of
(
Henc e
O
r)
(
O,
T¢ O
b
r)
a
h
=
r
d
c
(
) ab
(
b
o
( r T)
.
and e d a dyad of
a
a
o
r) -c d
a
b
T
.
d
r
c
(
)
.
results of this article may be summed up as follows
Th e o re m : Th e direct product of any number of dyadics
or of any number of dya di cs with a vector factor at ei t her
end or at both ends obeys the distributive and associative
a ws of multiplication parentheses may be inserted or
omitted at pleasure B u t the direct product of any numb er
of dyadics with a vector factor at some other position than at
either end is not associative — parentheses are necessary to
give the expression a de fi nite meaning
ater it will b e seen that by making use of the conjugate
dyadi cs a vector factor whi ch occurs between other dyadics
may be placed at the end and hence the product ma b e
ma de to as sume a form in which it s associat ve
Th e
l
.
L
.
i
i
y
.
2 80
fi
De
VECTOR ANA L YSIS
Th e
n i tio n :
a vecto r r and of a vector r
respectively b the equations
y
.
(
r
d
b)
a
x
yd i
products of a d a a b nto
into a dyad a b are defi n ed
s ke w
a
r
b
(
x r)
,
(r x a ) b
x (a b)
.
skew pro uct of a dyad and a vector at either end
dyad Th e ob vious e x tension to dyadics is
Th e
is
a
.
x
r
i
direct product of any number of dyad cs
multiplied at either end or at b oth ends by a vector whether
the multiplication be pe rformed with a cros s or a dot is
associative B u t in case the vector occurs at any other
position than the end the product is not associative That is
Th wre m
:
Th e
.
.
(e
(
-T
)
,
OT) x
(e
)
o
r
= O
s
r
x
(
O
s)
=e
r
r
o
-O
e
O) ¢
Furthermore
s
o
r
x O
an
.
d
can have no other meaning than
r
o
s
x s,
x s,
T
the e xpressions
s,
,
VECTOR ANA L YSIS
282
Degre e s
a dic s
D
f y
u l li ty
N
f
o
o
l
y
was shown ( Art 1 01 ) that a dyadic cou d alwa s
be reduced to a sum of three terms at most and t his reduction
can b e accomplis hed in onl y o n e way when t h e antecedents
or t h e consequents are specifi ed In particular cases it ma
be possib le to re duce the dyadic further to a sum of two
terms or to a single term or to z ero Thus let
It
.
,
y
.
.
O=
l + bm +
a
c n
y
are coplanar one of the three ma be e xpressed
in te ms of the other t wo as
If 1, m,
r
n
l =
x
m + yn
.
y i
d ad c has been reduced to two terms If 1 m n were
all collinear the dyadi c would reduce to a single term and f
they all vanished the d adi c would vani sh
Th e o re m : If a d ad c O be e x pressed as the sum of thr ee
terms
Th e
,
.
y i
y
i
,
.
O=
l + bm +
a
o n
of which the antecedent s a b c are known to be non-coplanar
then the dyadic may b e reduced to the sum of two dyads
when and only when the consequents are coplanar
Th e proof of the fi rst part of the theorem has just been
given To prove t h e second part suppose that the d adic
could be educed to a sum of two terms
,
,
,
.
y
r
.
O = dp +
e
q
and that the consequents l m n of O were non-coplanar
'
Th s supposition leads to a contrad ction
m n
Fo r et
be the s ste m reciprocal to l m 11 That is
i
,
y
,
mx
n
l
m
n
[
]
i
,
,
.
l
.
,
.
’
,
LINEA R VE C TOR FUNC TI ONS
r
2 83
vec t o s l m n e xist and are non -coplanar be cause
have
been
a
s
sumed
to
be
non
coplanar An y vector
n
1 m
ma b e e xpressed in terms of them as
Th e
’
’
,
y
.
l
ut
a nd
o
-m
l
l =m
’
m
o
i
m
e
u
Hence
’
n
o
o
n
’
n
’
m
xa
+ yb +
’
= 1,
n
z c
.
g ving to r a suitab le value the vector
equal to an vector in space
By
B
ut
r
,
,
,
B
'
y
r may be made
O
.
O
This
shows that O r must b e coplanar with d and e Hence
O r can take on o nl y those vecto r values which lie in the
plane of d and e Thus the assumption that l m n are non
coplanar lea ds to a contradi ction Hence I m 11 must be
coplanar and the theorem s proved
Th e o re m : If a d ad c O be e x pre ss ed as th e sum of thr ee
terms
.
o
,
,
.
i
y i
i
y i
,
.
,
.
of wh ch the ante cedents a b c are known to be non-coplan a r
the d ad c O can be reduced to a sin gle dyad when and onl
when the consequents 1 m n are collinear
Th e proof of the fi rst part was given above
To prove
the second a t supp o se O could be e xpresse a s
,
,
y
,
,
,
.
pr
d
O
z
dp
.
T= a l x p + b m x p +
c n
x u
.
VECTOR A NAL YSIS
2 84
From
the second equation it
postfactor for any vecto r
where
a
’
,
’
b
c
,
t
From
’
is
evident that
is the reciprocal system to
a
,
b, 0
T
us e d
a
i
g ve s
o
the fi rst e xpression
Hence
y l
must be z ero for every value of r tha t is for ever va ue of
Hence
y z
,
,
,
cc ,
.
l x p
z
m
m
m s
n
X p = u
Hence I m and n are all parallel to p and t h e theorem has
been demonstrated
If the three consequents 1 m 11 had been known to b e non
coplanar in stead of t h e three antecedents the statement of
the theorems would have to b e altered by interchanging the
words a n t e c e de n t and c o n s e qu e n t throughout
There is a fu r
ther theorem dealing with the case in whi ch both antecedents
and consequents of O are coplanar Then O is reducible to
the sum of two d ads
D e n i tio n : A dyadic which cannot be reduced t o
the sum of fewer than three dyads is said to be c o mp le te A
dyadi c which may be reduced to the sum of t wo dyads but
cannot b e reduced to a single dyad is said to b e p la n a r In
case the plane of the antecedents and the plane of the con
sequents coincide when the dyadic is e xpressed as the sum of
two dyads t h e dyadi c is said to be u n ip la n a r A dyadic
which may b e reduced t o a single dyad is Said to be lin e a r
In case the antecedent and cons e uent of tha t dyad a re col
,
,
.
,
,
,
.
fi
y
.
.
.
,
.
,
.
q
.
VECTOR ANA L YSIS
286
y i
a d c O us ed as a
d
f
prefactor reduces any ve c tor r to the l ine of the an tecedent
of O In particular any vectors perpen di cular to the con
sequent o f O are reduced to z ero Th e dyadic O used as a
postfactor reduces any vector r to the li ne of the consequent
of O In particular any vectors perpen di cular to the ante
cedent of O are thus reduced to z ero
If O is a z e r o dy a dic th e ve c to rs 8 a n d t a re bo th z e ro n o
m a tte r wh a t th e va lu e of r m a y be
D e n i tio n : A p la n a r dyadic is said to possess on e de gree of
A li n e a r dyadic is said to possess t wo degree s of
n u lli ty
A z ero dyadi c is said to possess th re e degre e s of n u l
n u l l i ty
l i ty or c o m p le te n u lli ty
Th e o re m : Th e di rect product of t wo complete dya dics
is complete ; of a complete dyadi c and a planar dyadi c
planar ; of a complete dyadic and a lin e ar dyadic linear
Th e o re m : Th e product of two planar dyadics is planar
e x cept when t h e plane of t h e consequent of the fi rst dya dic
in the product is perpendicular to the plane of the a n t e c e
dent of the second dyadic In this c a se the product reduces
to a linear dyadic — and only in this case
O ; bu t
se que n t o
no
va lu e s
o the r
Th e
.
.
.
.
.
.
fi
.
.
.
.
,
,
.
.
.
L et
O=
Q
Th e
vector
T
s
r
a
1
b
1
O
:
a
2
b 2,
T
o
.
takes on all value s in the
and 0 2
a; 0
vector
s
’
8
’
= O
O
o
3
1
y
c
z
.
takes on the values
s
9 4
+
{
cc
(b 2
°
cl
)
plane of
0
1
L INEAR VECTOR FUNCTI ONS
L et
:c
where
and
9
,
’
a
a:
(b l
0
a:
“2
(3
1
y
1
a
2,
y (b l
)
1)
’
2 87
’
Ff
! (b z
-
These
equations may always b e solved for a: and y when
’
any desired values cc and y are given — that is when s has
any des ired value in the plane of a 1 and e , — unless the
determinant
’
'
,
.
B
y
b
ut
bl
°
b2
°
Chap II ,
x
b
( .
°
1
b2
°
2
2
this is merely the product
.
.
1
bl
x
b .)
<c l
0
>
< c .)
.
vector b 1 b 2 is perpendicular to the plane of the con
sequents of O ; and 01 02 to the plane of t h e antecedent s of
T
Their scalar product va ni shes when and only when t h e
vectors are perpendicular that is when the planes are per
C ons equently 8 may take on any value in the
p e n dic u l a r
plane of a 1 and a 2 and O T is therefore a planar dyadi c
unless the planes of b 1 and b 2 01 and c 2 are perpendicular
If ho w ever b 1 and b 2 01 and c 2 are perpendi cular 8 can take
on only values in a certain line of the plane of a 1 and
and
hence O T is linear Th e theorem is therefore proved
T he o re m : Th e product of t wo linear dyadi cs is l inear
e x cep t when the cons equent of the fi rst factor is p e rp e n
In thi s case the
dic u l a r to the antecedent of the second
product is z ero — and only in this case
The o re m : Th e product of a planar dyadic into a linear is
linear e x cept when the plane of the consequents of t h e
planar dyadi c is perpendicular to the ant ecedent of t h e linear
dyadic In this case the product is z ero — and only in t his
case
Th e o re m : Th e product of a linear dyadic into a planar
dyadic is linear e xcept when the consequent of the linear
Th e
x
,
.
,
’
.
,
.
’
’
.
.
.
.
.
.
VECTOR ANA L YSIS
2 88
dyadic is perpendicular to the plane of the antecedents of
the planar dyadic In th s case the pr o duct s z ero — and
only in this case
It is immedia t ely e vident t hat in the cases mentioned the
produc ts do reduce to z ero It is not quite so apparent that
Th e proofs are
t hey can reduce to z ero in only those cases
si m ilar to t h e one given above in the case of two planar
dyadics They are left to the reader Th e proo f of t h e
fi rst theorem stated page 2 86 is al so left to the reader
i
.
i
.
.
.
.
.
,
,
Th e I de mfa c tor ;
fi
1
.
Re c ip ro c a ls
f Dy a dic s
Conj
u a te s
g
a nd
o
dyadic applied as a prefactor or as
a postfactor t o any vector al ways yields that vector the
dyadic s said to be an ide mfa c to r That is
De
n i t io n :
If a
i
.
if
O
r
or if
r
O
r for all values of
r
r
,
for all values of r
,
y
then O is an idemfactor Th e capital I is us ed as the s m
bol for an idemfacto r Th e idemfactor is a c o mp le te dya di c
Fo r t here can b e no di rection in which I r vanishes
Th e o re m When e xpressed in nonion form the idemfactor is
.
.
.
o
.
3
3
( )
Hence all ide m fa c to rs are equal
To prove that the idemfac t or takes t h e form ( 3 3 ) it is
merely necessary to apply the idemfactor I to the vecto s
i
1: respectively
et
.
j
,
,
.
I=
L
a
“
a
1
g
In t h e t h e o ry
d in a ry a l e bra
or
.
of
r
u
ii +
a
21
i
j+
a
31
ki +
dya d ic s
Th e
n o ta
the
i
t on
m
lz
i j+
j
22 j
a
82
a
ls
“
kj
gg
ik
k
j
23
a
as
kk
i d e fa c t o r I pla ys a mi le
is in t e n de d t o s u
e s t t h is
.
l
a na o
a na l o
gg
o us
y
.
to
un
i t y in
VECTOR ANAL YSI S
290
l
the fi rst place since O is t h e idemfactor it is a comp ete
dyadic Hence the antecedents a b c are non-coplanar an
possess a set of reciprocals a b c
et
In
,
,
.
’
'
By
y
r
h pothesis
Then
’
,
r
,
.
O
L
r.
r
for all values of r that is for a ll values of a y 2
correspondi ng coe ffi cients must be equal That is
,
,
,
,
.
m =b
y
’
n
,
Hence
,
.
c
z
’
.
i
d
an two dyadics and f the pro uct
1
O T is e ual to the idemfactor ; then the product T O
when the facto s a re taken in the revers e d order is a so
e ual to the idemfactor
Th eo re m
q
q
:
IfO and T be
d
r
l
,
,
.
L et
To
,
O
show
T
Tz
o
o
I
.
O= I
.
-( O T) = r -I = r
=
O
T
rO
O
r
)
(
r
,
,
Thi s
relation holds for all values of r A s O is complete
must take on all desired values Hence b y de fi nition
.
r
O
.
T
o
O= I
.
If the
product of two dyadics is an idemfactor that pro duct
may be taken in either order
De n itio n : When two d adics are so related that
their product is equal to the idemfactor they are said to b e
,
fi
.
y
,
1
of
i
Th is
m
t wo
inc o
m
d e f a c to r
.
m
m
it a t e s bo t h t h e dya di c s 4» a n d 11! t o b e c o pl e t e Fo r t h e pro d u c t
ple te dya dic s is in c o ple te a n d h e n c e c o uld n o t b e e qua l t o th e
n ec e ss
.
L INEA R VECTOR FUNCTI ONS
291
l
nota t ion used for reciproca s in ordinary
alge b ra is emplo ed to denote reciprocal dyadics That is
re c ip roc a
ls
1
.
Th e
y
.
I
—1
T
Th e o re m
:
T
a nd
O
,
I
_1
w
O
Reciprocals of the same or equal dyadics are
equal
et O and T b e two given equal dyadics O
their reciprocals as de fi n ed above B y h pothes s
L
.
y
.
a
42
os
—
—
1 —
-1
-1 —
0
O (
.
a
O
T,
:
-
1
.
O
.
O
I
l —
O
‘
Hence
O
:
1
.
I
‘
l —
O
1
T
1
.
T
:
O
.
—l
O
—1
2
.
O
O
.
-l
T
.
T
.
,
-l
,
I,
T
I
‘
T
I
“
d
1,
«H
‘
O
an
‘
-1
7
9
-l
o
i
I
T,
:
n
,
"
I
= T
“
l
4
.
reciprocal of O is the dya di c whose antecedents are the
eciprocal system to the cons equents of O and whose c o n s e
quents are the reciprocal syste m to the antecedents of O
If a complete dyadi c O b e written in the form
r
Th e
.
O
its reciprocal is
O
‘
bm
:
I
l
’
a
’
’
m b
o n,
’
’
n c
’
6
3
( )
.
Fo r
the direc t products of a complete dyadic
d adics T and 9 are equal as dya di cs then T and
1 A i
pl t dy di h
l
( fi i t ) ip
Th e o re m
into
two
y
:
If
n
nc o
m
e e
a
c
a s no
n
e
re c
ro c a
.
O
Q
VECTOR A NAL YSIS
292
r
are equal If the product of a dyadic O into two vecto s
and 3 (whether the m ul tiplication b e pe rformed with a dot
or a cross) are equal then the vectors r and s are e ual
That is
r
-
.
q
,
.
,
if
O
a nd
if
a nd
if
This
o
O
Q,
T= O-
then
T = O,
= O
then
r
a
r
s,
o
the n
y
y
=
s,
s.
r
y
ma be seen b multiplying each of
through b the reciprocal of O
,
T
‘
l
O
O
'
T
o
1
'
T
o
-1
T
T:
:
0
T g
a
-O
1
q
reduce the last e uation proceed as follows
an
vector
y
I
I
r = t
s
t To
x
,
y
x
o
t
Henc e
q
e uation s
the
I
o
.
Le t t b e
,
17
.
r
is an vector is equal to 8
Equations ( 3 7 ) give what is e uiva ent to the aw of
celation for complete dyad cs Complete d adi cs ma b e
canceled from either end of an e x pression just as if the
were sc a lar quantities Th e cancelation of an incomplete
dyadic is not admissib le It corresponds to the cancelat on
of a z ero factor in ordi nar algebra
Th e o re m
Th e reciprocal of the product of an
numb er of d a di cs is equal to the product of the rec proca s
taken in the op p o s ite order
It will be s u f
fi cient to give the proof for the c as e in wh ch
t h e ro uct c ons s ts of t wo d a d c s
To s how
As t
,
q l
.
i
.
y
l
y
i
.
.
y
y
l
.
y
i
i
.
p d
y
i
y i
.
VECTOR ANAL YSIS
294
G eometricall
y the transformation
r
= Or
’
r
is a reflection of space in t h e jk-plane This t ans formati on
replaces each fi gure b y a symmetrical fi gure symmetrically
si t uated upon the opposite s ide of the k-plane Th e trans
formation is sometimes called p e rve rs io n Th e idemfac t or
has a l so a doub l infin ite system of square roots of the form
.
j
y
.
.
T
G eometricall
,
jj
ii
kk
.
y the transformation
r
’
= Tr
is a reflec t ion in the i ax is This transformation replaces each
fi gure b y its equal rotated ab out t h e i-a x is through an angle
of
Th e idemfactor thus possesses not only two square
roots ; b u t in addition two doubly in fi nite systems of square
roots ; and it wil be seen ( Art 1 2 9 ) that these are b y no
means all
Th e c o nj
u a te of a dyadic has been de fi ned
A
9
9
r
t
g
(
)
as the dyadic ob tained by interchanging the antecedents and
consequents of a given dyadic and the nota t ion of a sub script
C has b een employed
Th e equation
-
.
l
.
.
.
’
.
h as b een
demonstrat ed Th e following theorems concerning
conjugates are useful
Th e o re m : Th e conjugate of the s u m or di fference of t wo
dyadics is equal to the sum or difference of the conjugates
.
.
,
T
¢
:
t
(
)
C
=
w
e
: l:
T0
.
conjugate of a product of dya di cs is equal
to the prod uct of the conjugates taken in the op p o s i te or er
The o re m
:
Th e
d
.
L INEAR VECTOR FUNCTI ONS
ill
295
w be s u fli c ie n t to demonstrate t h e theorem
the product contains two factors To show
It
in
case
.
m
0
n
(i
r
r
H ence
O
)
(
wc wm
:
'
-( T
-O
Oo
z
T = TC (r
r
o
r,
-O)
= TO
T) O = TC
O
(
-a
r
.
O0
.
.
y
i
conjugate of the power of a d adic s the
power of the conjugate of the dyadi c
Th e or e m
Th e
.
2b
y
Thi s
is a corollar of the foregoing theorem Th e e xpression
O 2 may b e interpreted in either of t wo equal ways
Th e o re m : Th e conjugate of the reciprocal of a dyadi c is
equal to the reciprocal of the conjugate of the dyadi c
.
.
.
.
1
7
1
( 6
Fo r
Th e
the
(T
A
R
A
To = ( T
Ia
o
I
:
.
y
idemfactor is its own conjugate as ma be seen from
nonion form
.
I=i i +
ji + k k
O0
H
enc e
(
¢
)
C
—l
$0
:
W
)
(
I
C
0
To
.
Hence
expression O0 may therefore be interprete d in either
of two equivalen t ways — as the reciprocal of the conjugate
or as the conjugate of the reciprocal
D e n i tio n : If a dyadic is equal to its conjugate it is said
If it is equ al to the negative of its con
uga te
to be s e lf-c o nj
Th e
"
fi
1
.
,
.
2 96
jugate it
dyadi cs
is
,
s
VECTOR A NAL YSIS
i
a d to be
a n ti se
fc o n u ga te
j
l
Fo r
.
fconjugate
se l
-
.
Fo r
anti-self-conjugate dyadics
r
-O
O
—
O
r,
o
OO
—
.
dyadic may b e divided in one and
way into two parts of which one IS se f-conjugate and the
other anti-self-conjugate
An y
Th e o r e m
l
.
Oa )
—
.
$00
¢
0
¢a
+ $2
—
Q
Hence the part %( O O0) is sel f-conjugate ; and the
-se f-conjugate
O
O
anti
Thu s the div sion has bee n
0)
H
accompl shed in one way
et
i
l
,
L
.
i
.
(O
¢ a)
O
(
To )
T
T
’
W
O
T
”
.
”
.
Suppose
y
it were possible to decompose O in an other wa
into a self-conjugate and an anti -self-conjugate p a rt
et
then
L
.
T
Whe e
(
r
O
Hence if ( O
T
(
"
’
(T
’
:
O)
’
Q)
Q)
Hence if ( O
conjugate
”
(
O
+ 9
)
(
’
9
is self-conjugate
T
(
H
Q)
T
”
Q
,
Q)0 = $
is self-conjugate
"
is anti -self-conjugate
.
Q
:
9
0,
Q
9
0.
.
QU
QO
‘
‘
o
0
Q
‘
is an ti -self
VECTOR A NAL YSIS
2 98
§(
O — Oc )
o
O
r
r
x
r,
X ¢
x
o
conjugate dyadi c O possesses one
degree of nullity It is a uniplanar dyadic the plane of
”
whose consequents and antecedents is perpendicular to O
the vector of O
This theorem follows as a corollar from equa t ions
Th e o re m : A n y dyadic O may b e b roken up into two parts
of which one is self-conjugate and the other equivalent to
minus one half the vector of O used in cross multiplication
Th e o re m : An y
”
a n t is e l f.
x
,
y
.
.
l
1
or symbolically
TX X
2
.
( 45)
.
vector 0 used in vector m ul ti
linear vector function Fo r
An y
plication de nes a
fi
.
Hence it must be possible to represent the Operator c x as a
dyadic Thi s dya dic will be uniplanar with plane of its
antecedents and consequents perpendicular to 0 so that it
will reduce all vectors parallel to c to z ero Th e d adic ma
be found as foll ows
.
,
.
Ry
3
1
( )
(I x
)
o
,
{(I x
r
I
Hence
and
-I
I
c
-( c
x
=
I
r
) }
{I
r
I) c
I
r
x
(
)
.
-( I x c )
This may be sta ted in words
x c
r
c
-r
r
y
r
.
(
c
x I)
r
,
y
L INEAR VECTOR FUNCTI ONS
29 9
vector c used in vector multiplication with
a vector r is equal to the dyadic I x c or c I used in direct
mul tiplication with r If e precedes r the dyadics are to be
us ed as p re f
a c t o rs ; if c foll ows r as
dyadics
o
f
a
r
s
t
c
t
o
s
T
h
e
p
I x c and c
I are an t i -self-conjugate
In case the vector 0 is a unit vecto r the application of the
operator 0 to any vector r in a plane perpendicular to c is
equivalent to turning r t hrough a posi t ive righ t angle ab out
the ax is 0 Th e dyadi c c x I or I c where c is a unit vector
therefore turns any vector r perpendi cular t o 0 t hrough a
right angle ab out the line 0 as an ax is If r were a vector
lying out of a plane perpendicular to c the e fle c t of t h e dya dic
I woul d b e to annihilate that component of r whi ch
I x c or c
is parallel to c and turn that component of r whi ch is p e rp e n
dic u l a r to c through a right angle a b out 0 as a x is
If the dyadi c b e applied t wice the vecto rs perpendicular to
r are rotate d through t wo right angles They are reversed in
direction If it be appli ed three times they are turned thr ough
three right angles A pplying the operator I c or c I four
times b rings a ve c tor perpendicular to 0 b ack t o its original
osition Th e powers of the dyadic are therefore
Th e o re m : Th e
x
.
x
x
,
.
.
x
.
.
x
.
.
x
.
.
p
x
.
—
—
I
(
— c c
)
,
I >
<c
— c c
c
x
,
x I
.
thus appears that the dya di c I c or c x I ob eys the same
law as fa r as its powers are concerned as the scalar imaginary
1 in alge b ra
V
I is a quadrantal vers or only for
Th e dyadic I c or c
vectors perpendicular to c Fo r vectors parallel to c it acts
as an annihilator To avoid this c fi e c t and obtain a true
It
‘
x
.
x
.
.
VE C TOR A NA L YSIS
8 00
quadrantal versor for all vectors r in space it is mere
sary to add the dyad c c to the dyadic I x c or c x I
ly
ne ces
.
X
— I,
X
—
2
3
X,
X = I,
4
X =X
5
i
.
dyad c X therefore appears as a fourth root of the
idemfactor Th e quadrantal versor X is analogo u s to t h e
imaginary V 1 of a scalar algeb ra Th e dyadic X is com
s ts of two parts of whi ch I x c
and
consi
s
anti
self
l
e
t
e
p
conjugate ; and c c self-c onjugate
Ifi
k are thr ee perpendicular unit vectors
Th e
.
i
.
,
j
,
.
,
I >
< i
i x I = k j—
jk
—
k i,
=k >
i — ij
<I = j
,
k
y
,
as ma be seen by multiplying the idemfacto
j
I
ii
ii
r
k 1:
into i
and k successively These e x pressions represent
quadran t al ve rs o rs ab out the ax is i k respectivel comb ined
with annihi lators along those ax es They are equivalent
when u sed in direct multiplication to i
k x respectively
,
.
,
,
j
y
,
.
,
—
x
,
,
,
i
(j
—
ij
),
e pression
is an idemfactor for the plane of i and
b u t an annihilator for the direction k
In a similar man
ner the dyad k k is an idemfactor for the direction k but an
Th e
j
,
x
4
k
(Ix )
.
,
VE C TOR A NAL YSIS
302
115
L
et
]
vector
f Dy a dic s
Re du c t ion
t o N o r ma l For m
o
y i
any complete d ad c and let
Then t h e vector r
O be
r be
i
un t
a
’
.
’
r
= Or
ll
i
is a linear function of r When r takes on a v a lues co us s
tent wi t h its b eing a un i t vector that is when the termin us
of r describ es the surface of a un it sphere — the vector r
varies continuous l and its terminus describes a surface This
1
surface is closed It is fact an ellipsoid
Th e o re m It is always possi b le to reduce a complete d a di c
to a sum of thr ee term s of which the antecedent s among
themselves and the cons equents among themselves are mutu
ally perpendic u lar This is called the n o rm a l fo rm of O
.
,
’
y
,
.
y
.
.
.
.
O
a
’
’
b jj
i i
’
k k
c
.
r d i d
demonstrate the theore m consider the su fac e escr be
To
r
’
= Or
.
this is a closed surface there must be some directi on of r
’
which makes r a max imum or at any rate gives as great
a value as it is possib le for r to ta k e on
et this d rection
of r b e called 1 and let the correspondin g direction of r
’
the direction in whi ch takes on a value at e a st as great as
any — b e called a Consider ne xt all the values of r which
li e in a plane perpendicular to i
Th e corresponding values
of I lie in a plane owing to a fact that O is a linear vector
As
’
L
’
.
r
,
r
i
’
l
.
r
.
’
1
Th is m a y b e
p ro ve d a s fo ll o ws
’
r z
H
e nc e
g
r
.
dn
non on
d
d es c rib e s
re e
is t h e
.
e
He n c e
llipso id
.
’
r
mq
fo r
B y e x p re s s in g ‘
lfin
e
-1
r
—l w
f ( Tc
r=1
i
r
-1
=
¢
r
a
,
'
o
qua t io n r
ua d ri c s urfa c e
th e e
'
a
.
?
’
r
Th e
z
] is
on
se e n
.
t o b e o ft h e se c o n
ly c l o se d q ua dric
f
d
s ur a c e
L INEAR VECTOR FUNCTI ONS
O f these
3 03
function
values of r one mus t be at least as great
as any other Call this b and let t h e correspondi ng directi on
Finally choose k perpendicular to i and
of r b e called
upon t h e positive side of plane of i and
et 0 b e the
value of r whi ch corresponds to r k Since the dyadic O
changes i k into a b c it may b e e xpressed in the form
.
j
.
'
j
j L
.
.
’
,
j
.
,
,
,
O
i
a
b
j
k
c
.
rema ins to show that the vectors
above are mutually perpendicular
It
b,
a,
as determin ed
c
.
r
’
o
dr
’
’
r
o
i
a i -d r
+ b j+
c
k)
+ b j+
c
k)
+
r
’
r
,
d r,
o
b jdr
o
r
+
’
o
c
k
-d r
.
When r s parallel to i r is a maximum and hence must be
’
i
,
y
perpen dicular to d r Since r is a unit vecto r d r s alwa s
perpendicular to r Hence when r is parallel to i
’
.
.
r
’
o
b
j dr
+
r
’
c
-d r =
k
j
0
.
further d r is perpendic ul ar to r c vanishes and if
d r is perpen di c ul ar to k r b vanishes
Hence when r is
parallel to i r is perpen di cular to b oth b and 0 B u t when
r is parallel to i r is parallel to a Hence a is perpendi cular
to b and 0 Consider nex t the plane of and k and the
plane of b a n d c
et r b e any vector in the plane o fjand k
If
’
,
’
,
o
c
,
.
’
.
,
’
.
.
L
.
( b j+
r
’
o
j
.
,
dr
’
r
’
When r takes the value
di
hence is perpen cular
b
o
jr
,
to
c
j dr
k)
+
r
r
,
’
c
c
k
is a max imum in this plane and
Since r is a un t vector it is
dr
’
’
.
i
VECTOR
3 04
r
A
NA L YSIS
r is
perpendi cul ar to d
Hence when
is perpendic ul ar to and
j
.
ll l j
para e
to
,
dr
,
k
-d r
.
j
Hence r c is z ero B u t when r is parallel to r ta kes t h e
value b Consequentl b is perpen di cular to 0
It has therefore been shown that a is perpendicular to b and
C onsequentl the three
c and that b is perpendicular to 0
antecedents of O are mutu ally perpendic ul ar They ma be
’
’
denoted b i
Then the dyadi c O takes t h e form
k
’
y
.
.
’
,
.
y
.
,
y j
y
.
’
,
.
,
O=
where
a
’
’
b jj+
i i
c
’
k k,
are scalar consta nts positive or negative
Th e o re m : Th e complete d adic O may alwa s be
reduced to a sum of three dyads whos e antecedent s and
whose cons equents form a right-handed rectangul ar system
of unit vectors and whose scalar c o e fli c ie n t s a e either a
positive or a ll negative
a ,
b,
y
c
.
y
r
ll
.
O
:
z iz
(
a
5
3
( )
’
’
i i + b jj
proof of the theorem depends upon the statements
made on page 2 0 that if one or thr ee vecto s of a righ t -handed
system be reversed the resulti ng s stem is left-hande d but
if two be reversed t h e system remains right-handed If then
o n e of the c o e fii c ie n t s in
5
2
s
negative
the
d
rections
of
the
( )
other two ax es may be reversed Then all the coeffi cients
are negative If two of the coe ffi cient s in ( 5 2) are negative
the directions of the two vectors to which they belong ma
be reversed and then the coeffi cien t s in O are all positive
Hence in any case the reduction to the form in which all
the coe ffi cients are positive or a are negative has been
performed
As a l imitin g c ase between that in which the coe ffi cients
a re a
o s it v e a n d t ha t n wh ch the
a re a ll nega t v e comes
Th e
r
y
,
i
,
i
.
.
y
.
,
.
ll
.
ll p i
i i
y
i
VECTOR A NAL YSIS
3 06
O0
O
O,
z
a
:
O
’
i i
a
z
s
’
ii
O
O
a
2
I
(b
(
O
z
a
z
I
(
j
c
s j
z
c
2
O
a
z —
O
z
2
a
a
Z —
O
a
’
z
k k
kk
’
,
.
O0,
’ ’
=
i i +
kk
b
(
:
’
O, =
.
I = ii + j
j
O2
’
’
’
) jj
z
2
z
l)
.
’
’
jj
2
(c
’
k k,
a
2
)k
’
k
’
,
i = 0
’
o
) jj
l)
2
o
(o
—
z
z
d
)
k k,
i = 0
.
i
were not parallel ( O 2 a I) wo ul d annih late
two vectors i and i and hence ever vector in their plane
2
2
a
I) would therefore possess two degrees of nullit
(O
and b e linear B u t it is apparent that f a b c are di fle re n t
this dyadi c is not linear It is planar Hence i and i must
be parallel In like manner it may b e shown that and
k and k are parallel
Th e dyadic O therefore takes the form
If i
a nd
i
’
2
y
’
y
.
i
.
.
,
,
j
.
.
’
j
’
,
’
.
O=
where
a
,
b,
a
i i + bj
j+
c
kk
are pos itive or negative scalar constants
c
fi
Do u ble Mu ltip lic a tio n
.
1
product of two dyads is
t h e s c a la r quantit obtained by multipl ing the scalar product
of the antecedents by t h e scalar product of the consequents
Th e product is denoted b y inserting two dots between the
De
Th e do u ble do t
n i t io n :
y
y
.
dya ds
“
a
This
Th e
p rin te d
y
: c
d=
a
-c
b
d
( 56 )
.
product evidentl obeys the commutative law
a
1
b
re s e a rc h e s
fo r t h e
fi
rs t
m
of
t
i
b
z c
dz
Pro fe ss o r Gibbs
e
.
c
d
u
po n
z a
D
b,
o u bl e
M ult ip lic a t ion
a re
h e re
L INEAR VECTOR FUNCTI ONS
3 07
and the distributive law both with regard to the dyads and
with regard to the vectors in the dyads Th e doub le dot
produc t of two dyadics is ob tain ed b y multiplyin g the prod
u c t out formally accor di ng to the distri b utive law into the
sum of a number of double dot products of dyads
.
.
T=
a
c
2
1
d1 +
h2
c
2
d2 +
03
d3 +
b
3
3
a
3
(
°
1
d1
ez
aa
c 3
_
a
3
1
b1
:c l
d1
a l
bl
z c 2
d2 +
a
2
b2
261
d1 +
a
2
b2
3 02
d2 +
d1
a
a
bgz
d2
a gba z c l
c z
b3
fi
bi
i c a
ds i
a ,
b,
z c a
d3 +
a
ba
i cs
ds
l
a
'
d3 +
'
0
d2 + a 3 3
product of t wo dyads is the
dy a d of which the antecedent is the vector product of the
ant ecedents of t h e two dyads and of which the consequent is
the vector product of the consequent of the two dyads Th e
product is denoted b y inserting t wo crosses between the
d ads
De
n i ti o n :
Th e do u ble
c ro ss
.
y
b x d
.
This
l
y
i
5
( 7)
product a so evidentl ob eys the commutat ve law
VE C TOR ANAL YSIS
3 08
d
d
d
and the distrib utive law both with regard to the dyads an
wi t h regard to t h e vectors of which the dyads are compose
Th e doub le cross product of two dya di cs is therefore de fi n e
as the formal e xpansion of the product accordi ng to the
distrib utive law into a sum of double cross products of
dyads
.
.
If
and
i ( 01 d1 +
c 2
+c
+
a 3
b3
§
c l
d1 +
u
b l x d1 + a 1 x
+
+
a
>
< 03 11 3 >
< d1 +
8
c2
b z x d3 +
b2 >
< d2
>
<c l
d3 +
-
a 3
a2
3
d2
a
3
m
>
< c 2 b3 >
< d1
double dot and doub le cross products of
two dyadics obey the commutative and distrib utive laws of
multiplication B u t the double products of more than two
dyadics ( whenever they have an meaning) do not obe t h e
associative l a w
Th e o re m
:
Th e
y
.
y
.
O
O
(
O
z
T
T
O
=
T
T§ O
§
m ww
y
z
wm
i
.
theorem is su ffi cientl evident w thout demonstration
.
VECTOR A NAL YSIS
31 0
the factors is reversed each scalar triple product cha nges
sign Their product therefore is n o t a l t e re d
A dya di c O may be multipl ied by itself w th doubl e
cross
et
i
i
.
.
L
.
O=
a
l + bm +
c n
l x m +
<
+ b >
+
>
<a
c
n
products in the main
equal in pairs Hence
Th e
l >
<n
c
m x m + b x
<b
m x l + b >
a
>
<
a
di agonal
x
m +
vanish
>
<
c
.
m x
c
n
c
Th e
x n
n
.
r
othe s are
.
m x
If a , b ,
c
and l
,
m,
n
n
+
c
are non —coplanar this
2
[
a
b
l x m)
x a
c
]
l
m
[
n
a
l
l
(
]
,
’
ma y
+ b m
.
60
( )
be written
’
c
r
n
r
)
,
product O 35O is a species of power of O It may be re
garded as a square of O Th e notation O2 will be employed
to represent thi s product after the scalar factor 2 has been
stricken out
Th e
.
.
2
(b x
c
mxn +
c
l x m)
>
<a
6
1
( )
triple product of a dyadi c O e xpressed as the sum of
three dyads with itself twice repeated is
Th e
O
O2
:
O=
(b
i
Oz O= 3
n
O
l x m)
x c
y
e xpanding this product ever term in whi ch a letter is
repeated vani shes Fo r a scalar triple product of t hree vec
In
.
L INEAR VECTOR FUNCTI ONS
i
to rs two of wh ch are equal is z ero
reduces to three terms only
O2
:
O=
[ b o a ] [ m n l]
O2
O
:
:
p
[c
a
b]
3
[a
b
Hence the product
.
[ n l m]
c
31 1
[
a
b
c
]
l
m
n
[
]
] [l m n]
l
m
n
[
]
y
.
triple roduct of a d ad ic b y itself twice repeated is
equal to s ix times the scalar t riple product of its antecedents
multiplied by the scalar triple product of its consequents
Th e product is a species of cu be
It will b e denoted b y O3
after the scalar factor 6 has been stricken out
Th e
.
.
.
O
i
l
O
g
z
[
a
b
c
]
I
n
[
n
]
2
6
( )
.
be ca led the se c o n d of O ; and O3 the third of
O the following theorems may be stated concerning t h e
seconds and thir ds of conjugates reciprocals and products
Th e o re m Th e second of the conj u gate of a dyadic is equal
to the conjugate of the second of that dyadic Th e third of
the conjugate is equal to the third of the dyadic
If O2
,
,
,
,
.
.
.
w
( a
s
:
63
( )
$3
second and third of the reciprocal of a
dyadic are e ual respectively to the reciproca s of the second
and third
The o re m
:
Th e
q
l
.
—
«0 9
(
D
«
O
2
2
(
O
3
—1 _
l
l
a
1
fig
—1
1 + bm
a
l
l
Q2
$3
l
m b +
[
c
]
]
11 0
l
b
1
n
l
m
[
]
1
-1
o n
’ ’
a
.
VECTOR ANAL YSIS
31 2
[
a
b
c
[l m n ]
]
l
( w )2
—1
B
[
ut
a
Hence
(
b
c
[
]
l
b
l
c
]
—l
m
’
1
$8
[
b
a
]
c
n
n
’
]
O2
( O )2
“
l
m
l
[
and [ 1
n o
)
n o
l
’
1
O2 )
a
mb
a
mb
a
l
l
(
l
]
1
l
m
n
[
]
.
4
.
l
m
n ,
[
]
1
[
1
( O )3
‘
Henc
(
e
Og)
[a
l
"
(
a
’
0
’
’
b
O
]
b
‘
c
l
n
l
m
[
]
] [l
’
’
m
O3
)a
,
"
1
.
r q
second and thi rd of a product a e e ual
respecti vely to the product of the seconds and the product of
Th e o re m : Th e
t h e t hirds
.
(O
T) 2 = O 2
o
9
Choose
03
-T2
$3 T3
:
y
,
q
an three non-coplanar vectors 1 m n as cons e uents
of O an let m n be th e ante cedents of T
d
,
’
,
’
.
,
O
(
e
x f+
>
<
c
fx d +
a
a
>
<b
d x
e,
l >
< m,
d x
Hence
Henc e
O2
e
(
O
o
x f+ c >
<a
T) 3 =
[
a
b
c
]
fx d+
d
e
f
[
]
a
>
<b
c
.
dx
e
.
VE C TOR ANAL YSIS
31 4
O2 = b >
<c m x
n
O3
+
c
>
<a
[
b
c
a
]
l >
< m,
n
m
l
[
]
.
necessary and suffi cient con di tion that a
dyadic O be complete is that the third of O be d fferent from
z ero
Fo r it was shown ( Art 1 06) that b oth the antecedents and
the consequents of a complete dyadi c are non-c oplanar
Hence the two scalar triple produc t s whi ch occur in O3
cannot vanish
Th e o re m : Th e necessar and su ffi cient condition that a
dyadic O be planar is that the third of O shall vanish but the
second of O shall not vanish
It was sho wn ( Art 1 06 ) that if a dyadic O be planar its con
sequents l m 11 must be planar and conversely if the c o n s e
quents be 0 0planar the dyadi c is planar Hence for a planar
dya di c O8 must vanish B u t O2 cannot vani sh Since a
b e have b een assumed non-coplanar the vectors b x c c
a
a x b are nonHence if O2 vani shes each of the
c oplanar
vectors m x n n x l 1 x m vanishes — that is l m n are col
linear B u t thi s is impossible since the d adic O is planar
and not linear
Th e o re m : Th e necessar and suffi cient condition that a
non vanishing dyadic be linear is that the second of O and
cons equently the third of O vanishes
Fo r if O be linear the consequents l m n are collinear
Hence their vector products vanish and the consequents of
O2 vanish
If con versely O2 vanishes each of its conse uents
must b e z ero and hence these consequents of O are collinear
Th e vanishing of the third unaccompanied by the vanish
ing of the second of a dyadic implies one degree of nullity
Th e vanish n g of the second implies two deg ees of n ul li t
Th e o re m : Th e
i
.
.
.
y
.
.
.
,
,
.
x
.
.
,
,
,
,
,
.
,
y
,
.
.
,
,
,
y
-
,
,
.
,
.
,
.
,
q
,
.
,
i
,
r
y
.
.
L INEAR VECTOR FUNC TI ONS
i i
y
lli y
van sh ng of the dyadic i t self is complete nu t
res ults ma be put in tab ular form
Th e
31 5
.
Th e
.
O3
O3 = O,
O3 = O,
is complete
O
0,
O2
O2 = O,
.
0,
O is p la n a r
O
0,
.
O is lin e a r
y
.
follo ws imme di ate ly that the third of an anti -self-conjugate
dyadi c vanishes ; b u t the second does not Fo r any such
d adic s planar but cann ot be linear
It
y i
.
.
N on io n Fo rm
If O
D
e te r mi n a n ts
.
conjugate of
they are ar ange
d agon al Thus
O
ul 1
ii +
a
“2 1
ji
“2 2
a 31
ki
+
i j+
a
ll
“
12
“
32
f
n ts o
a
Dya dic
ik
13
k
l
2s
kj
a
kk
33
.
has the same scalar coeffi cients as O but
s mmetricall with res pect to the main
r d y
i
I n va ri a
.
be ex pressed in nonion form
O=
Th e
1
y
,
.
O0
:
a
11
“
a
ii
a
i
j
12
la
21
a
31
i k,
32
1 k,
a
jj
“
k
j
23
a
“2 2
ki
i j+
33
kk
.
second of O may be computed Take for instance one
term
et it b e required to fi n d the coe ffi cient of i jin O2
What terms in O can ield a doub le cross product equal to
i
Th e vector product of the antecedents must b e i and
the vector product of the cons equents must b e
Hence the
antecedents must be and k ; and the conse uents k a n d i
These ter ms a re
Th e
.
L
y
j
.
j
q
“as
o
“3 1
1
m
Th e
re s u l ts
de t e r in a n ts
,
,
.
g
h o ld
k1
x
kk
o
mg
“2 1 “3 34
“ 3 1 “2 3
0
.
.
,
1
0
”
g
l y fo r de t e r i n a n t s o ft h e t h i rd o rde r
e bra
A
t
e
u
l
h
M
h
r
o
u
t
i
s
i
d
ipl
l
r
r
s
o
e
h
r
e
h
of
on
j
.
.
Th e
e
xt i
e ns o n
to
31 6
j
VECTOR A NAL YSIS
Hence the term in i in
O2
is
“ 3
2
Th
is is the rst minor of
fi
i
Th s
i
l
l
sa
“
“
m
the determi na n t
21
a
i
i
minor is tak en w th the negative sign That s the
coe ffi cient of i in O2 is what is termed the c ofa c to r of the
coeffi cient of i in the determinant Th e cofa cto r is merely
the fi rst minor taken with the positive or negative sign
accordin g as the sum of the subscripts of the term whose
fi rst minor is under consideration is even or odd
Th e c o
fi c ie n t of any dyad in O2 is easily seen t o be the cofactor of
ef
the correspon di ng term in O Th e cofacto s a re denote d
generally by large letters
j
j
.
,
.
.
r
.
.
a
a
i
r
rs
the cofacto of a
is
the cofactor of a
is the cofactor of
“
“
a
21
i
With th s notat on the second of O becomes
A 2 1 ji + A 2 2
Aal k i
Th e
value of the third of
as the sum of thre e dya ds
O
“
21
j
“
31
“4
“
.
A 32 k j
O
Az s k k
A3 3 k k
.
y ri i
may be obta ined b w t ng
19 1 4
"
22 l
“3 2
k“
2a j
“s a
l0k
“
“
O
VECTOR ANALYSIS
31 8
Ifthe
determinant be denote d by D
A2 1
A
11
_
If T is
A31
“
77
D
A1 2
A2 2
D
D
A1 3
A2 3
1
1)
0
A 82
'
k
°
’k
D
88
y
a second d adic given in nonion form
T = bu i i + bu i
j
as
bl s i k,
”2 3 311 ,
ba s k k ,
b,, k j
ba l k i
y dily be fo nd
the product O T of the two dyadics ma rea
by actuall performin g the mul tiplication
y
T
T
”1 1
“
“
13
1
“
“
22
“
23
”
“3 3
”1 3
”2 1
“2 3
”3 2 ) j
j
”1 1 1
“
“
32
”2 1
“
33
11
”1 1
”2 3
“2
”1 2
2
”2 3
“
12
“ 3
2
”2 3
33
”1 2
“1 3
”2 1
“
22
”2 2
“2 3
”2 3
“3
”3 1
“3 2
”3 2
“
”3 3
Since
33
”3 3
2
”2 2
>k 11
”1 3
“2 1
1
”2 2
”3 3” k
“3
“
”2 2
>1 k
“2 2
”1 2
“
32
”3 3
13
”3 011 1 4
“
“1 2
“
”3 1 >
l1
”1 2
“
”1 2
“1 2
”1 3
”3 2 ) k l
$1 ?
”3 1 ) 1 1
“1 3
”3 2 ) 1 j
”1 1
”
12
”2 1
u
:
the third or determinant of a product is equal to the
product of the determinants the law of multiplication of
de t erminants follows from ( 6 5) and
,
L INEAR VECTOR FUNCTI ONS
“
”1 1
“
”2 1
“
”3 1
“
”1 1
“
”2 1
“
”3 1
“
”1 1
“
”2 1
“3
”3 1
11
21
31
“
11
”1 2
“
”22
“
“
21
”1 2
“
22
”22
“3 1
”1 2
“
32
”2 2
12
31 9
12
22
32
13
23
13
”3 2
“
11
”1 3 1
“
12
”2 3
“
13
”3 3 :
“
23
”3 2
“
21
”1 3
“ 2
2
”2 3
“ 3
2
”3 3 ,
“
33
”3 2
“
”1 3
“
”2 3
“
”3 3
31
"
“
32
33
3
6
7
( )
°
rule may be stated in words To multiply two deter
minants form t h e determi nant of which the element in the
wit h ro w and n t h column is t h e sum of the produc t s of t h e
elements in the m t h row of the fi rst determinant and n t h
column of the secon d
Th e
.
.
If
O2 = b x
l >
<m
e
.
Then
c
Hence
Hence
I O2 !
IO 2
(
x
a
[
2 1 93
A
A
A
A
A
A
A
A
A 33
a
a
b
x b]
n
m
x
[
l
m
n]
[
2
11
O3
“
.
determinant of the cofactors of a given determinant of
t h e third order is equal to the square of the given determinant
1 22 ]
that is
A dyadic O has thr ee scalar invarian t s
three scalar quanti t ies which are independent of the form in
which O is e x press e d These are
Th e
.
.
T3
,
”a ,
the scalar of O the scalar of t h e second of O and the thi rd
or determinant of O If O b e e xpressed in nonion form these
quantities are
,
,
.
VE C TOR
3 20
$8
A
NAL YSIS
11
23
A 33
A 22
A1 1
3
“
“2 2
“
“
13
12
“
23
22
“
“ 2
3
33
r
No
matter in terms of what right-ha nded ectangular system
of these unit vectors O may b e e xpress ed these quantities are
the same Th e scalar of O is the sum of the three coeffi cients
in the main di agonal Th e scalar of the second of O is t h e
sum of the fi rst minors or cofactors of the term s in t h e
main diagonal Th e third of O is the de t erminant of the
coe ffi cient s These three invariants are b far the mos t
important that a dyadi c O possesses
The o re m : A n y dyadic satis fi es a cub ic equa t ion of whi ch
t h e three invariants OS O2 3 O8 are the coeffi cients
.
.
y
.
.
B y (6 8)
(
.
,
,
O
(
O
A
DC
O
(
:
“
“
3 “
12
13
— 27 “
22
23
“
“
“
Hence
(
O— c
)3
= Os
— cc
Ow
9”
33
32
-a
—
l
'
2
— az 3
as may b e seen by actually p e rfo rm in g t h e e xpansion
'
(
O
— T
—
I) 2 O
(
x
l)0
= Os
— x
.
— a s
S
O2 5 + s
This
equation is an identi t y holding for a ll values of the
scalar ac It therefore holds if in place of th e scalar a the
dyadic O which depends upon nine scalars be sub stituted
That is
3
2
O
=
O
l )a ( O
O I)3
O
I O3
O Oz s i O OS
(
'
.
,
,
.
o
B
ut
.
the te rms upon the left are identically z ero
$
2
$2 8 O
$8 1 = O
°
.
Hence
3 22
VECTOR A NAL YSIS
p
dyadics are equal when they are equal as o erato rs
upon all vectors or upon three non-copl a nar vec to rs That
is when
T r for all values or for three non
O r
coplanar values of r
r T for all values or for three non
O
r
coplanar values of r
T r for all values or for thre e non
O r
s
s
coplanar values of r and 3
Two
.
,
,
,
.
y
A n y li near
y
i
vector function ma be represented b a dyad c
Dyads obey the dis t ributi ve law of multipl ication w th
regard to the two vectors composing the dyad
+
i
.
l +
c
O
0
i
Multiplication by a scalar is associati ve In v rt u e of these
two laws a dyadic may be ex panded into a sum of nine terms
by mean s of the fundamental dyads
.
,
O=
a
_
a
u
i i,
ij
i k,
,
13
ii
k i,
k
ii +
ki +
a
a
1k
j
,
.
k k,
m i j+
32
a
k j+
13
a
i lg
33
kk
.
If two
dyadi cs are equal the corresponding coe ffi cients in
the i r expans ions into nonion form are equal and conversely
.
L INEA R VECTOR F UNCTI ONS
3 23
dya dic may b e e x pressed as the sum of three dyads of
whi c h the an t ecedents or the consequen t s are any t hree
given non -coplanar vectors This e x pression of the dyadic is
uni que
Th e symbolic product a b kn own as a dyad is the most
general produc t of t wo vectors in whi ch multiplication b y a
scalar is associative It is called t h e indeterminate product
Th e product imposes fi ve condi t ions upon t h e vectors a and
Their directions and the product of their lengths are
b
determined by the product Th e scalar and vector products
are functions of the indeterminate product A scalar and
a vector may be obtained from any dyad c b y inserting a dot
and a cross b etween the vectors in each d ad This scalar
and vector are functions of t h e dya di c
An y
.
.
.
.
.
.
i
.
y
.
.
direct product of two dyads is the dyad whose ante
cedent and cons equent are respectively the ant ecedent of the
b
t
rst
dyad
and
the
consequen
of
the
second
multiplied
fi
y
the scalar produc t of the conse uent of the fi rst dyad and
the antecedent of the second
Th e
q
.
(
a
b)
0
(
e
d)
(b
o
c
)
direct product of t wo dyadics is the formal e x pansion
i
c
ord
ng
to
d
stri
utive
law
of
the
product
nto the
b
i
t
h
e
a c
Th e
i
,
,
VECTOR A NAL YSIS
3 24
sum of products of dyads Direct multi p lication of dyadics
or of dyadics and a vector at ei t her end or at both ends ob e s
the distri b utive and associative laws of multiplicat ion Co n
sequently such e xpressions as
y
.
.
w wf
0
0
SO
,
Q
‘
wr
‘
,
y
written without parentheses ; for parentheses ma
b e ins erted at pleasure without altering the value of the
product In case the vector occurs at other positions than
at the end t h e product is no longer associative
Th e skew product of a dyad and a vector ma be de fi ned
b the equation
r
a
b
r
b
x
a
X
( )
ma y b e
.
y
.
y
,
2
8
( )
i q
skew product of a dyadic and a vector s e ual to the
formal expansion of that product into a sum of products of
dyads and that vector Th e statement made concerning t h e
associative law for direct products holds when the vector is
connected with the d adi cs in skew multiplication Th e
e xpressions
Th e
.
y
-T
e
,
O
o
T x r,
.
e
o
i
s,
r
-O
x
s,
e
x s
( 29 )
y
may b e written w thout parentheses and parentheses ma b e
inserted at pleasure without altering the value of the product
Moreover
-O
.
,
O
o
(
r
T
x
.
the parentheses cannot be omitted
Th e necessar and suffi cient condition that a d ad c ma
be reduced to the sum of two dyads or to a single dyad or
to z ero is that when e x pressed as the sum of three
dyads of which the antecedents ( or consequents ) are known
B
ut
y
.
,
y i
y
VECTOR ANA L YSIS
326
Ifthe
product of t wo complete dyadics is equal to the idem
factor the dyadics are commutative and either is called
the reciprocal of the other A complete dyadic may be
canceled from either end of a product of dyadics and vectors
as in ordinar algeb ra ; for the cancelation is equivalent to
mul t iplication b y the reciprocal of that dyadic Incomplete
dyadi cs pos s ess no recipro c als They correspond to z ero in
ordinary algeb ra Th e reciprocal of a product is equal to the
product of the reciprocals t aken in inverse order
.
y
.
.
.
.
O
(
.
51
—1
-
T
0
l
O
—l
( 3 8)
.
conjugate of a dyadic is the dya di c o b tained b y in ter
changing the order of the an t ecedents and consequents Th e
conj u gat e of a product is equal to the product of t h e con
u a t e s taken in the Opposite order
jg
Th e
.
.
4
0
( )
conjugate of the re c iprocal is equal to the reciprocal of
the conjugate A dyadi c may b e divided in one and o nl y
one way into the sum of two parts of which one is self
conjugate and the other anti self—conjugate
Th e
.
-
.
—
a
>
3
4
( )
.
dyadi c or t he anti -self-c onjugate
part of any dyadic used in direct multipli cation is equivalent
to minus one-half the vector of tha t dyadi c used in skew
m u ltiplication
O x r
r
An y
i
ant self-conjugate
,
,
,
)(
r
$0)
A
I
X $10
4
4
( )
dyadi c of the form c x I or I x c is anti-self-conjugate and
used in direct multiplication is equiva ent to the vector 0
u sed in s kew m ul tiplication
l
.
L INEA R VECTOR FUNCTI ON S
(
c
x I)
3 27
r,
a
-O
.
x
dya di c c x I or I c where c is a unit vec t or is a quad
rantal versor for vec t o rs perpendi cular to c and an annih ilato r
for vectors parallel to c Th e dyadic I c c c is a true
quadr an t al versor for all vectors Th e powers of these dyadics
behave like t h e powers of t h e imaginary unit V l l as may
Applied to the
b e seen from the geome t ric interpretation
unit vectors i k
Th e
,
x
.
.
—
,
,
j
.
,
i x I = k j—
I >
< i
Th e
vector
(a
b) x
x
jk ,
e tc
(
.
49)
in skew multiplication is equivalen t
I in direct multiplication
3
x
b
to
.
— a
(
r
a
— a
x b)
(
x
a
x b)
r
-(b
a —
b)
a
b
0
5
( )
-r
1
5
( )
b)
.
complete dyadi c may b e reduced to a sum of three
dyads of whi ch the anteceden t s among t hemselve s and t h e
consequents among t hem s elves each form a right -handed
rectangular system of three unit vectors and of whi ch the
scalar c o e flic ie n ts are all positive or all negat ive
A
.
3
5
( )
A
n in c o m
s
is
called
the
normal
form
of
the
dyadic
Thi
u t one or more of
b
dyad
c
may
reduced
to
th
s
form
i
b
e
i
l
t
e
e
p
the c o e fli c ie n t s are z ero Th e reduc t ion is unique in case
the constants a b c are difi e re n t In case t hey are not
ferent the reduc t ion may b e accomplished in more than
di f
one wa y An y self-conjugate dyadic may b e reduced to
the normal form
.
.
,
.
,
.
in whi ch the consta nt s
a
,
b,
c
y
55
( )
are not necessaril positive
.
VEC TOR A NAL YSIS
3 28
i
yd
double do t and doub le cross mul t iplicat on of d
is de fi ned by the equations
Th e
a
a
b
b i
d=
z c
c
d=
a
a
>
<
b
c
o
o
s
( 56 )
d,
5
( 7)
b x d
c
a
.
double dot and doub le cross multiplication of dyadics
is o b tained b y e xpanding the product formally accordi ng to
Th e
t h e di stri b utive law into a sum of products of dyads
doub le dot and doub le cross multiplication of dyadics is com
mutative but not associative
O n e -half the doub le cross product of a d ad c O b i t sel f
is called the second of O If
Th e
,
.
,
y i
.
y
.
O2
l xm
.
(
61 )
O n e -third
of the double dot product of the second of O a n d O
is called the th ird of O and is equal to the product of the
scalar triple product of the antecedents of O an the scalar
triple product of the cons equent of O
d
.
é
O; O
O8
:
O=
[
a
b
c
]
m
l
[
n
]
.
second of the conjugate is the conjugate of the second
Th e third of the conjugate is equal to the third of t h e
original dyadic Th e second and third of the reciprocal are
the reciprocals of the second and third of the second and
third of a dyadic Th e second and third of a product are the
roducts of the seconds and thi rds
Th e
.
.
p
.
.
T
( o le
0
(
92
16
(
r
$3 2
so
2
¢2
.
r.
a
.
VECTOR A NAL YSIS
330
Prove the statements made in Art 1 06 and t h e con
verse of the statements
T Q then
Show tha t if 9 is complete and f O Q
7
G ive the proof by means of theor
O and T are equal
developed prior to A rt 1 09
T O t hat
D e n i t io n : Two dyadics such that O T
8
is t o say t wo dyadics that are commutative — are said to b e
Sh ow tha t if any num b er of dyadi cs are h o m o ge
ho m o lo go u s
neons to one another any other dyadi cs whi ch may b e ob t ained
from them b y addit ion sub traction and direct mul tiplication
are homologous to each other and to the given dyadi cs Show
al so that the reciprocals of homologous dya di cs are homolo
—1
1
O
gous J
ustify the statement that if O T or T
which are equal be called the quotient of O by T then the
rules governing addition sub t raction multiplication and
divi s io n of h o m o lo go u s dyadics are identical with the r ul es
governi ng these opera t ions in ordinar algeb ra — it being
understood that incomplete dyadics are analogous to z ero
and the idemfactor to unity Hence the algeb ra and higher
analysis of h o m o lo go u s dya di cs is practicall identical with
that of scalar quantities
Show tha t ( I x c ) O = c >
9
< T and ( c x I) O = e x O
10
Show that whether or not a b c be coplanar
6
.
.
i
.
.
'
y
.
.
fi
.
.
.
:
,
.
,
,
,
.
‘
o
.
,
,
,
,
,
y
,
y
.
,
.
o
.
o
.
,
.
,
and
are copl a nar use the ab ove relation to prove
the law of sines for the triangle and to ob t ain the relation
with sc a lar coeffi cients which e xists between three coplanar
vectors This may b e done b y multipl ing the e uation b a
u nit normal to the plane of a b and c
11
.
If
a,
b,
c
y
.
,
12
.
,
q
y
.
What is the condition which mus t sub sist between the
i
coe ffi cients in the e xpansion of a dyadic into nonion form f
L INEAR VECTOR FUNCTIONS
331
the dyadic b e self-conjuga te ? What if the dyadi c be anti
self-conjugate
,
Prove the statements made in Art 1 1 6 concerning the
numb er of ways in which a dyadic may be reduced to its
normal form
13
.
.
.
necessary and s uffi cient condition that an anti
self-conjugate dyadic O b e z ero is t hat t h e vector of the
dyadic Shall b e z ero
14
.
Th e
y
.
15
.
Show
that if
O
be an dyadic the product
O
O0
is
self-conjugate
Show how to make use of the relation O
0 to
16
demonstrate that the antecedents and consequents of a self
conjugat e dyadic are the same ( Art
.
x
.
.
17
.
Show
that
(172
i O2
O3 2 O
and
y
that if the double dot product O O of a d adic
by its elf vanishes the dyadi c vanishes Hence ob tain the
condi t ion for a linear dyadic in t h e form O2 O2 O
-OZ f
1 9 Sh o w t h a t
18
.
Show
,
.
.
.
.
20
.
T)3 = O3 + (172
Sh o w t h a t
T + O : T2 + T3
:
that the scalar of a product of dyadics is
cha nged by cyclic pe rmutation o f the dyadi cs That is
21
.
Sho w
.
(
O
m a
fi
?
a
O
w
a
r
-
(
r
o
s
.
un
C
H
AP TER VI
R OTA TIO NS AN D STRAIN S
y
the foregoing chapter the a n a ly tic a l theor of
dyadics has b een dealt wi t h and b rought to a state of
completeness which is nearly fi nal for practical purposes
There are however a numb er of new questions which presen t
themselves and some old ques t ions which present t hemselves
under a new form when t h e dyadic is applied to physics
or geome t ry Moreover it was for the sake of t h e applica
tions of dyadics that the t heor of them was developed It is
then t h e o b ject of the present chapter to supply an e x tended
application of dyadics to the theory of rotations and strains
and to develop as far as ma appear necessary the further
analytical theory of dyadics
That the dyadic O may b e used to denote a transformation
of space has already b een men t ioned A kno wledge of the
precise nature of thi s tran s formation ho wever was not needed
at the time Con s ider r a s drawn from a fi x ed origin and r
as drawn from the same origin
et now
IN
.
,
,
y
.
.
y
,
,
.
.
,
,
.
.
r
’
’
L
,
= Or
.
This
equation therefore may be regarded as defi ning a trans
format ion o f the points P of Space Situated at the terminus of
’
r into t h e point P situated at the terminus of r Th e origin
remains fi x ed P oints in the fi nite regions of space remain in
the fi nite regions of Space A n y point upon a l ine
’
,
.
.
.
r
b e c o me s
a
nt
i
o
p
r
’
= b +
x a
= Ob +
:c
a
O-
.
VECTO R A NAL YSIS
3 34
is important to notice t hat the vector 8 denoting a plane
area is not transformed into the same vector 3 as it woul d
This is evident from the fact that in
b e if it denoted a line
t h e latter case O acts on 3 whereas in the former case O2 ac t s
upon 3
To Show that volumes are magni fi ed in the rat o of O3 to
unity choose any three vectors (1 e f whi ch dete rmi ne t h e
volume of a parallelopiped [ d e f] Ex press O wi t h the vec
tors which form the reciprocal system to d e f as consequents
It
’
.
i
.
,
,
.
,
O
c
,
.
’
f
.
i
dyadi c O changes d e f into a b c ( whi ch are d fferent
from the a b 0 a b ove unless d e f are equal to l m
Hence the volume [ d e f] is changed into the volume [a b
Th e
,
,
,
,
’
,
,
,
O3
[a
b
c
,
,
[a
b
d
e
f
O
[
] 3
]
0
,
]
[ d e f]
Hence
’
.
.
y
ratio of the vol um e [ a b c ] to [ d e f] is as O3 is to un i t
B u t the vectors (1 e f were a n y three vectors which deter
mine a parallelopiped
ence all volumes are chang ed by
the action of O in the same ratio and this ratio is as O3 is to 1
Th e
,
.
H
,
.
.
Ro ta tio n s
a
bou t
Fi xe d P o in t
a
y
Ve rs o rs
.
i
necessar and su ffi cient cond tion tha t
a dyadi c represent a ro ta tio n about some axis is that it be
reducible to the form
Th e o re m
1 2 5]
j
:
Th e
where
and i
systems of unit vectors
et
i
L
’
’
k’
,
j
,
k
dd
are two right-han e
.
xi
x
+ y j+
i
’
’
yj
z
k
z
k
’
.
1
( )
rectan gular
Hence if
R O TA TI ONS A ND
S
TRA INS
3 35
j
is reduci b le to the given form the vectors i k
are changed into t h e vec t ors i
k and any vector r is
changed fr om its po s ition relat ive t o i k into the s ame posi
k
tion relative t o i
Hence b y the transformation no
change of Shape is e ffected Th e s t rain reduces to a rota t ion
k
Conversely suppose the
w hi ch carries i k into i
b ody su ffe s no change of Shape — that is suppose it su b jected
Th e vectors i
k mus t b e carried into ano t her
t o a ro t a t ion
right-handed rectangular system of unit vectors
et these
he i
k
Th e dyadic O may therefore be reduced to the
form
O
j
’
,
j
’
r
,
j
j
,
’
’
,
j
,
’
.
.
’
,
,
j
’
’
,
j
,
.
,
,
.
’
,
,
’
,
.
’
,
L
’
.
’
fi
De
A
n i t i on :
+k k
.
y
d adi c which is reducible to the form
’
'
’
jj
i i
k k
and which consequently represents a rotation is called a
ve r s o r
.
conjugate and reciprocal of a versor are
equal and conversely if the conjugate and reciprocal of a
dyadic are equal the dyadic reduces to a versor or a versor
multiplied b the negative Sign
Th e o re m
:
Th e
,
L et
y
.
¢
=
=
i
l
l
i
1
k
k
+
+
jj
C
l
r l
O
—l
—
r
OC
.
Hence the fi rst part of the theorem is proved
second part let
O
3
1
b
j
c
k,
O
O
a a
+ b b +
c c
o
OC =
=L
.
To
prove the
VE C TOR A N AL YSIS
336
q
Hence ( Art 1 08) the antecedents a b c and t h e cons e uents
Hence ( page 8 7) they
a b 0 must b e reciprocal systems
must b e either a right -handed or a left-handed rectangu lar
system of unit vec t ors Th e left-handed system ma be
changed to a right-handed one b y p re fi xin g the negative
Then
Sign to each vector
,
.
,
,
.
,
y
.
.
_
O_
third or determinant of a versor is evidently equal t o
u nity ; that of the versor with a negative Sign to minus one
Hence the criterion for a versor may be stated in the form
Th e
.
,
O
0
O0 =
L
2
( )
inasmuch as the determinant of O is plus or minus one
if O OG = I it is only necessar to state th at if
Or
y
,
O
o
O0 = I,
O is
There
a ve rs o r
.
are two geometric interpretations of the transforma
tion due to a d adic O such that
y
$
¢3 = l ¢ l
0
O=
i
i
transformation due to O s one of rotation combined w th
’
re ection in the ori gin Th e d adi c i i + + k k causes a
rota t ion ab out a de fi nite ax is it is a versor Th e negative
Sign then reverses the direction of ever
vector in Space and
replaces each fi gure by a fi gure symmetrical to it with respect
’
’
to the origin B y reversing t h e d rections of i and the
syste m i
k still remains right-handed and rectangular
but the dyadic takes t h e form
Th e
fl
.
y
’
y
’
,
j
i
.
’
j
,
,
:
.
’
—
O
jj
’
’
k k,
VECTOR A NA L YSIS
338
—
jk)
4
( )
.
— ii
=
k
k
I
.
ij +
.
k j—
Hence
O=ii +
y
jk
q (I
c os
I x i
z
—
i i) +
.
s in
i
I
x
q
5
( )
.
i
more
generall
in
place
of
the
axis any axis denoted
If
by the unit vecto r 3 be taken as the axis of rotation and if as
b efore the angle of rotation about tha t a x is b e denoted b q
the d adic O whi ch accomplishes the rotation is
y
y
O=
a a
+
c os
,
6
( )
I
q (
how that this dyadic actually does a ccomplish the
rotation apply it to a vec t or r Th e dyad a a is an idemfactor
for a ll vec t ors parallel to a ; b u t an annihi lator for vectors
perpendicular to a Th e dyadic I — s e is an idemfac t or
for all vectors in the plane perpendic ul ar to a ; b u t an
annihilator for all vectors parallel to a Th e dyadic I x a
is a quadrantal versor ( Art 1 1 3 ) for vecto rs perpendicular
to a ; but an annihilator for vectors parallel to a If then
To
S
.
.
.
.
.
r
b e p a ra ll e l t o
a
(D .
r
a a
r
o
z
r
.
Hence O leaves unchanged all vectors ( or components of
vectors ) which are parallel to a If r is perpendic ul ar to a
.
Hence the vector r has been rotated in its plane through the
angle q If r were any vector in space its component parallel
to a s u fi e rs no change ; but its component perpendicular to a
is rotated about a through an angle of q degrees Th e whole
vector is therefore rotated about a t hrough that angle
et a be given in terms of i k as
.
.
L
,
a a
— a
2
1
-
1 1 1 a
0
0
j
.
,
0
1
a
2
°
rj + a
0
1
a
1
3
k
R O TA TI ONS A ND
S
TRA INS
3 39
jk
+
a
a
3
I x
ki +
1
= 0i i
a
M
O
{
+
a
2
a
jl
a
{a
a
ga z
{
0
+
{
a
+
{
a
{
a
a
2
3
s
z
1
(
3
1
(
l
(1
(1
2
2
a
3
—
n
a
kj
2
a
i j+
3
on
a
-
2
,
i k,
z
s
a
k k,
k,
(1
1
1
a
a
a
a
d
-c o s
— 008
3
(1
—
1
(l
—
a
(l
1
( +
q)
-d
cos
a
cos 9} i i
2)
c os
sin q } i j
q)
q)
—
a
— a
1
Sin
k
j
qj
sin q }
2
ki
kj
cos q ) + c o s q }
kk
.
( 7)
be written as in equat ion ( 4) the vecto r of
and the scalar of O m a y be foun d
If O
O
.
O8 = 1 + 2
c os
q
.
ax is of rotation i is seen to have the direction of O
the negative of t h e vector of O This is true in general
Th e direction of the ax is of rotation of any versor is the
negative of the vector of O Th e proof of this statement
An y versor O
depends on the invariant propert y of O
may be reduced to the form ( 4 ) by ta king the direction of i
Th e
x,
.
.
.
x
.
VECTOR A NAL YSIS
34 0
coincident with the direction of the a xis of rotation After
this reduction has b een made the direction of the ax is is seen
B u t O is not altered b y the
t o b e the nega t ive of O
reduction of O to any particular form — nor is the ax is of
rotat ion altered b y such a reduction Hence the direction of
O the dire c
t h e a x is of rotation is always coincident with
tion of the negative of the vecto of O
Th e tangent of one-half the angle of version q is
.
x
x
.
.
r
sin
1 +
.
q
c os
-OX
x/ Ox
( 8)
1 + OS
q
tangent of one half the angle of version is therefore
determined when the values of O and O , are known Th e
vector O and the scalar OS which are invariants of O deter
mine completely the versor O
et
be a vector drawn
in t h e di rection of the a x is of rotation
et the magni t ude
of
be equal to the tangent of one -half the angle q of
version
Th e
-
.
,
x
,
.
L Q
,
.
Q
.
L
.
.
Q
;
1 +O
Q
o
ta
n
z
éq
.
y
vector determines the versor O completel
Q will b e
call ed t h e ve c to r s e m i t a n ge n t of ve rs io n
B y ( 6 ) a versor O was e xpressed in terms of a unit vector
parallel to the ax is of rotation
Th e
.
.
-
.
.
O=
+
a a
c os
q (I
— a a
)
+
s in
l
q
x
a
.
Hence if G be the vector semi-tangent of version
Q Q
.
+
c os
+
q
a
si
There
>
<
Q
V e -a
1
0
( )
is a more compact e xpression for a versor O in terms
of the vector semi tangent of version
et 0 be any vector in
Space
Th e version represented by Q carri es
-
.
L
.
.
c
—
Q X
c
in t o
c i
Q x
c
.
VE C TOR ANAL YSIS
342
and
c
c
-( I
M ul t ip ly b y
c
I x 0)
—
H ence
th e
-1
.
i
(c
dyad c
Q
carr ies the vector 0 Q x 0 into the vector 0
x o no matte r
what the value of 0 Hence the d adic O determines the
version due to the vector semi-tangent of version
carries the vector 0 — 0 0 into
Th e dyadic I + I X
c
I
+
Q
Q
(
)
y
.
Q
Q
.
.
x
.
—
y
<c
Q >
c
(
Hence the d adic
I+ I x Q
1 + Q
I
I
(
Q
x Q)
“
carries the vector 0 Q x 0 into the vector 0 if c be p e rp e n
dic u l a r to Q as has been supposed
C onse uentl the d ad c
.
1 + Q Q
q
,
y
y i
°
r
di
produces a rotation of all vecto s in the plane perpen cul ar
If however it b e applied to a vector x Q parallel to Q
to
the resul t is not equal to x Q
Q
.
,
,
.
R OTA TI ONS AND
i
S
TRA INS
Q
34 3
ob viate th s di ffi culty the dyad Q whi ch is an annihilator
for all vectors perpendicular to
may b e added to the nu
m e ra t o r
Th e versor O may then be written
To
Q
.
,
.
1 + Q
a
+ l
( 1 0)
Q
)
QQ
( I x Q)
Hence substitut
QQI
—
.
in g
(
Th
1
—
e
is may be e panded
x
1 + Q
noni on form
in
Q
(1 + a
2—
b
a
bi
i
]
0 k.
2
—
+b
(1
c
1
1 28
If a
L et
.
2
— a 2
+ (2 a
Q
a
2
b
— a 2—
2
2 a )ik
b +
2
0
(1 1 )
) kk
“
c
2
y
is a uni t vector 3 d adic of the form
O
2
( 1 2)
I
a a
a biq u a dra n ta l versor Tha t is t h e dyadi c O turns t h e
points of space ab out the axis a through two right angles
This may be seen by setting q equal to 71 in the general
e xp ession for a versor
is
.
,
.
r
O=
a a
+
c os
q (I
— a s
)
or it may be seen directly from geometrical cons iderations
Th e dyadic O leaves a vector parallel to a unchanged but re
verses ever vector perpendicular to a in direction
Th e o re m : Th e product of two biquadrantal ve rs o rs is a
ve sor the ax s of whi ch is perpend cular to the axes of the
.
y
r
.
i
i
VECTOR A NAL YSIS
344
b iquadrantal
and the an gle of which is twice the
angle from the axis of the second to the ax is of the fi rst
et a and b be the ax es of two bi uadran tal ve rs o rs Th e
product
ve rs o rs
L
q
9
y
— I
2
b
b
(
)
:
o
.
.
s e —
2
(
I)
y
is certainl a versor ; for the product of an tw o ve rs o rs
is a versor Co nsider the common perpen di c ul ar to a and b
—
2
I reverses this perpendi cular
a
iqua
rantal
versor
a
d
b
Th e
in direction ( 2 b b — I) again reverses it in direction and con
se q uently b rings it b ack to its origin al position Hence the
produc t 9 leaves th e common perpendicula r to a and b u n
changed 9 is therefore a rotation about this line as axis
.
.
.
.
.
Qa =
.
Th e
a
.
2
b
b
(
—
I)
o
2
(
cosine of the an gle from
-Q
o
e
2 b
z
-a
b
o
I)
—
3 3
a — a
o
a
to
a
Q
.
o
= 2 (b
a
o
a
o
= 2 bb
o
a — a.
is
a
)
2 —
1 =
cos
Hence the angle of the versor Q is equal to twice the angle
from a to b
Th e o re m : Conversel any given versor may be ex pressed
as the product of two biquadrantal ve rs o rs of which the ax es
lie in the pla ne perpendicular to the ax is of the given versor
and include between them an angle equal to one half the
angle of the given versor
Fo r let I? b e the given versor
et a and b be unit vectors
perpendi cular to the ax is — Q of thi s versor Furthermore
let the angle from a to b b e equal to one ha lf the angle of
this versor Then b y the foregoing theorem
.
y
.
,
L
.
.
.
x
.
.
O=
( 1 4)
2
( bb
resolution of ve rs o rs into the product of two b iq u a d
rantal ve rs o rs a fi o rds an immediate and Simple method for
compounding two fi nite rotations about a fi x ed point
et
O and T be two given ve rs o rs
et b b e a unit vector per
Th e
.
L
.
L
VECTOR A NA L YSIS
34 6
O8 = 4
T= 4
c
h
o
c
(
T
o
O=4
c
T
(
o
4
z
a
o
h)
c
b
o
2
(o b)
—
c a
O) x = 4
2
xb
Hence
a
o
QZ X Q I
a
b
2
—
x
c
)
1
z —
a
1
+ I,
a a
,
.
c
’
b
-b
'
'
B
h
b x
Q2
’
—
c c
e
+ I,
x b,
-a
c
1,
— 2 0c
c
o
a
2 —
2 b b
—
b
Tx = 4
Ts
a
o
c
c
[
-c
a
a
o
b
c
]
b b
b
-c
°
ut
b
Hence
.
a
< Q1
Q2 >
Hence
Ql
0l
,
Q2
(
O
b b
Hence
Hence
o
c
1
QI
e
-b b
a
o
o
c
e
h b
o
x b)
a
+
>
<
Q,
s
a
a
-b
a
Q2 x
a
xb
e
a
.
b b
.
c
a
.
c
a
o
b b
o
c
a
.
b b
Q2
,
QI
Q" A T Q 2 1 11
'
‘
.
1
b
.
.
b
c
ROTA TI ONS
AN D S
TRA INS
34 7
This
form ul a gives the composition of two fi nite rotations
are b o t h in fi n it e s i
If the rotations b e in fi nitesim a l Q 1 and
mal N eglecti ng in fi n it e s im a ls of the second order the for
mula reduces to
.
.
Q3
QI i Q2
‘
"
.
i
in fi nitesimal rotations comb ine accord ng to the law of
vector addi tion This demonstrates the paral lelogram law for
angul ar velocities Th e subject was treated from di fferent
standpoints in Arts 51 and 6 0
Th e
.
.
.
.
Cyc li c s Righ t Te n s o rs , To n ic s ,
,
If
the dyadi c
O
a nd
Cy c lo to n i c s
y
be a versor it ma be written in the
form ( 4 )
—
k
q ( j
s in
+
k
j)
.
axis of rotation is i and the angle of rotation about th at
ax is is q
et T b e another versor with the same ax is and
’
an angle of rotati on e ual to q
Th e
L
.
T
y
:
q
ii +
c oS
q
’
.
ji
C
+ k k) +
s in
q
’
—
k
( j
jk )
.
M u ltipl ing
O
T
:
T
O
:
ii
(jj+
c os
k k)
jk )
( 1 6)
>
This is the res ul t which was to be e x pected
the product of
two ve rs o rs of whi ch the ax es are coincident is a versor with
the same ax is and with an an gle equal to the sum of the
angles of the two gi ven ve rs o rs
If a versor b e multiplied by itself geometric and anal tic
cons iderations alike make it evident that
sin ( c
'
k
( :
+g
—
.
y
.
,
—
and
s in n
—
k
q ( j
jk)
jk)
,
.
VECTOR A NAL YSIS
34 8
O
k
j
n
jj
the o t her hand let Ol equal
—
T
k
hen
j
O
i
i
cos
sin
O
q I
(
and
kk;
q
e ual
O2
.
”
q
product of i i into either OI or O2 is z ero and into itself is
Hence
O
rn
O
cos
s
O
ii
q 2)
q I
(
1
l
O
O
n cos
O
sin
c os
O
ii
q
q
q ,
l
,
Th e
ii
.
”
"
”
:
:
”
”
”
fl-
"
dyadic OI raised to any po wer reproduces itself OI OI
Th e dyadic O2 raised to the second power gives the negative
of OI ; raised t o the third power t h e negative of O2 ; raised
to the fourth power OI ; raised to the fi ft h power O2 and so
on ( Art
Th e dyadic OI multiplied by O2 is e ual to
Hence
O2
Th e
“
.
.
,
q
,
,
.
.
O =ii +
”
B
cos
q O1 +
"
n c os
"
‘
1
q
a
si
2
ut
E quating
for
O
and
OI
O2
in these two e xpressions
"
c os n
s rn n
coe ffi cients of
q
=
n
q
—
c os
n
cos
”
‘
1
q
(
2!
n
q
s rn
q
y
Thus
l
n —
(
n —
) cos
1
—
2
the ordinar expansions for cos n q and Sin n q are
ob tained in a manner ver Similar to the manner in which
they are generally obtained
Th e e xpression for a versor may be generali z ed as follows
et a b c b e any three non -coplanar vectors and a b
the
reciprocal system Consider the dyadic
y
.
L
.
,
’
,
,
’
,
.
O
:
a a
’
+
cos
’
q (b b +
c c
’
)
+
s in
q
(c
b
’
1
7
( )
VECTOR A NAL YSIS
3 50
a displacement of t h e
radius vector r
an e llip tic ro ta tio n th r ough a
sector q from its si m ilarity to an ordi nar rotation of which
it is the projection
De n i tio n A dya di c O of the form
th e who le
e llip s e
Such
1
i
s to 2 7 r
q
may b e called
a s
.
fi
y
.
O
a a
:
’
+
cos
’
q (b b +
c c
sin
’
)
(
g
c
b
’
—
b
c
’
1
( 7)
)
is called a c yc lic dya di c Th e versor is a Special case of a
cyclic dyadic
It is evident from geometric or analytic considerations that
the po wers of a cyclic dyadic are formed as the powers of a
versor were formed b y mul t ipl i ng the scalar q by the power
to which the dyadic is to be raised
.
.
y
,
,
.
O
If the
"
a a
cos
’
n
q
(b b
’
c c
’
)
sin n q ( c
scalar q is an integral sub-m ul tiple of
2
2
b
7r,
’
,
'
71
9
i
i
th at is f
p
it s possible to raise the dyadic O to such an integral ower
namely the power m that it becomes the idemfactor
,
,
,
O
m
=I
may then be regarded as the m t h root of the idemfactor
In like manner if q and 2 n are commensurab le it is possi b le
to raise O t o such a power that it becomes equal t o t h e idem
factor and even if q and 2 are incommensurab le a po wer of
O may be found which difi e rs by as little as one pleases from
t h e idemfac t or
Hence any cyclic dyadic ma be regarded as
a root of t h e idemfactor
O
.
71
y
.
g
.
i d e n t t h a t fix i n t h e re s u l t o ft h e a ppli c a t i o n o f T t o a ll ra dn ve c t o rs
in a n e llip s e p ra c t i c a l l y fi x e s it fo r a ll ve c t o rs in t h e pl a n e o f b a n d c
Fo r a n y
ve c t o r in t h a t pla n e m a
u l t ipl e o f a ra di us ve c t o r o f
y b e re a rd e d a s a s c a l a r
t h e e l lips e
1
It is
ev
.
g
m
.
fi
De
R O TA TI ONS A ND
n it i o n :
O=
w here
transform
Th e
a
ii + b
S TRAINS
ation rep resented
jj+
c
the
by
( 1 8)
kk
are posi t ive scal ars is called a p u re s tra in
dyadic itself is called a right t e n s o r
A right te n sor may b e facto red into three facto rs
b,
a ,
3 51
c
Th e
.
.
b
k
k
i
i
i
i
c
+
+
j
+
j
j
j
j
j
)
(
(
Th e o rder in which these fa ctors occur is immate ial
transformation
o
z
a
.
r
(i i +
jj+
c
k k)
o
k k)
.
.
Th e
r
is such that the i and jcomponents of a vecto r remain u n
altered but t h e k-c omponent is altered in the rati o of c to 1
Th e transforma t ion may t herefore b e described a s a s t retc h or
elongation along t h e dir ec t ion k If the cons t ant c is greater
t han uni t the elongation is a tru e elonga t ion : b u t if c is less
t han unit the elongati on is really a compression for the ra t io
of elongation is less t han uni t y B etween these two cases
comes the case in which t h e constant is uni t
Th e lengths
of the k-c omponents are then not a ltered
Th e transformation due t o t h e dyadi c O may b e regarded
the successive or simultan eous elongation of t h e com
as
n ts of r parallel to i
and
k
respectively
in
the
ratios
n
e
o
p
a to 1
to 1 c to 1 If one or more of the cons t an ts a b c
is less than unit t h e elongation in tha t or t hose di rections
If one or more of the constants is
b ecomes a compression
uni t y components parallel to that di rection are not altered
Th e di rections i
k are called t h e p ri n c ip a l a xe s of the s tra in
Their directions are n o t altered b y t h e stra in whereas if the
cons t an t s a b c be different ever other direction is al t ered
Th e s c alars a b c are known as t h e p ri n c ip a l r a tio s of
.
'
y
y
.
,
y
.
.
.
,
b
,
y
,
j
,
,
.
,
.
,
,
,
tio n
,
,
.
,
,
e l o n ga
j
.
y
,
,
.
it wa s seen that any complete dya di c
reducib le to the normal form
In Art
.
.
115
wa s
SS
VECTOR A NAL Y I
3 52
y
where a b c are positive constants Thi s expression ma be
factored in to the product of two dyadi cs
,
.
,
.
o
(
:t
:
a
’
’
'
i i + b jj+
’
a
1
9
( )
)
+ k k)
factor
Th e
k k
’
'
i
:
c
'
i i
’
+
’
'
k k
jj
which is the same in ei t her metho d of factoring is a versor
k
T
i
turns
the
vectors
i
k
into
the
vectors
his
j
It
versor may b e represented b y i ts vector semi-tangent of
,
j
.
'
'
,
,
,
'
.
ve rs io n a s
1
1
+
JJ+
’
k k
1 +i
o
1
’
’
+ JJ+ k
o
k
other factor
Th e
a
ii + bj
j+
c
kk
is a right tensor and represents a pure strain In the fi rst
'
case the strain has t h e lines i j k for principal ax es : in
the second i k In b oth cases the ratios of elongat ion are
the same — a to 1 to 1 c to 1 If the negative sign occ u rs
b efore the product the version and pure strain mus t have
associated with them a reversal of direc t ions of all vectors in
space — t hat is a perversion Hence
The o re m A n y dyadic is reduci b le to the product of a
versor and a right tensor taken in either order and a pos itive
or negative sign Hence the most general transformat ion
representab le b y a dyadic consists of the product of a rota
tion or version ab out a de fi nite ax is through a de fi nite angle
accompanied b y a pure strain either with or without pe rver
sion Th e rotation and strain may be performed in either
order In the two cases the rotation and the rat ios of elonga
t ion of t h e s t rain are the same ; b u t the principal a x es of t h e
strain differ according as it is performed before or after the
.
,
,
j
,
.
,
,
,
,
.
b
.
,
.
.
.
’
’
,
3 54
A NA L YSIS
VE C TOR
b
componen ts are stretched in the rati os a to 1 to 1 c t o 1
If one or more of t h e cons t ants a b c are negative the com
b
c are re
n e n t s parallel to the correspon di ng vector a
o
p
versed in direction as well as changed in magnitude Th e
t onic may b e fac t ored into thr ee factors of which each
s t re t che s t h e components parallel to one of the vectors a b c
b u t leaves unchanged the components p a rallel to the other
,
,
.
'
,
,
,
,
.
,
t wo
,
.
value of a ton ic O is not altered if in place of a b c
any three vec t ors respectively collinear wi th the m b e s u b
sti t uted provided of course that the corresponding changes
which are nece s sar b e made in t h e reciprocal system a b c
B u t with the e x ception of this change a dyadic which is
e xpressib le in the form of a tonic is so e xpressib le in only
one wa y if the constants a b c a re different If t wo of the
constants say 6 and c are equal any two vectors coplanar
with t h e corresponding vectors b and 0 may be substituted
in place of b and c If all the consta nts are equal the tonic
reduces to a constant multiple of the idemfa ctor An y three
non -00planar vectors may b e taken for a b c
Th e product of two tonics of which the a x es a b c are the
same is commuta t ive and is a tonic wi t h these ax es and
with scalar coeffi cients equal respectively to the produc t s of
t h e correspondi ng coe f
fi c ients of the t wo dya di cs
Th e
,
,
~
y
,
’
'
,
,
,
,
.
,
,
.
.
,
.
,
,
,
.
O
Th e
o
97
:
7
11
-O =
a
1
a
2
a a
’
4 b
b
generaliz ation of t h e cyclic dyadic
(
c
b
’
—
b
c
’
)
( 23 )
’
.
ROTA TION
S AND S TRAI S
355
N
non-coplanar
where a b c are three
vectors of which a b c
is the reciprocal sys t em and where t h e quantities a b c are
positiv e or negative scalars This dyadic may be changed
into a more conve n ient form b y de t erm ining the posit ive
scalar p and the positive or negative scalar q ( which may
always b e chosen be tween the limits 1 7r) so that
,
,
’
’
,
,
,
,
’
,
.
b =p
c os
=p
s in
c
q
q
.
a nd
Then
O
a a a
:
Th
’
+ 19
cos
b
b
q (
’
c c
’
)
+p
s in
q
(
o
h
’
( 25)
is may be fact red into the product of three dya
di cs
o
O=
{
a a
(a
a a
’
cos
’
bb
’
q (b b
c c
’
’
) (
c c
’
)
a a
’
’
+p bb +p
sin q ( c b
c c
’
)
’
order of these factors is immaterial Th e fi rst is a tonic
which leaves unchanged vectors parallel to b and 0 b u t
stretches those parallel to a in the ratio of a to 1 If a is
negati ve the stretching must be accompan ed b y reversal
It leaves
Th e second factor is also a to ni c
in direction
unchanged vectors parallel to a but stretches all vectors in
the plane of b and c in the ratio p to 1 Th e th rd is a
cyclic factor Vectors parallel to a remain un changed ; but
radii vectors in the ellipse of whi ch b and c are conjugate
se m i -di ameters are rotated through a vector such that the
area of the vector is to the area of the whole ellipse as q to
O ther vectors in the plane of b and 0 may b e egarded
2 7r
adi vectors of the elli ps e
a s scalar m ul tiples of the
Th e
.
i
.
.
.
.
i
.
.
.
r i
r
.
356
Defin i tio n
O
’
e a
a
i
A
:
+ p
SS
VECTOR A NAL Y I
y
d adic whi ch is reducib le to the form
cos
b
b
q (
’
c c
’
)
sin
+p
(c b
q
’
( 2 5)
ow ng to the fact that it combines the properties of th e
c clic d adic and the tonic is called a c yc lo to n ic
Th e product of two c y c l o t o n ic s whi ch have the same three
vec t o rs a b c as antecedents and the reciprocal s stem
for con sequents is a third c cloto mic and is com
b
a
c
mu tative
y
y
.
,
’
,
’
’
y
y
,
,
,
.
=
a
=
a
’
O
:
1
2
a a
a a
ill
'
’
+p1
c os
+p2
cos
-O = a
+ 19 1
q1
q2
a a
a
+ 101 1 )s
Re du c ti o n
(b b
’
’
g1
sin
l
+p1 pz
c os
+ 92 )
(91
s in
a
di
a
D
cs
t
o
C
f y
o
( q1
(e h
q2
(
c
(c
b
’
b
’
—
—
b
c
c
)
’
)
+ q2)
( 26)
'
n o n ic a
b
’
l Fo rm s
y
general an dyadic O may be reduced
ei ther to a toni c or to a c y c l o t o n ic Th e dyadi cs for whi ch
the reducti on is impossible may b e regarded as li miting cases
which may be represented to an desired degree of a pp ro xi
mation by tonics or c y c l o t o n ic s
From th s theorem the importance of the tonic and c cl o
to nic whi ch have been treat ed as nat ural generaliz ations of
the right tensor and the cyclic dyadic ma be seen Th e
proof of the theorem including a d scus sion of all t h e
special cases that may arise is long and somewhat tedious
Th e method of proving the theorem in general however is
patent If three di rections a b c may be found wh ch a re
left unchanged by the application of O th en O must b e a
to nic If only one such directi on can be found the e e x ist s
a plane in which the vectors suffer a change such as that due
to the c y c l o to n ic and the d adi c ndeed proves to be s uch
Th e o re m
:
In
.
y
i
y
.
i
,
y
.
,
.
.
,
i
,
.
,
y
i
r
.
3 58
An y
A NAL YSIS
VE C TOR
value of
which satis fi es this equation wi l
x
that
y
That
(O
x
l )3
l
be
such
0
.
is to sa the dyadic O x 1 is planar A vector per
O
n dic u l a r to its consequents is reduced to z ero
ence
e
p
leaves such a di rection unchanged Th e further di s c u ssion
of the reduction of a d adic to the form of a tonic or a cyclo
tonic depends merely upon whether the cubic equation in se
has one or three real roots
Th e o re m If the cu b ic equation
,
.
.
y
H
.
.
x
$3
$8 4 x $2 3
2
“
1
has three r e al roots the dyadic
to a tonic
:
1
O
may in general be reduced
.
For
let
a
ac
:
be the three roots of the equation
O—
a
I,
O
—
L
=
Th e
.
b I,
O
c
y i
d ad cs
—
c
I
are in general planar
et a b c be respectively three
vectors drawn perpendicular to the planes of the consequents
of these dyadics
.
,
,
.
(O
I)
a
O,
(
(D — b I)
-b
= 0,
(
O
a
O
a
O b
O
-c
a a
b
b,
c
&
,
the roots a b c are distinct the vectors
00 planar
Fo r suppose
m a
n b
If
,
,
.
(O
— c
I)
a,
are non
R O TA TI ONS
m O
a —
O
m
m
(
.
m
a
n
a a,
N
O
o
O
o
b
b = 0
— n c
b = bb
.
.
a
(a
m = Oor
Cons e
+
c a
S TRAI S
AN D
n
(b
n
= Oor b =
a
quently if the vectors
c
.
are coplanar the roots are
not distinct ; and t herefore if the roo ts are distinc t the
vectors a b c are necessarily non-coplanar In case the roots
are n o t di s t inct it is still always possi b le t o choose three
non coplanar vectors a b c in such a manner t hat the equa
tions ( 30) hold This b eing so there e x is ts a system a b c
reciprocal t o a b c and the d adic which carries a b c into
a a
b b c c is the to n ic
b,
a,
c
,
,
,
.
,
-
,
,
y
,
,
’
,
,
.
,
’
,
’
,
,
,
O=
The o re m
If the
:
a a a
’
bbb
’
c c c
.
q
cub ic e uation
x2
y
OS +
cc
(Dz s
y
has one real root the d adic O ma in general be reduced to
a cycloto mic
Th e cub ic equa t ion has one real root
This must b e posi
tive or negative a c cording as O3 is positive or negative
et
the root be a D ete rmine a perpendicular to the plane of
the conse uents of O a I
.
L
.
.
q
.
.
(
D ete rmine at!
O
—
a
(
O
I)
o
a
= 0
.
also so tha t
a
’
—
a
I) = O
and let the lengths of a and a be so adjusted that a a l
Th s cannot be accomplished in the special case in whi ch a
i
’
'
z
.
S
L et
VECTOR A N AL Y IS
3 60
and a are mutual ly perpendi cular
the plan e perpendicular t o a
'
.
b
y
be an vector in
’
.
8”
O—
(
I)
a
b = 0
0
.
Hence (
Hence O h is
I) b is perpendic ul ar to a
2
3
m
O
perpendicular to a In a si ilar manner
b O h and
3
1
O
b etc will all be perpendicular to a and lie in
O
b
one plane Th e vectors O h and b cannot be parallel or O
wo ul d have the direction b as well as a unchanged and
thus t h e cubic woul d have more than one real root
z
Th e dyadic O changes a O h b into O a O b O b re
Th e volume of the parallelopiped
s p e c t ive ly
O— a
’
o
o
.
’
.
—
o
,
,
’
"
.
,
,
,
o
.
.
o
,
,
o
,
,
.
O
[
B
c
OZ
a
O
O
h
o
Oa
ut
Hence
a
a a
b)
a
o
b]
h
.
.
= O3
a
vectors O h O h b all lie in the same plane Their
vec t or products are parallel to a and to each other Hence
Th e
Z
o
o
,
.
,
’
a
Inasmuch
Z
O
(
as
and
a
x(
h)
o
b
1
=p
2
O
l
‘
vectors b 2
2
a
-1
bz
l
z
b3 + b1 = 2
b 1 + b _1
2
n
n
b2
b
b
x b
.
( 3 2)
.
p
z
2
and b 1 are parallel
b
O
z
‘
O
‘
Oz
2
,
n
b1
L et
.
.
b1 + b2 = 2
b -1 + b _2
—
2
o
-b
bl x bz,
b2 + b = 2
Then
O3
b _2 _ p
b
bs
Th e
= O3
Ob
1
o
b)
,
p
b3 = p
O
have the same sign let
OS
“
.
n
n
b3
b
b,
etc
etc
.
.
3
3
( )
SS
VE C TOR A NAL Y I
3 62
There
i
remain two cases in wh ch the reduction
s impossible as can b e seen b y look ng over the proof In
the fi rst place if the c o n s t a n t n used in the reduction to cyclo
tonic form b e i 1 the reduction falls through In the second
lace if the plane of the antecedents of
i
1
i
,
.
p
.
O—
q
I
a
r
l
and the plane of the conse uents a e perpendicu ar the
’
vectors a and a used in the reduction to c y c l o t o n ic form are
’
perpendicular and it is impossible to determine a such that
a
a sha l l be unity Th e reduction fall s through
’
.
.
If
= i
”
L et
b _l + b 1 = i
1,
bl
b _1
Choose
0
y
Consider the d a di c
b1
z
T=
T
'
ill
o
2
b =b
—
—
O
= O
a
a
a
.
.
(b b
p
’
b
-b
,
=
O
h
p
c c
c
’
,
O
~
a a
a
b _l
+p
—
Hence
o
b = p b i—p h 1
’
o
a
a
o
z h
“
2b
’
)
+p
-c
.
c
bl
( 3 7)
.
Th e
y
transformation due to this dyadic may be seen best b
factoring it into three factors which are independent of the
order or arrangement
O
(
:
a
a a
’
’
+ b b +
c c
’
)
(
1
In
In t h e s e
one
case
dya d i
t he s e
wh i c h t wo
mg
a nd
i
a
t wo
t wo
cs
o
c as e s
o
f th e
ro o
o ft h e ro o t s o f
be
se e n
a re
ts a
e
d
re
dis t i n c t Th e
in a ry p a rt o f t h e
.
bb
’
mg d q lil mi i g lyi g b
i m gi y
i
i
li m i
g
d
k
i g pl
i m gi y
i pp
i g
il
it w l
ma y be
ft h e
a a
’
a r e
re
t
as
a
na r
m a y be
t wo
a
c a
n
a nd
na r
h
bi c e qu a t io n
o th e r c a s e
c a se s
t h e to n
de
re
ro a c
cu
in t h e
a nd
ua
t
th e t o n
tha t th e
ar
as ta
ro o ts o
ft h e
n
C
n
c
in wh
n
c
e
h a s t h re e
t h re e
t we e n t h e
c
h
a ce
y c l o t o n ic
e a c h o the r
.
of
a
ll t h e
m
re a
t he
c
W)
l
.
ro o t s .
Th us
y c l o t o n ic in
ro o
mg
t s a re
re a
l
i th e r by t h e p u re
be c o i n z e ro o r by
e
S TRAINS
R O TA TI ONS A ND
3 63
Th e fi rst
facto r represents an e l ongation in the dir ection a in a
ratio a to 1 Th e plane of b and c is undisturbed Th e
second factor represents a stretching of the plane of b and c in
the rat o p to 1 Th e last factor takes the form
.
i
.
.
I+
1
( +
1
( +
b
c
(I
o
)
’
)
c
y
h
’
ma ,
ma
cc
b
M
c
b =
’
)
zc
b
a o
'
a s,
cc c
.
d adic of the form I o h leaves vectors parallel to a and c
unaltered A vector x b parallel to b is increased by the vec
tor c mul tiplied b y the ratio of the vector a xb to b In other
words the trans forma t ion of points in Space is such that the
plane of a and c remains fi x ed point for point b u t the points
in planes parall el to that plane are shift ed in the di rec t ion c
by an amount proportional to the di stance of the plane in
which they lie from the plane of a and 0
A dyad c reduci b le to the form
Defin itio n
A
’
.
.
i
.
I+
is called a
c
dyadi c or
transformation which it causes
general dyadic
is
l
and the geometri ca
calle a s h e a r Th e more
s h e a re r
s h e a ri n g
ll
h
’
ll
d
.
( 3 7)
i
wi al so be ca ed a shearing dyad c or s he a rer Th e trans
formation to which it gives rise is a shear combined with
elongations in the direction of a and is in the plane of b and c
— 1 instead of n
If n
+ 1 the result is much the same
Th e dyadic then b ecomes
.
"
.
.
,
O
O:
(a
a a
’
bb
:
’
a
a a
c c
’
— e h’
’
)
{a
a
’
-
p
(b
b
’
(I
SS
VECTOR A N AL Y I
364
facto rs are the same ex cept the second whi ch now re p re
sents a stretching of the plane of b and 0 comb ined with a
reversal of all the vectors in that plane Th e shearing dyadic
O then represen t s an elongation in th e direction a an elonga
tion comb ined with a reversal of direction in the plane of
b and c and a shear
Suppose that the plane of the antecedents and the plane of
the consequents of the dyadi c O — a I are perpendi c ul ar
et
these planes b e taken respectively as the plane of and k and
the plane of i and k Th e dyadic then takes t h e form
Th e
.
,
.
,
j
j
O
coeffi cient B must vanish
— a
I
—
B I=
L
.
O
Th e
.
(
Fo r
.
B i + A
—
y
other wis e the d adic
—
j
E h)
q
is planar and the scalar a B is a root of the cubic e uation
With this root the reduction to the form of a tonic may b e
carried on as b efore N othing new arises B u t if B vanishes
a new case occurs
et
.
.
This
L
.
may be reduced as follows to the form
a
where
a
-b
’
z
a
Square T
-c
’
Tz
z
b
z —
’
b + b
c
’
o
o
c
z
’
A D ki =
a nd
a c
b
’
.
H nce must be chosen parallel to k ; and
Th e dyadic T ma then be trans formed into
e
.
a
y
T=AD k
—
0i + Dj
A D
A D k,
b =A
j
c
+ A ji
’
,
.
l
parall e to
i
.
VECTOR A NAL YSIS
366
i
i
more systematic treatment of the var ous k nds
of dyadics which may arise may be given by means of the
Hami lton -Cayle equat ion
A
y
O
Os I = 0
Ts O + O 2 5 O
2
3
and the cubic equation in
x2
Os
3 —
x
x
+ Oz g
x —
i
O3 = 0
.
m
are
the
roots
of
th
s
cubic
the
Ha
ilton
Cayle
b
equation may be written as
If a
,
,
c
(
O
— a
-( O
I)
y
4
0
( )
yy
If, however,
the cubic has only one root the Hamilton -Ca le
equat ion takes the form
(O
— a
2
I) ( O
4
1
( )
general the Hamilton Cayley equation which is an equ a
tion of t h e third degree in O is the equation of lowest degree
which is satis fi ed by O In general therefore one of the above
equations and the corresponding reductions to the tonic or
c y c l o t o n ic form hold
In special cases however the dyadic
O may satisfy an equation of lower degree
That equation
of lowest degree which may b e sat is fi ed b y a dyadi c is called
its c h a ra c te ris tic equation Th e follo wing pos sibilit ies occur
In
-
.
.
,
,
.
.
—
O
(
O
(
—
a
I) O
(
a
I)
.
o
(
2 —
O
O
(
—
O
(
— a
a
b I)
2p
I)
O
(O
o
c os
I) O
(
(O
(
—
—
-( O
O
(
q
b I)
2
R OTA TI ONS AND
S
3 67
TRA INS
b
e
the
rst
c
se
the
dyadic
is
a
tonic
and
may
reduced
a
fi
In
to the form
O=
In the second case
reduced to the form
’
a a a
+ bb b
’
+
dyadi c is a
the
c c c
’
.
l
o to n i c
y
c
c
and may
In the third case t h e dya di c is a simple shearer and may
reduced to the form
O=
a a a
’
+
b(
b b
’
+
c c
’
)
+
c
be
be
’
b
.
the fourth case the dyadic is again a tonic Two of the
ratios of elongation are the same Th e following reduction
may b e accomplished in an in fi nite numbe r of ways
In
.
.
.
O=
a
a s
’
b(
+
bb
’
+
the fi fth case the dya di c is a complex shearer and may be
so e xpressed that
In
T
:
the six th case the dyadic
may be reduced to the form
In
O=
a
I +
c
b
' —
.
a
(
is
a s
'
again a simple shearer which
+ bb
'
+
c c
'
)
+
c
b
’
.
the seventh case t h e dyadic is again a tonic which ma
reduced in a doub ly infi nite numb er of ways to the form
In
These
y
be
seven are the only essentially di fferent form s which a
dyadic may take There are then only seven really different
kinds of dyadi cs three tonics in which the ratios of elonga
tion are all difi e re n t two alike or all equal and the cyclo
tonic toge t her wi t h thr ee mi t ing cas es the two simple and
the one complex shearer
.
li
,
.
,
,
,
VECTOR
3 68
Su mm a ry
r
ANA L YSIS
f
o
Ch a p te r VI
t ansformation due to a dyadic is a linear h o m o ge n e o u
strain Th e dyadic itself gives the transformation of the
points in space Th e second of the dyadic gives t h e trans
form ation of plane areas Th e third of the d ad c gives the
ratio in which volumes are changed
Th e
.
y i
.
.
.
D
l
= ¢
3
1k
y
necessary and suffi cient condition that a d adic re p re
sent a rotation about a de fi nite ax is is that it be reducible to
the form
Th e
or
tha t
or
th a t
OO0 = I
O3 = + 1
T T0 = I
T3 > 0
0
y
y
necessar and suffi cient condi tion that a d adic repr o
sent a rotation combined with a transformation of reflection
b which each fi gure is replaced by one symme t rical to it is
that
Th e
y
O=
or
’
+ k k)
OOC =
th a t
O3
.
or that
O
O0
1
—
O3
0
r
3
( )
.
dyadic of the form ( 1 ) is called a ve sor one of the form
a perversor
If the a x is of rotation of a ve sor be chosen or the i-a xis
the versor reduces to
A
r
.
O = ii +
cos
j+ k k )
q (j
O = ii +
If an
y unit vector
O=
Th e
a a
9.
q
( 4)
- k
k
q ( j j)
(I
5
( )
is directed along the ax is of rotation
+
r
cos
sin
c os
q
(I
i
axis of the ve sor coinc des in
i on wi h
di rect
t
Ox
.
37 0
VECTOR A NAL YSIS
y
duct of two c clic dyadics whi ch have the same antecedents
’
b c and consequen ts a b c is obt ained b y adding their
a
angles q A cyclic dyadi c may b e regarded as a root of the
idemfactor A dyadic reducib le to the form
’
,
’
,
.
.
l
O=
ii + bj
j+
a
c
kk
where a b c are positive scalars is called a right tensor It
represent s a stretchin g along the principal a xis i k in the
ra t io a to 1 to 1 c to 1 which are called the principal rati os
of elonga t ion Thi s transformat ion is a pure strain
A n y dyadic may b e e x pressed as the product of a versor
a right tensor and a positive o r n e ga t ive sign
,
,
,
b
,
j
.
,
,
.
.
,
,
O
i
:
(a
.
.
’
i i
’
’
’
b jj
c
’
k k
’
) (i
’
i +
’
jj k
’
k)
or
( 1 9)
Consequently
any linear homogeneous strain may be regarded
as a comb ination of a rota tion and a pure strain accompanied
or unaccompanied b y a perversion
Th e immediate generali z atio n s of the right te nsor and the
cyclic dyadic is to the tonic
.
O
a a a
b
’
and c y c l o t o n ic
bb
’
c c c
’
—
b
c
or
( 23 )
)
—
where
An y
=
+
P
Vb +
z
c
z
and
ta n
y
l q
2
—
p
—
b
c
’
) ( 2 5)
b
b
+
p
d adic in general may be reduced either to the form
and is therefore a tonic or to the form
and is
t herefore a c c l o t o n ic
Th e con di tion that a d adic be a
y
tonic is that the cubic equati on
y
,
.
x
3
«9s
+ a ,, x
a8 = o
R OTA TI ONS
AN D S
TRA INS
371
shall have three re a l roots Special cases in whi ch the
eduction may b e accompl ished in more ways than one arise
when the equation has equal roots Th e condi t ion that a
dyadi c b e a c y c l o t o n ic is that this cubic equation shall ha ve
only one real root There occur two li mitin g cases in which
the dyadi c cannot b e reduced to c y c l o to ni c form In thes e
cases it ma be written as
r
.
.
.
y
.
O =
a
a a
’
+
+ p
c
b
’
and is a simple shearer or it takes the form
,
3
8
( )
y l
and is a comple x shearer Dyadics ma be c a ssi fi ed accord
in g to their characte ri stic equa t ions
.
O
(
(O
—
a
I)
— a
l)
b I)
(
o
O
— c
I) = O
(O
y
O
(
I)
a
O—
a
(
I)
O
(
b
O
-( O —
a
0
simple shearer
bl) = 0
special tonic
I)
2
I) 3
0
complex shearer
2
0
special simple shearer
(O
a
I)
O
a
I)
(
tonic
c cloto mic
2
g
(
-( O —
0
pecial tonic
S
.
C
H
A P TER VII
M ISCELL AN E OU S A PPLI CA TIO NS
If O
Qu a dric Su rfa c e s
y
q i
be an constant dya di c the e uat on
r
O
r
const
.
y
is quadratic in r Th e constant in case it be not z ero ma
be divided into the dyadic O and hence the e uation takes
the form
1
O r
r
r O 2
r = 0
( )
q
,
.
,
,
.
y
dyadi c O ma be assumed to be self-conj u gate Fo r if
T is an antiself-conjugate dyadic the prod u ct r T r is
identicall y z ero for all values of r Th e proof of this state
ment is left as an e x ercise B y Art 1 1 6 any self-conjugate
dyadi c is reducible to the form
Th e
.
,
.
.
.
O_ i
i i
a
2
i
b
a
m
Hence the equation
r
O
k k
jj
,
i
.
c
y
r
a
1
y
represents a quadric surface real or imaginar
Th e dif
ferent cases which arise are four in number
If the
signs are all positive the quadric is a real elli psoid If one
sign is negative it is an hyperboloid of one shee t ; if two are
.
.
,
.
VECTOR A NAL YSIS
3 74
Hence the most general quadratic e xpression may be reduced
to
O
r where O is a constant dyadic A a constant vector a nd C
a constant scalar Th e d adic may b e regarded as sel f
conjugate if desired
A make a change of or g n
To be rid of the linear term
’
b replacing r b y r t
’
y
.
,
,
r
.
y
ii
,
.
(
’ —
r
’
r
O
o
equal
.
’
(
—
r
o
+
l
Since O
O
t)
t
’
r
’
r
— t +
)
O
o
o
(
’ —
r
’
—
r
r
’
t)
-O
o
o
A+
-O
t + t
o
t
A— t
o
r
is se f-conjugate the second and thi rd terms a e
Hence
'
r
If now O
.
-A
O
is complete the vec t or t may b e chosen so that
1
§
A= O
o
t
or t
Q
Hence the quadric is reducible to
r
’
O
l
—
t he
O
r
-A
.
central form
const
’
1
.
case O is incomplete it is u n iplanar or u n ilinear
O is self -conjugate
If A lies in the plane of O or in
of O as the case may be the equation
In
.
1
2
A=
w
o
t
is solub le for t and the reduction to central form is still pos
sib le B u t unless A is so situated the reduction is impossib le
Th e quadric surface is not a central surface
Th e discussion and classi fi cation of the var ous non-central
quadri cs is an interesting e x ercise It will not be taken up
he re Th e present ob ject is to develop so much of the theory
.
.
.
.
.
i
Q UADR I C S URFA CE S
3 75
of quadr ic surfac es as will be useful in applications to mathe
m a ti c a l physics wi t h especial reference t o non -iso t ropic media
Hereaft er therefore the central u a dr cs and in parti cul ar the
ellipsoid will be discus sed
Th e tangent plane may be found b di f
ferentiation
q i
.
y
.
r
dr
Since O
O
o
+
r
o
-r =
O
o
r
O
o
1
.
.
dr = 0
o
.
is self conjugate these two terms are e qual and
-
O
dr
r
O
.
increment d r is perpen di cul a r to O r Hence O r is
normal to the surfac e at the e x tremity of the vector r
et
this norm a l be denoted b y N and let the u nit normal be u
Th e
L
.
.
.
N = T
L et
r
~
be the vector dr awn fr om the origin perpendicular to
the tangent plane p is parallel to n Th e perpendicular
distan ce from the origin to the tangent plane is t h e square
root of p p It is also equal to the squ are root of r p
p
.
.
.
Hence
r
o
=
p
r
P
P
B
ut
H ence inasmuch as
r c os
r
p
'
‘
P
o
=
p
p
p
.
P
_
P
P
-O -r
o
r
‘
P
-N
z
l
.
y
q
and N are parall el the are e ual
,
O
-r = N
.
VECTOR A NAL YSIS
37 6
page 1 08 it was seen t hat the vecto r which ha s the dire c
tion of t h e normal to a plane and which is in magnitude equal
to t h e reciprocal of the di stance from the origin to the plan e
may be ta ken as the vector co ordi nate of that plane Henc e
the above equation shows th at O r is not merely normal to
the t angent plane b u t is also the co ordi nate of the p lane
That is the length of O r is the reciprocal of the distance
from the origin to the plane tan gent to the elli psoid at
the e xtremity of the vector r
Th e equation of the ellipsoid in plane c o tirdin a t e s ma be
found b eliminating r from the two e uations
On
.
'
,
.
,
y
.
y
q
r
O
o
O
r
Hence
r
=
-O -e
l
“
-O
l,
z
=N
r
o
.
-O
-O -O I -H
N
o
I
‘
-r
.
-l
N
z
‘
.
O
1
.
q
Hence the desired e uati on is
N
O
o
‘
5
—
Let
a
2
I
-N = 1
.
’
+
b
ii +
s
j+
0
2
kk
.
and
where a v w are the reciprocals of the intercept s of the
plane N upo n the ax es i k Then the ellipsoid may be
written in either of the two forms famili ar in Cartesian
geometry
,
,
,
.
j
,
.
3 78
r
vecto s
dyadi c O
Th e
th e
VECTOR ANALYSIS
Let
.
a
r
vecto s
Th e
b,
a,
=O
’
’
a
,
b
Fo r
’
,
a
a
o
’
0
o
are changed into
c
b = O
'
,
O
o
c
O
O
b, O
o
a
c
y
b
’
c
a,
b,
c
.
= 1,
a
-b
dyadic
h,
a,
form the system reciprocal to
c
Th e
o
O
-O -c = 1
= 0,
’
c
o
b -0
’
O
a
o
,
=b
o
O
c
o
= 0,
= O
.
may be therefore e x pressed in the forms
O
O
’
b b
a a
+ b b
a s
—l —
’
’
’
’
c c
’
c c
.
for convenience the three directions a b c be called a
system of three conjugate radii vectors and if in a si m ilar
manner the three tangent planes at their e x tremities be called
a system of t hree conjugate ta ngent planes a number of
geometric theorems may b e ob tained from interpreti ng the
invariants of O A system of three conjugate radii vecto s
may be obtained in a doubly infi nite number of wa s
Th e volume of a parallelopiped of which three concurrent
edges con sti t ute a system of three conjugate radi vectors is
constant and equal in magnitude to the rectangular p arallelo
piped constru cte d upon the thr ee semi -ax es of the ellipsoid
Fo r let a b c be any system of three conjugate ax es
If
,
,
,
,
,
y
i
.
r
.
.
,
Th e
.
,
determ inant
pendent of
th e
or third of O is an in variant
form in which O is e xpressed
‘
I
.
Q UA DR I C S URFA CES
B
ut
3 79
if
O3
Hence
[
“
a
1
a
b
c
2
]
This
b
2
a
b
c
”
.
c.
demons trate s the theorem In l ike manner by inter
1
O
O
and
b
it
is
possi
le
to
show
that
O
re t in
S
S
p
g 3
Th e sum of the s uares of the radii vectors drawn to an
ellipsoid in a system of three conjugate di rections is constant
and equal to the sum of the Squares of the semi-ax es
Th e volume of the parallelopipe d whose three concurrent
edges are in the di rections of the perpe n di culars u pon a system
of thr ee conjugate tangent planes and in magnitude equal t o
the reciprocals of the d stances of those planes from the
center of the ellipsoid is constant and equal to the reciprocal
of the parallelopiped co n structe upon the semi -ax es of the
ellipsoid
Th e sum of the squares of the reciprocals of the thre e per
n di c u l a rs dropped from the origin upon a system of three
e
p
conjugate tangent planes is constant and equal to the sum of
the squares of the reciprocals of the semi -ax es
If i
k b e three mutuall perpen di cular unit vectors
.
q
“
,
,
.
,
i
d
,
.
,
L et
j
y
,
O
aa ,
‘
l
o
i
.
+ j-O
"
1
j+
be th ree radii vectors
respectivel parallel to i k
a,
b,
y
c
,
a
-O
Hence
i
a
a
o
o
j
,
in
k
o
O
'
1
o
k
the ellipsoid drawn
.
=b
O
-O
o
i
+
o
a
j
o
b
O
-j
-O -b
k
+
c
.
O
-O
-k
o
c
the three terms in this e xpression are the squares of the
reciprocals of the radii vectors drawn respectively in the i
ence :
k d rections
B
ut
i
.
H
,
j
,
VECTOR A NAL YSIS
3 80
l
y
sum of the squares of the reciproca s of three mutuall
perpendicul ar radii vectors in an ellipsoid is constant An d
in a similar manner : the sum of the squares of the p e rp e n
dic u l a rs dropped from the origin upon three mutuall p e rp e n
di c u l a r t angent planes is constant
Th e equation of the polar plane of the po nt deter
1
a
mined b y the vector is
O3
s a = 1
1
( )
Th e
.
y
i
.
.
let s be the vector of a point in the polar plane Th e
vector of any point upon the line which joins the termin u s of
s and the termin u s of a is
Fo r
.
y
s
a
+
:c a
+ g
/
is point lies upon the surface
If th
s
y
ar
+ ma
t y
2
s
-
w
a
'
w
z
l
If the
terminus of s lies in the polar plane of a the two values
of the rati o m y determined b y this equation must be equal
in magn it ude and opposite in Sign Hence the term in m y
vanishes
Hence
O a = 1
s
.
.
o
o
is the desired equation of the polar plane of the terminus
of a
et a be replaced by z a Th e polar plane becomes
L
.
.
s
s
1
re a
mg
It is
l or i
-O
o
a
,
z
mm
,
l
z
1
id e n t ly i a t e ri a l wh e t h e r t h e
ina ry e llip so id o r h ype rbo lo id
ev
a
-O -z a
.
c e n t ra
m
l qu a dri c d e t e r i n e d by 4»b e
VECTOR A NAL YSIS
3 82
i
is any point outside of the qu a dric and f all the tangent
pla nes which pass t h rough A are drawn these pla nes envelop
a cone This cone touche s the quadric along a plane curve
the plane of the curve bein g the polar plane of the poin t A
Th e equation
Fo r let a be the vector dr awn t o the point A
of any tangent plane to the quadric is
If A
,
.
.
.
8
If this
0
0
1
r
.
plane contains A its equation is satis fi e d b y a Hence
the condi tion s which must be satisfi ed by r if its tangent
plane passes through A are
.
,
a.
r
T
o
O
l
1,
‘
1
r
.
points r therefore li e in a plane r ( O a ) 1 whi ch
on comparison with ( 1 3 ) is seen to be the polar p ane of A
Th e quad ric which passes through the curve of intersection
of thi s polar plane with th e given quadric and whi ch touches
the quadric along that curve is
Th e
l
(
If this
T
e
o
r
passes through the point A
(
Hence
r
(
O
a
O
r
o
s
O
-r
O
-a
.
,
— 1
r — 1
a O
a
) (
)
—
(
a
o
O
a
r
transforming the origin to the point A this is easily seen
to be a cone whose verte x is at that point
et O be an self-conjugate dyad c It is e xpres
sible in the form
By
L
y
.
O= A ii + B
B
’
,
C
jj+
i
.
Ckk
are positive or negative scalars
A < B
O— B I=
(
0— B
)
<
.
0
kk —
— A ii
B
(
)
.
Further
Q UADR IC S URFA CE S
3 83
’
Then
O
B I=c
+ a )(o
—
a a
c
+
c
and
=p
a
O= EI+
— a
c
= q
.
1
1
4
( )
2
dyadi c O has been ex press ed as the sum of a constant
multiple of the idemfactor and one ha l f the s u m
Th e
p q + q p
~
reducti on has assumed taci t ly tha t the cons tants A B C
are different from each other and from z ero
This e x pression for O is closely rela ted to the circular
sections of the quadric sur fa ce
O r r = 1
Th e
,
,
.
.
Substituting
the value of
B
L et
O,
-r -i- r -p
r
r
r
o
r
+
q
p
be any plane perpendi cular to
B
O
r
1
r
= l
becomes
.
n
p
.
q
n
o
r
o
By
substitution
r —
1 = O
.
This
is a sphere b ecause t h e te rms of t h e second order a ll
have th e same coe ffi cient B If the equation of thi s sphere
be sub tracted from that of the given quadri c th e resul ting
e q uation is that of a quadric which passes through the inte r
section o f the Sphere and t h e given q u adric Th e difference
.
,
.
18
q
o
r
(
r
o
p
Hence the Sphere and t h e qu a dric intersect in two plane
c u rves l i ng in the planes
=
r
r
and
0
n
p
q
y
.
o
z
.
VECTOR A NAL YSIS
384
y
i
Inasmuch
as these curves lie upon a sphere the are circles
Hence planes perpendi cula r to p cut the quadr c in circles
In like manner it may b e shown that planes perpendicular to
Th e proof ma be conducted as
q cut the quadric in circles
follows :
=
1
B r r + r r
p q
.
.
y
.
o
.
r is a radi us vector in the plane pas s ed through the center
of the quadric perpendi cular to p or q t h e term r p q r van
ishes Hence the vector r in thi s plane sat s fi es the equati on
If
i
,
.
B
r
r
o
=1
r
and is of constant length Th e section is therefore a ci cular
section Th e rad us of the section is equa l in length to the
mean semi—axis of the quadric
Fo r convenience let the quadric be an elli psoid
Th e con
l
s ta n t s A B
T
are
then
ositive
h
reciprocal
d
i
c
O
C
e
a
p
ma be reduced in a simila r manner
i
.
.
.
.
y
,
.
,
.
k
Then
dy
O
-l
and
d
I = ff— d d
— d
)
+
f
(
‘
3 86
VECTOR A NA L YSIS
i
the rotat on the section is a circle In li ke manner consider
the projection or shadow of the ellipsoid cast upon a plane
parallel to the mean ax is b y a point at an infi ni te dis tance
from t hat plan e and in a direction perpendi cular to it A s the
elli p s oid is rotated ab out it s mean ax is from t h e position in
whi c h the major a xis is perpendi cular to the plane of p ro j
ec
tion to t h e position in which the mi nor ax is is perp endi cul ar
to t ha t plane the Sh a d ow and the projecting cylinder have t h e
mean ax is of the elli psoid as one axis Th e other axis cha nges
from the min or ax is of the ellipsoid to the major and hence at
some stage of the rotation it passes through a value equ a l to
the mean ax is At thi s stage t h e shadow and projecting
cylinder are circul ar
Th e nece s sary and s u fii c ie n t condi tion that r be the major
or mi nor s emi-axi s of the sec t ion of the ellipsoid r O r 1
b y a plane passing through the center and perpendi cular to a
is that a r and O r be coplanar
.
.
,
,
.
.
.
'
Let
,
,
.
r
and
Difi e re n t ia t e
d
0
r
w
o
r
1
r
O
a
.
r
O
0,
Furthermore
r
if r is to be a major or minor ax is of the section ; for is a
max imum or a m in in u m and hence is perpendic ul ar to d r
These three equations show t hat a r and O
are all ortho
gonal to the same vector d r Hence they are coplanar
,
r
,
.
C onversely
may
Then
dr
be
if
.
[a
r
O
[a
r
O
r]
r]
O
.
0,
chosen perpendicular
d
r
.
o
r
to
0
.
t
l
heir common p ane
.
Q UA DR I C S URFA CE S
3 87
'
Hence r is a maximum or a m in in u m and is one of the prin
c ip a l semi axes of the section perpendi cula r to a
It is frequently an advanta ge to write the equati on
of an elli psoid in the form
.
t
in s t e a d
o
f
This
y
Tz
o
r
O
o
ma be done ; becaus e
r
o
r
o
l,
z
l
z
.
if
jj
ii
kk
z
c
ii
+
a
’
ii
kk
b
0
is a dyadic such that T is equal to O T may be re garded as
a s u are root of O and written as O i B u t it must b e re
memb ere d that there are o t her square roots of O for
3
q
.
.
i i
jj
kk
i i
jj
kk
y
y i
thi s reason it is necessar to bear in mi nd that th e square
root whi ch is meant b O 5 s that particular one whi ch has
bee n denoted by T
Th e equation of the e lli psoid may be wr tte n in the form
Fo r
i
t
T
o
o
T
o
r
L et r be the radius vect r of a unit
the sphere
r
’
o
is
’
r
’
1
l
z
.
,
phere
S
.
Th e
q
e uation of
VECTOR ANAL YSI S
38 8
If r
i
r it becomes ev dent that an ellipsoid ma y be
transformed into a unit sphere by applying the operator T
to each radi us vector r and vice ver s a the unit sphere may
b e transformed into an elli psoid b y applying the inverse O per
ator T I to each radi us vector r Furt hermore if a b c a re
a s ste m of three conjugate radii vectors in an ell ipsoid
’
T
o
,
,
.
y
’
‘
.
a
.
T -c ,
T
-b
Z
a
If for
w
o
z
a
the moment a
a
z
z
z
’
,
b
o
w
o
b
o
z
T
o
b
z
’
’
0
,
b
c
z
=
c
,
,
c
o
W
T2
o
C
O
0
a
denote respectivel
I,
Z
= 0
.
y
T
a
T
,
b,
’
c
r
’
Hence the th ee radii vectors a b c of the unit sphere into
which three conjugate radii vectors in the ellipsoid are trans
1
T
are mutually orthogonal They
formed b y the operator
form a right-handed or left-ha nded system of three mutuall y
perpendic ul ar unit vectors
Th e o re m : A n y elli psoid may b e transformed into an other
elli psoid b y means of a homogeneous strain
et the e uations of the ellipsoids be
’
’
’
,
,
“
.
y
.
L
q
.
.
-O -r -
r
and
z
l,
r
ii
r
means of the strain O i the rad vectors of the fi rst
ellipsoid are changed into the radii vec t ors r of a un it sphe e
By
r
’
r
’
= O4
r,
o
means of the strain T the radii vectors r of thi s unit
sphere are transformed in like manner into the radii vectors r
of the second elli psoid Hence b y the pro uct r is changed
By
’
’
i
d
.
r=
-
.
T
i
Ti
0
r
.
( 1 9)
390
Iftwo
VECTOR A NA L YSIS
l
y
,
L et the quadrics be
r
O
r
1,
and
r
T
r
1
Let
r
-r a n d s
= O
s
Then
ri
l
confoca quadrics intersect the do so a t ght a n g es
= O
1
"
o
s an
d
’
.
= Tr,
= T
r
1
“
q
the uadr ics may be written in terms of s and s
and
s
O
’
T
s
y
-s =
I
‘
I
"
s
where b the confocal property
—1
r
Ifthe
’
as
1,
’
1,
,
— I
p
r
.
quadrics inte sect at the condition for perpendicularity
is that the normals O r and T r b e perpend cular That is
i
,
.
s
B
ut
T
1
‘
:c S
In
o
s
’
s
T
’
‘
o
l
o
’ —
s
=
s
o
T
‘
s
T
(
+
8
o
—1
+
1)
a
s
o
9c s ,
l
s
o
’
=1
T
— s
‘
l
o
s
’
.
like manner
r —
O
‘
I
-S =
'
x s
o
2
s
a
T
“
’
o
s
o
s
-O
s
s
I
’
"
s
’
1
(O
—
o
s
’
(O
“
I
s
‘
l
ml)
-O
T
“
1
‘
1
)
s
a
s
’
= O
=
s
a
o
‘
1
-O
'
s
o
s
o
‘
1
s
’
o
’
s —
1
.
’
.
and the theo rem is proved
If the parameter n b e allowe d to var from
c
th e
c o to
resultin g confocal qu a drics will consist of three fam ili es of
which one is ellipsoids ; another hyp erb oloids of one sheet ;
a n d the third h
B y the foregoing
yp erb oloids of two sheets
y
.
o
,
,
.
Q UADR I C S URFA CE S
y
391
y
theorem each su rface of any one fam il cuts ever surface
of t h e o t her two orthogo n ally Th e sur fa ces form a triply
orthogonal system Th e lines of intersection of two famili es
-sheeted and t h e fa mi ly of two-shee t ed
say
the
family
of
one
(
hyperb oloids) cut orthogonally t h e other family — the family
of ellipsoids Th e points in which t wo ellipsoids are cut b y
these lines are called corre s pondi ng points upon the two e llip
It may b e Shown t ha t the ratios of the components of
s o ids
the radi us vector of a point to the ax es of the ellipsoid
thr ough that point are the same for any two corresponding
points
Fo r let any elli psoid b e given by the dyadic
.
.
.
.
.
ii
a
kk
:
1
$
7
.
neighb o rin g elli psoid in the family is represented by the
dyadic
Th e
i i
Z —
a
dn
T
“
I
2 —
6
O
"
dn
1
I dn
.
Inasmuch
as O and T are homologous ( see Ex 8 p 3 30)
dyadics they may b e treated as ordin a ry scalars in algeb ra
Therefore if te rms of order higher tha n the fi rst in d n b e
.
,
.
.
o
m l t t e d’
Th e
T
O
I dn
.
two neigh boring ellipsoids are then
r
r
O
1,
f
(
O + I ci
n
-
g
5
O dn) i
Oi
‘
O d n)
o
r
=
r
o
r,
-r
,
VECTOR A NAL YSIS
39 2
vectors f and r differ by a mul tiple of O r whi ch is
perpendicular to the ellipsoid O Hence the te rmi ni of f and
r are correspon di ng points for they lie upon one of th e lines
which cut the famil y of ell ipsoids orthogonally Th e com
:c and
i
o n e n t s of r and i in the direction i are r
p
Th e
.
,
.
at n
r
a:
Th e
Th e
a
.
1
1
Q
0
0
r
L
T:
a:
2
rati o of these components
dn
is
2
a
ax es of the ellipsoids in the direction are Va
Thei ratio is
i
r
V
z —
a
2
2
dn
an
d
dn
a
In lik e
manner
_
b
dn _
V
2
b
2
V0
2
3/
dfl
0
Hence t h e ratios of the components of the vectors 1 and r
drawn to correspondin g points upon two neighboring e lli p
fer at most by terms of the se cond order in al n
s o ids o nl y di f
from the ratios of the ax es of those ell ip s oi ds It foll ows
immedi ately that the ratios of the components of th e vectors
drawn to corresponding poin ts upon any two elli psoids sepa
rated by a fi n i t e variation in the parame t er n onl differ at
most b y terms of the rs t order in d n from the ratios of the
ax es of the ellipsoids and hence must b e identical with them
Thi s completes the demonstration
"
.
fi
y
,
,
.
.
Th e P r op a ga t io n
f
o
Ligh t in
Cry
s ta
ls
l
electromagneti c equations of the ether or of any
in fi ni t e i s o tropic medi um whi ch is transparent to e l e c tro ma g
ne t ic waves may be wri t t en in the form
1 43
1
]
Th e
To t re a t
Th e
g m m
g
m m
fo llo wi n d i sc u s s io n u s t b e re a rd e d a s a t h e
t h e s u bj e c t fro
t h e s t a n d po in t o f p h y s i c s wo u ld b e
a
i l
t ca
o ut o f
not
p h ys i c a l
pla c e
he
r
e
.
.
VE C TOR
3 94
D= A
Then
c os
A
NAL YSIS
(m
o
r
—
t)
n
where A and m are cons t ant vectors and n a constant scalar
Th e vi b ra ti ons ta ke place in
represen t s a train of waves
the direction A That is t h e wave is plane polari z ed Th e
wave advances in t h e direct ion m Th e velocit v of that a d
vance is th e quotient of n b y m the magni tude of t h e vector
If this wave is an elec t romagnetic wave in t h e medium
m
considered it must sat isfy the two equa tions of that medium
Sub s ti t ute t h e value of D in those equations
Th e value of V D V V O D ma y b e
D and V V O obtained most easily b y assum ing the direction i t o b e c o in c i
dent wi t h m m r then reduces to m i r which is equal to
mx
Th e variables y and z no longer occur in D Hence
.
y
,
.
.
.
,
.
.
.
o
o
,
,
.
.
.
B = A
V
V
V V
Hence
o
o
n
9D
l
z
V O
o
a
o
V
o
—
m
—
m
-V
V V
— i
-A
— m2
D
V
m
(
a:
O
2
i i
O
A
o
x
c os
(m
zc
-O -A
-D
—
-O -D
—
m
e
2
o
x
m
m
O
O
o
O
o
D
-D
.
J
D
.
Hence if t h e harmonic vib ration
tions ( 4 ) of the medium
D=m
m
(
c os
m
mm
n
2
m
(
-A sin ( n -r
M oreover
n
s in
m
x
D
-O -D
c os
-D —
-A = O
.
y
D is
to satis f the equa
mm
o
O
-D
THE
Th e
be t r
Th e
PR OPA GA T ON OF L GHT
I
I
CR YS TAL S
IN
39 5
latter equation states at once tha t the vib ra tio n s m us t
a n s ve rs e to the direction m of propagation of the waves
former equation may be put in the form
.
m
mm
m
o
n
2
Introduce
vector 8 is in the direction of ad vance m Th e magni t ude
of s is the quoti ent of m b y n Thi s is the reciprocal of the
velocit of the wave Th e vector 3 may therefore be called
the wave-slowness
O D= S s O D— s e D
Th e
.
y
.
.
.
o
This
o
.
may als o be written as
D=
Dividing b y
the scalar factor cos (m x
-A) x s s s O -A —
s s
o
-O -A
(7)
.
evident tha t t h e wave slowness 3 depends not at all
upon the phrase of the vib ration b u t only upon its directi on
Th e motion of a wave not plane polariz ed may b e discussed b
decomposing the wave into waves whi ch are plane polari z ed
et a b e a vector drawn in t h e di recti on A of the
displacement a n d let the magni tude of a be so determin ed
It is
y
.
L
t
hat
Th e
.
-O -a
a
= l
( 8)
.
equation ( 7 ) t hen b ecomes reduced to the form
O s
s
O s
s
s s a
o
a
c
O
-a
o
z
l
(9)
o
( 8)
.
These
y
are the equations b y which the discussion of the velocit
or rat her t h e s lowness of propagation of a wave in different
directions in a non-isotropic medi um may b e carried on
s
a O a = 3 s
a = s
a .
o
o
o
.
VECTOR A NAL YSIS
3 96
Hence the wave slowness s due to a displacement in th e
direction a is equal in magni tude ( b u t not in di rection) to the
radius vector drawn in the elli psoid a O a 1 in that
direction
a x O
s — a x s
s O s
-a )
-a S O s
s
0
.
o
o
z
B
ut
the fi rst term contains
a
x
-O
s
o
s
O
a
[a
i
o
i
tw ce and van shes
s
o
.
Hence
.
O
1
1
( )
wave-slowness 8 therefore lies in a plane wi th t h e
direction a of displacement and the normal O a drawn to the
ellipsoid a O a 1 at the terminus of a Since 8 is p e rp e n
dic ul a r to a and equal in magnitude to a it is evidentl com
p l e t e ly determined e x cept as regards Sign when the d rection
G iven the direction of displacement the line of
a is known
advance of the wave compatib le with the di splacement is com
l
e
l
determined
the
velocity
of
the
a
d
ance
is
likewise
v
t
e
p
y
known Th e wave however may advance in either direction
along that line B y reference to page 3 8 6 equation ( 1 1 ) is seen
to b e t h e conditi on that a shall b e one of the principal ax es of
th e ellipsoid formed by passing a plane through the e ll ipsoid
perpendicular to 8 Hence for any given direction of advance
there are two possib le lines of di splacement These are the
principal ax es of the elli pse cut from t h e ellipsoid a O a 1
b y a pla ne passed through the center perpendicular to the
line of ad vance To these statement s concerning the de t er
mi n a t e n e s s of s when a is given and of a when s is given just
such ex ceptions occur as are ob vi ous geometrically If a and
O a are parallel 8 may have any di rection perpendi cul ar to a
Th is happens when a is directed along one of the principal
ax es of the ellipsoid If s is perpendicular to one of the
c ircular sections of t h e ellipsoid a may have an d recti on in the
p ane of the sectio n
Th e
.
i
y
.
,
.
.
,
.
.
.
.
.
.
l
.
yi
3 98
VECTOR ANA L YSIS
L et
j+
x
Then
i + y
and
k
z
3
2
x
2
r
2
+ y +
z
2
.
i
the equation of the surface in Ca tesian co ord nate s
is
2
equation in
d rectly from
i
(
i
Cartesian
Th e
l —
O
‘
s
s
o
y
i
coord nates ma be obta ned
I+
s s
y
)8
_- O
.
determinant of this d adic is
a
z
x
2 —
b
means of the relation
forms
By
a
2
—
2
+
2
a
y
z s
l
y
2
2
3
y
2 —
3
a
2
+
y
b y
2 —
3
b
2
e
z
this assumes the
2
z
+
8
2
s
b
z
+
s
This
equation appears to be of the six th degree It is how
ever of only the fo u rth Th e terms of the six th order cancel
out
Th e vector s represents t h e wave-slowness
Suppose that a
plane wave polariz ed in the di rection a asses the orig n at a
.
.
.
p
.
i
THE
PR OPA GA T ON OF L GHT
I
I
IN
CR YS TAL S
399
certain instant of time wi th this slowness At the end of a
unit of t ime it will have travelled in the di rection 3 a distance
equ a l to the reciprocal of t h e ma gni t ude of s Th e plane will
be in this position represented b y the vector 3 ( page
.
,
.
If
s
n
i +
v
j+ w k
plane at the ex piration of the unit time cuts o ffintercepts
upon the ax es equal to the reciprocals of i t v w These
quantities are therefore the plane co ordinates o f the plane
They are connecte d with the co ordi na t es of the points in the
plane b y the rel ation
th e
,
,
.
.
c
=
w
z
L
+
y
If di fferent
plane waves po la riz ed in all possible di fferent
directions a be supposed to pass through the origin at the
same instant they wi ll envelop a surface at the end of a unit
of time This surface is known as the wave -surface Th e
pe rp endic ul a r upon a tangen t plane of the wave-surface is the
reciprocal of the slown ess a n d gives the velocit with which
the wave travels in that direction Th e equation of the wave
su rface in plane co ordi nates u v w is identical with the eq u a
tion for the locus of the terminus of the slowness vector s
Th e equati o n is
.
.
y
.
,
,
.
3
2
b
2
1
where 3
n
O
This may be wri tten in any of the
w
forms given previously Th e su rface is known as Fre s n e l s
Wa ve -Su rfa c e Th e equations in vector form are given on
page 3 9 7 if the variab le vector 8 be regarded as determi ning a
plane instead of a point
In an isotropic medium the di rection of a ra y of
li ght is perpendi cular to the wave-front It is the same as
the di rection of the wave s ad vance Th e velocit of the ra
2
2
z
2
.
’
.
.
.
.
’
.
y
y
VECTOR ANA L YSIS
4 00
is equal to the velocity of the wave In a non isotropic
medium this is n o lon ger true Th e ray does not travel per
-front — that is in t h e dir ection of the
to
wave
t
h
e
a
r
n di c ul
e
p
wa ve s advance A n d the velocity with whi ch t h e ray travels
is greater t ha n the velocit of the wave In fact whereas the
wave -front travels o ff a l ways tangent to the wave -surface the
ray travels along the radi us vector drawn to the point of tan
-planes
-plane
T
h
e
nc
of
wave
wave
envelop
e
t
h
e
th e
g
y
wave-surface ; the termini of the rays are situated upon it
Thus in the wave-sur face the radi us vector represents in mag
n i t u de and direction the velocity of a ray and the p e rp e n
dic ul ar upon the tangent plane represents in magnitude and
direction the velocity of the wave If instead of the wave
surface the surface which is the locus of the e x tremit of the
wave slowness b e considered it is seen that the radius vector
represents t h e slowness of the wave ; and the perpendicular
upon the tangent plane t h e slowness of the ray
et v be the velocit of the ray Then s v 1 because
the ext rem ity of v lies in the plane denoted by 8 Moreover
the condi tion t h at v b e the point of tangency gives d v per
In like manner if s b e the slowness of the
p e n di c u l a r to s
ray and v the velocity of the wave 8 v 1 and the cond tion
of tangency gives cl s perpendi cul ar to v Hence
-
.
.
,
’
y
.
.
,
,
.
.
y
.
L
y
,
’
.
’
.
’
.
’
’
’
i
.
’
,
’
.
s
and
s
v
v
y
’
v ma be e xpressed in terms of
=
a
da = 2
s
o
8
ds O
0
8
-a
— s ds
M ultiply b y
a
O
-d a
a
=
¢
a
0
— S
o
O
’
d
o
and
a , s,
— 8
O
o
-a
o
=
s
= 0,
8
0
a
s s
¢
O
0
s
’
o
d v = 0,
as follows
a’
ds +
s
-s
-O -d a
O
-d a
.
and take account of the relations
0and a
a
s
Then
s
‘
o
o
.
.
a
s
O
and
VECTOR ANA L YSI S
402
and the normal to the elli psoid the wave -slowness and the
1
on the other
O
ellipsoid
It was seen t hat if s was n ormal to one of the cir
c u la r sections of O the displacement a coul d take place in any
direction in the plane of that section Fo r all directions in
t hi s plane the wave -slowness had the sa me direction and the
Hence t h e wave-surface has a singular
s ame magnitude
plane perpendi cular to s Thi s plane is tangent to th e surface
along a curve instead of at a single point Hence if a wave
travels in the di rection 8 the ray travels along the elements of
t h e cone drawn from the cen t er of t h e wave-surface to this
curve in whi ch the singular plane touches the s urface Th e
t wo directions 3 whi ch are normal to the circ ul ar sections of O
are called the p rim a ry Op tic a xe s These are t h e ax es of equal
wave velocities but un equal ray velocities
In like manner v b eing coplanar with a and O a
,
,
—
.
.
.
.
.
.
.
.
’
O
[
o
a
’
v
a
]
last equation states that if a plane b e passed through
1
O
the center of the ellipsoid
perpendicular to v then a
which is equal to O a will b e di rected along one of the prin
Henc e if a ray is to take a de fi n i t e
c ip a l a x es of the section
di rection a may have one of t wo d rections It is more con
ve n ie n t however to regard v as a vector determining a plane
Th e fi rst equation
O
a
v
a
0
]
[
Th e
’
“
’
,
i
.
’
.
’
.
’
stat es that a is the rad ius vector drawn in the ell ipsoid O to
the point of tangency of one of the principal elements of t h e
cylinder circumscribed ab out O parallel to v if b y a principal
element is meant an element passing through the e xtremi ties
of t h e major or mi nor ax es of orthogona l plane sections
of th at cylinder Hence given the di rection v of the ray the
t wo possib le directions of displacement are those radi i vectors
’
’
.
,
VAR IA B LE DYADICS
4 03
of the ellipsoid whi ch lie in the principal planes of the c ylin
der circumscrib ed ab out t h e elli psoid parallel to v
If the cylinder is one of the two circula r cylinders which
may b e circumscrib ed ab out O t h e direction of displ a cement
may be any di rection in t h e plane passed through the center
of t h e elli psoid and containing the common curve of tangency
of t h e cylinder wi t h t h e elli psoid Th e ray-velocity for all
these directions of di splacement has the same di rection and
the same magni tude It is therefore a line drawn to one
of the sin gular points of t h e wave -surface At this singular
oint there are an in fi nite num b er of tangent planes envelop
ing a cone Th e wave -velocity may be equal in magni tude
and direction to the perpendicular drawn from the o ri gin to
any of these planes Th e directions of the ax es of the two
circular cylinders c irc um s c rip t ib l e ab out the ellipsoid O are
t h e di rections of eq ua l ray-velocity b u t unequal wave velocity
They are t h e radii drawn to the singular points of the wave
surface and are called the s e c o n da ry op tic a xe s If a ray
travels along one of the secondary optic ax es the wave planes
travel along the elements of a cone
’
.
.
p
.
.
.
-
.
.
.
Va ri a ble Dy a dic s
Th e Dife re n tia l
.
a nd
In t egra l Ca lc u lu s
Hi t herto the dyadi cs considered have b een constant
Th e vectors which entered into their make up and t h e scalar
coe ffi cients which occurred in the e x pansion in noni on form
ha ve b een const ants Fo r t h e elements of the theory and for
elementary applications the s e constant dyadi cs suffi ce Th e
intro ducti on of variab le dyadi cs however leads to a s im p lifi c a
tion and uni fi cation of the di fferent ia l and integral calculus of
vectors and furthermore variab le dyadi cs b ecome a necessity
in t h e more advanced applications for instance in the theory
of the curvature of surfaces and in the dynamics of a rigid
body one point of which i s fi x ed
.
.
.
,
p
,
,
,
.
VECTOR A NAL YSIS
4 04
Let W be a
L
ector function of position in space
et r b e
the vector drawn from a fi x ed origi n to an point in space
y
v
x
Ol
Hence
d
ly
—
i + y
j+
OW
M
9
W= d r
z
.
.
k,
SW
SW
sy
x
s
'
z
9W
9W
ly
E
Oz
o
expression enclosed in the braces is a dyadi c It thus
appears that the di fferential W is a linear function of d r the
differential change of position Th e antecedents are i k
and the consequents the fi rst partial derivatives of W with re
spect of x y z Th e ex pression is found in a manner precisel
analogous to de l and will in fact be denoted by V W
Th e
.
,
,
.
,
,
j
,
,
y
.
.
VW—
9
,
r
W
O
9 m
W
OW
9y
Oz
'
1
( )
2
W d r VW
( )
This equation is like the one for the di fi e re n t ia l of a scalar
func t ion V
dV
dr V V
Then
d
.
.
:
may be regarded as de fi ning
nonion form V W b ecomes
It
VW —
.
.
rr
M
.
9X
OY
a
9
x
eg
aX
a
z
”
VW
x
sy
.
aY
.
If
e xpanded into
VECTO R A NAL YSIS
4 06
y
an attempt were made t o appl the operator V s ym b o li
cally to a scalar function V th re e times the result would be a
sum of twenty -seven terms l ike
If
,
9 V
9 V
3
,
”
x
9
This
3
9
x
9 y 9
z
is a tria dic Three vectors are placed in jux taposition
wi thout an Sign of mul tipli cation Such e x press ions will
not b e discussed here In a similarmanner if the operator V
b e applied twi c e to a vector function or o n c e to a dya d ic func
tion of posit ion in space the result will be a triadic and hence
ou t s ide the limit s set to t h e di scussion here Th e operators
V x and V may however be appli e d to a d adic O to ield
respectively a dyadi c and a vector
y
.
.
.
,
,
.
y
y
.
9
V
o
¢ =
Qy
a:
9 O
l
o
a
x
OO
+ 1
0
9y
If
where
u,
v
,
w are
V
v X O
o
vector functions of position in space
O=V
CW
9
9g
Oz
v
r
o
u
i + V
a
17
.
n
Oz
O
9
a
9 w
+
o
v
j+ V
-w
k
,
.
OW
9
a:
a v
9 W
9 y
Oz
( 8)
a similar manne the scalar operators ( 3 V) a n d ( V
may be applied to O Th e resul t is in each case a dyad c
In
.
i
V)
,
4 07
VA RI AB LE D YA DI CS
a T
9 a)
V
(
.
9
V) O
2
9
O
x
2
9
+
9 (17
O
2
9 y2
2
9
+
9
O
2
z
'
operators a V and V V as applied to vector func
tions are no longer necessaril y to be regarded as sin gle Oper
a t o rs
Th e individual steps ma be carried out by means of
the dyadi c V W
Th e
y
.
.
=
V
a
W
(
)
a
-V) W = V
(V
-V W
.
when applied to a dyadi c the operators cannot b e inte r
re t e d as made up o f two successive steps wi t hout ma k ing u se
p
of the triadi c V O Th e parent heses however may b e removed
wi t hout danger of confusion just as they w ere removed in
case of a vector function b efore the introduc t ion of the dyadi c
Formul ae sim ilar to those upon page 1 76 may b e given for
di fferentiatin g products in t h e case that the di fferentiation
B
ut
.
.
l e a d t o dya di c s
x
.
w -V
w)
V
(
V
o
v
—
V
w) = V
v
-Vw —
-w +
-v w +
V
v
-V w -l- V w v
V W
v
o
o
,
v,
-V
w
-V
O,
.
V
V x V x O
:
VV
-O —
V
e tc
.
principle in these and all similar cases is that enun
cis ted before nam ely : Th e operator V ma be t eated s m
Th e
,
y r
y
VECTOR A NAL YSI S
4 08
as a vector Th e differentiations which it implies
must b e carrie d out in turn upon each factor of a roduct
to which it is appli ed Thus
b o lic a lly
p
.
.
—
v
w
.
Hence
A gain
Hence
A
was
seen
( rt 7 9) that if C denote a curve of
]
whi ch t h e initial point is t and the fi na l point s r the li ne in
te gra l of t h e deri vative of a scalar function taken along the
curve is equal to the di fi e re n c e between the va ues of tha t
function at r a nd r
1 48
It
i
.
o
l
o
.
f
a
In
like manner
V V = V ( r)
r
f
a
r
V W=
f
dr
It
o
V
V ( ro )
.
W(t )
W ( r)
o
,
W= 0
.
may be well to note that the integrals
f
dr
-V W and
i
i
d
are b y no means the same th ng V W is a dyad c Th e
ve ctor al r cannot be placed arbitraril upon either si e of it
.
y
.
.
VECTOR ANA L YSIS
41 0
surface integrals are taken over the complete bounding
surface of t h e region t hroughout whi ch the volume integrals
are taken
N u merous form ul ae of integration by parts li ke those upon
page 2 5 0 might b e added Th e reader wil l fi n d no di ffi cul t y in
ob t ain ing them for himsel f Th e integrating operators ma
also b e e x t ended to other cases To the potentials of scala r
and vec t or functions the pote ntial P o t O of a dyadi c may b e
added Th e N ewtonian of a vector function and the apla
c ia n and Max wellian of dyadics may b e de fi ned
Th e
.
y
.
.
.
L
,
,
.
.
N ew
r
W
rs
W(x
2,
9
2,
d 17 2 ,
r3
x $ 0132
r
Q
r
Ma x O
z
,
3
a
)
d
1)
12
( x2
,
y”
dc
2
analytic theory of these integrals may be developed as
Th e most natural way in which the demonst rations
b efore
may b e given is b y considering the vector function W as the
sum of its components
Th e
.
.
,
W= X i +
Y j+ Z
k
q
and the dyadi c O as e x pressed with the constant conse uents
i k and variab le antecedents u v w or vic e ve rs a
,
j
,
These
,
,
,
,
matters will b e left at thi s point Th e object of e u
te ring upon th em at all was to in di cate the natural e x t ensions
whi ch occur when variab le dyadi cs are considered These e x
te nsions di ffer so sli ghtly from the simple cases which have
.
.
THE CUR VA TURE OF
S
URFA CES
41 1
gone before that it is far better to leave the details to b e worked
out or assumed from analogy whenever they may be needed
rath er than to attempt to develop t hem in advance It is su fh
cient merely to mention what the e x te ns ions a re and how the
may be treate d
y
.
.
Th e C
u rva
f
tu r e
O
Su rfa
ces
]
There
are t wo difi e re n t methods of treati ng the cur
In one the surface is e x pressed in pa ra
va t u re of surf a ces
met ic form by three equations
.
33
= fr
(
)
“a v
,
z = fs ( u
y = f2
r
= f u,
(
r
)
,
.
This
is analogous to the method followed ( Art 5 7) in dealing
wi t h curvat ure and torsion of curves and it is the meth od
employe d b y Fehr in the b ook to whi ch reference wa s made
In t h e second method t h e surfa ce is e x pressed b y a single
equation connecting the variab les a y z — thus
.
.
'
,
F
(
as,
z
y,
)
,
0
.
latte r method of treatm ents a ffords a simple applicat ion of
M oreover the
t h e differen t ial cal c ul us of variab le dyadi cs
dyadi cs le a d naturally to th e most important resul t s connected
wi t h the elementary t heory of surfaces
et r be a ra di us vector drawn from an arb itrary fi x ed
origin to a variab le point of t h e surface Th e increment d r
lies in the surfac e or in the tangent plane drawn to the surface
at t h e terminus of r
Th e
,
.
L
.
.
.
s
dl
'
O
VF
Z
- O.
‘
Hence the derivative V F is collinear with the normal to th e
surface M oreover inasmuch as F and the negative of F when
1
M uc h
m
,
.
o f wh a
e s pe c ia ll y t ru e
t
fo ll o ws
o ft h e
t re a t
m g
p ra c t i c a lly fre e fro
e n t o f e o d e t ic s A rt s
is
,
.
the
u se
1 55-1 5 7
.
of
dy a di c s
.
Th is is
VECTOR A NAL YSIS
41 2
y
equated to z ero give the same geometric surface V F ma be
considered as t h e normal upon either side of the surface In
case the surface belongs to the famil de fi ned b y
,
y
l
F
(
a
y
'
,
,
z
.
const
)
.
the norma V F lies upon that side upon which the consta nt
inc reases
et V F be epresente d b N the magn i tude of
which may b e denote d b y N and let n be a uni t norma l drawn
in the direction of N Then
.
L
r
y
,
.
N
e
2 =
=
N
VF
N
o
V F,
y
l
q y
is the vector drawn to an point in the tan gent p ane at
the te rminus of r s — r and n are perpend c u la Conse uentl
the equation of the tangent plane is
If s
i r
,
6
—
.
=
V
F
&
fl
li i
and in like manner the equation of the normal ne s
(s
)
X
r
+ kVF
r
s
=
VF
O,
q
where k is a variable parameter These e ua tions ma y be
translated into Cartesian form and give th e famil iar esul t s
Th e variation d n of the un i t normal to a su rface
plays an important part in the theor of cu vatu re d a is
perpendic ul ar to 11 because 11 is a unit vector
.
y
r
.
r
.
.
VECTOR A NA LYSIS
41 4
the surface
form
is possib le ( Art
It
.
r
to educe
1 1 6)
.
O
to the
5
( )
j
y
where i and are two perpendi cula r unit vectors l ing in the
tangent plane and a and b are positive or negative scalars
’
’
.
a
vectors i
of the surface
n
= d
r
o
j and the sc
(
a
’
’
fi + b
ars a b vary from point t o point
Th e dya dic O is variab le
Th e conic r O r
1 is called the in di catri x of the
surface at the point in question If this conic is an ellipse
that is if a and b have the same Sign the surface is convex at
the point ; b u t if t h e conic is an hyperb ola that is if a and b
ha ve opposite signs the surface is concavo-convex Th e curve
r O r 1 may b e regarded as approx imately equal to the
intersection of t h e surface with a plane drawn parallel to the
tangent plane and near to it If r O r be set equal to z ero
the resul t is a pair of s t ra ight lines These are the a s ym p
totes of the conic If they are real t h e coni c is an hyperb ola ;
if imaginary an ellipse Two directions on the surface which
are parall el to conjugat e diameters of the conic are called con
j u gate directions Th e directions on the sur fac e which coin
cide with the directions of the principa l ax es i
of the
indi c atrix are known as the principal di rections They are a
spe c i a l case of conjugate directions Th e directions upon the
surface which coincide with the direc t ions of the asymptotes
of the indi catrix are known as as ymptotic directions In case
the surface is conve x t h e indicatri x is an ellipse and th e
asymptotic direc t ions are imagina ry
In Special ca s es the dyadi c O may b e such that the c o e f
fi
ciente a and b are equal O may then be reduced to the
form
’
Th e
,
’
al
,
.
.
.
,
,
,
,
,
.
.
.
.
,
.
.
’
,
.
.
.
,
.
.
¢ =
a
a
f+
rn
j
’
THE CUR VA TURE OF
S
URFA CE S
j
41 5
in an in fi nite num b er of ways Th e directions i and may b e
any two perpendi cular di rections Th e indi catrix b ecomes a
circle A n y pair of perpendicular di ameters of thi s circle
give principal directions upon the surface Such a poin t is
called an u m bili c Th e surface in the neighb orhood of an
um b ili c is convex Th e asymp t otic directions are imaginary
In another special case the dyadi c O becomes lin ear and redu
Cib le t o the { 0 1 m
a i i
’
.
’
.
.
.
.
.
.
f f
.
indicatrix consists of a pair of parallel lines perpendicular
to i Such a point is call ed a parab olic point of the surface
Th e further di scussion of these and other special cases will b e
omitt ed
Th e quadric surfaces afford e x ample s of t h e vari ous kinds
of points Th e ellipsoid and the hyperb oloid of two sheets
are conve x Th e indicatri x of point s upon them is an ellip s e
Th e in
Th e hyperb oloid of one sheet is concavo -convex
Th e indicatri x
di c a t ri x of points upon it is an hyperb o a
of any point upon a sphere is a circle Th e points are all
Th e indicat rix of any point upon a cone or cyli nder
u m b ilic s
is a pair of parall el lines Th e points are parab olic A sur
face in general may have upon it points of a ll ty pes — ell iptic
hyperb olic parab olic and um b ilical
A li ne of princ ipal curvature upon a surface is a
1 52 ]
curve whi ch h a s at each point the dire c tion of one of t h e prin
c ip a l ax es of the indi catri x
Th e direction of the curve at a
point is always one of the principal direc t ions on the surface at
that point Through any given point upon a surface two per
n di c u l a r lines of prin c ip a l curvature pass
e
hus
the
lines
T
p
of curvat ure divide the surfac e into a system of in fi ni te s i
mal rectangles A n asympt o t ic line upon a surface is a curve
which has at each poin t the direc t ion of the asym ptotes of the
indicatri x Th e direc t ion of the curve at a point is al ways
one of the as mptotic directions upon the surface Th rough
Th e
’
.
.
.
.
.
.
l
.
.
.
.
.
.
,
.
,
,
.
.
.
.
.
y
.
VECTOR A NA L YSIS
41 6
p
given point of a surface t wo asymp t otic lines ass These
lines are imaginary if the su rface is convex Even when real
they do n o t in general intersect at righ t an gles Th e angle
between t h e two a s ymptotic lin es at any point is b isected by
the lines of curvature which pass through tha t point
Th e necessary and su ffi cient condition that a curve upon a
surface b e a line of principal curvature is that as one advances
along that curve the increment of d n the uni t normal to the
s u rface is parallel to the line of advance Fo r
any
.
.
.
.
,
,
.
d
n
= O
-d r
-d r = ( a
dr
a:
y
Then
evi dently d n and d r are parallel when and onl when
’
d
is parallel to i or j Th e state ment is therefore proved
It is frequen t ly taken as the de fi n i tion of lines of curvatu e
Th e di fferent ial equation of a li ne of curvature is
r
’
r
.
dr = 0
a
.
.
( 6)
.
A nother
method of statement is that the nor mal to the surface
the increment d n of t h e normal and the element d r of the
surface li e in one plane when and only when t h e element at r
is an element of a lin e of principal curvature Th e differential
equa tion t hen b ecomes
,
,
.
[
y
11
a
d r] = 0
n
( 7)
.
necessar and su ffi cient condi tion that a cur ve upon a
surface be an asym ptoti c l ine is that as one advances a l ong
t hat curve the increment of the unit norm al to the surface is
perpendic ul ar to the line of ad vance Fo r
Th e
,
.
dn = d r
da
If t hen d n
-d r
z
dr
o
O
O
-d r
.
is z ero d r O d r is z ero Hence d r is an
asymptotic direction Th e state ment is t herefore proved It
dr
.
.
.
VECTOR A NAL YSIS
41 8
0:
Hence
C
C
d
a
a
cos ( i
a
cos ( i
2
2
a
t
’
dr
+ b
d r)
,
’
d r)
,
a
e
dr
r 2
)
-d r
b
cos (j
d r) ,
b
sin ( i
d r)
’
2
,
2
’
,
1
0
( )
.
r
interpretation of this formula for the cu vature of a
normal section is as follows : When the plane p turns about
the normal to the surface from i to the curvature C of the
plane section varies from the value a when the plane passes
’
through the principal direction i to t h e val ue b when it
passes through the other principa l d rection
Th e val ues
of the curvature have algeb raically a max imum and minimum
in the directions of the principal li nes of curvature If a and
b ha ve unl ike signs that is if the surface is concavo -conve x
at P there ex ist two di rections for whi ch the curvature of a
normal section vanishes These are the as mptotic directions
Th e sum of the curvatures in two normal secti ons
at ri ght angles to one another is const ant and independent of
the actual position of those secti ons Fo r the curvatu e in
one section is
Th e
j
’
’
,
i
,
j
’
.
.
,
,
y
,
.
.
r
.
C1
and in
a
(308
2
(i
’
,
d r)
b
sin ( i
2
’
,
d r) ,
section at right angles to th is
C2
a
sin ( i
2
’
d r)
,
b
cos ( i
2
’
,
d r)
.
Hence
which proves the statement
It is easy to show that t h e invariant O s is e ua to the pro
z
duc t of th e curvatures a and b of the lines of principal c urv
ature
q l
.
.
q
Hence the e uation
a
x2
O,
a:
b
O2 5
THE CUR VA TURE OF
S
URFA CES
41 9
is the quadratic equation whi ch determines the principal c u rv
atures a and b at any poin t of the surface B y means of this
equation the scalar quantit es a and b may b e found in terms
of F ( x y z )
i
,
.
.
,
n n
)
—
I
(
VVF
o
n u
)
N
VVF
(
n u
2
—
VVF+
n n
VVF
o
-n n -V V F ) S
-V V F
V
V
F
(
)S
Hence
(
=
(
n n
o
n n
.
v
V V F)S
n u
N
v
r
v
(
(
n n
-V V F ) S
H ence
V
.
r
m
v
z
VF
o
V F,
VVF=
n n :
-V F
V
Th ese
n n
VF VF
z
N
3
VF
V
VVF
n
o
u
.
VVF
VF
X
F
J
y
y
i
ex p e ssions ma be written out in Cartesian co ord na tes
but the are e x tremel long Th e Cartesian ex pressions for
O2 5 are even long er
Th e vector e x p ession may be o b tained
y
r
.
.
as
,
fo ll o ws :
(I
“
11 1
02
V
V
F
)2
(
I
(
— un
g
)
2
7
A
I
(
Hence
VF
.
F
V
V
)2
(
N
G iven
4
y
— nn
.
)2
VF
VF VF
:
N
( V V F) 2
.
4
L
an curve upon a surface
et t be a unit
tangent to the curve n a unit norma l to the su face and m a
.
,
r
VECTOR A NAL YSIS
4 20
vector de fi ned as n x t Th e three vectors n t m constitute
an i k system Th e vecto r t is para l lel to the element at
Hence the cond tion for a line of curvature becomes
,
j
,
i
r
,
,
.
.
Hence
t x d
n
= 0
m
n
z
o
d
.
o
-a n
n
m
o
a
n
.
+
n
o
a m
.
= 0
.
-d m = 0
.
t x d m = 0,
1
6
( )
d m x dn = 0
.
p
increments of m and of n and of r are all arallel in case of
a line of principa curvature
A geodetic lin e upon a surface is a curve whose osc ul ating
plane at each point is perpendicular to t h e surface That the
geodetic li ne is the shorte st line whi ch can be drawn b e t ween
t wo points upon a surface may be seen from the follo wi ng
considerations of mechanics
et the surface be smooth and
let a smooth elastic string which is constrained to lie in the
surface b e st retched between any two points of it Th e string
actin g under its own tensions will take a position of e q u ili
b rium along the shortest curve which can be drawn upon the
surface between the two given points Inasmuch as th e
string rs at rest upon the surfac e the normal rea ctions of the
s u rface must lie in the osculating plane of the curve Hence
t hat plane is normal to the surface at ever point of the curve
and the curve it self is a geodetic line
Th e vectors t and d t lie in t h e osculati ng pla ne and deter
mine that plane In case the curve is a geodetic the normal
to the osculating plane lies in the surface and consequentl is
perpendicul ar to the norm al n Hence
Th e
l
.
.
L
.
.
.
y
.
.
.
,
.
y
422
VECTOR A NAL YSIS
i
given surface is la d o ff Th e te rmin us P of this normal lies
upon the surface of a sphere If the normals to a sur face at all
points P o f a curve are thus const ructed from the same origin
the points P will trace a curve upon the surfa ce of a unit
Sphere
This curve is called the spherical image of the given
curve In like manner a whole re gion T of the surface ma
Th e region T upon
b e mapped upon a region T the sphere
the sphere has b een called t h e ho do g a m of the region T upon
the su rface If d r b e an element of a re upon the surface the
correspondin g element upon the uni t sphere is
’
.
.
,
’
y
.
.
’
’
.
r
.
T
dn
dr
o
.
be an element of area upon the surface the corre
d
element
upon
the
sphere
is
a
where
s p o n din
A
r
t
(
g
If d a
,
’
.
d a = O2
’
O=
O2 =
a
Hence
b i x
d
a
da
.
a
’
’
o
a
’
=
j
’
j
i x
b
n n
’
-d a
a
b
n u
.
.
tio of an element of surface at a point P to the area of
its h o do gra m is equal to the product of the principa l radii of
curvat ure at P or to the reciprocal of the product of the prin
c ip a l c urvatures a t P
It wa s seen t hat the measure of t urning to t h e right or left
is m d t If then C is any c u rve drawn upon a s u rface the
total amount of turn ing in advancing along the curve is the
inte gral
Th e
ra
.
.
.
m
dt
.
2
( 0)
any closed curve thi s integral ma y b e evaluated in a
manner analogous to that employed ( page 1 9 0) in the proof
of Stokes s theorem C onsider two curves C and C near
Fo r
’
’
.
C UR VA
TH E
TURE
RFA CES
42 3
SU
OF
to ge t her Th e variation which the inte gral u ndergoes when
t h e curve of inte gra tion is changed from C to C is
.
’
B
f
m
-d t
f
8
z
m
(
d
8
f
-d t =
m
(m
-8 t)
f
8m
o
f
-8 t
= dm
dt
+ m
f
dm
—
-d t
em
d t)
o
o
f
m
+
.
8 dt
d 8t
o
8t +
f
(m
d
integral of the perfect di fferential d ( m
when taken around a closed curve Hence
St )
Th e
-8 t)
.
vanis hes
.
f
S
m
-d t
z
f
dm
dt
o
—
f
dm
dt
o
.
Th e ide m fa c t o r is
Sm
o
dt = 8m
o
l
o
d t = 8m
for t cl t and 8 m m vanish
be effected upon the term at m
.
f
S
m
o
f
dt=
B y di fi e re n t ia t in g
8
m
(
o
n
the relations
o
n n
dt
o
similar transformation ma
Then
8t
A
.
-d t
n
m
—
n
dm
o
u
0
and
—
m
n
n
-B t )
t
.
0
it is
seen that
Hence
S
8
f
m
f
m
-d t
f
=
m
(
y
-3 n
t
-d n
—
o
f
du t
o
-Sn )
dn x d n
.
VECTOR ANAL YSIS
424
x
ferential 8 n
Th e di f
the h o do gra m
r
represents the element of a ea
upon the unit sphere Th e inte gra
dn
l
.
f
n
o
in
dn x d n
epresents the to ta l area of the h o do gra m of the s tri p of
surface which li es between the curves C and C
et t h e
curve C start at a point upon the surface and spread out t o
any desired siz e Th e total amount of turni ng whi ch is re
qu red in making an infi nite simal circuit about the oint is
2
Th e to tal variation in the integral s
r
’
.
i
i
ff
m
dt
o
f
m
z
da
n
B
p
.
d
ut
-d t
’
H
—
2
7r.
,
if H denote the total area of the h o do gra m
H ence
f
m
-d t
H = 2
H +
= 2
7r —
L
H
.
,
7r
f
m
o
dt = 2
rn
area of the h o do gra m of the region enclosed
closed curve plus the total amount of turning along that curve
is equal to 2 7r If the surface in ques t ion is convex t h e area
upon the sphere wil l appear positive when the curve upon the
surface is so describ ed that the enclosed area appears positive
If however the surfac e is concavo-convex the area upon the
s phere will appear negative
Thi s matter of the sign of the
h o do gra m must be taken into accoun t in the state ment made
above
Th e
.
"
.
,
,
.
.
VECTOR A NAL YSIS
4 26
Vibra tio n s
a rm o n ic
H
a nd
B ive c to rs
differential equation of rectilinear harmonic
Th e
motion is
—
n
z
z
.
d y
l
integral of this equation may be reduce b a suitab e
choice of the constants to the form
Th e
A
mz
This
sin
t
n
.
represents a vibration back and forth along the X-axis
ab out t h e point x 0
et the displacement be denoted b
Th e equation may b e written
D in place of :c
L
.
y
.
D= i A
C onsider
This
D
i
A
s in n
sin
n
t
t
cos
.
m
x
.
y
is a displacement not merel near the point
but along the entire ax is of a
.
At
poi nts
2 h
x
rr
,
m
0
se
where
is a positive or negative integer the di splacement is at a ll
Th e equation represents a stationar
t imes equal to z ero
wave with nodes at these points A t points midwa between
these t h e wave h a s points of max imum vib ration If th e
equat ion b e regarded as in three variables x y z it re p re
sen t s a plane wave the plane of which is perpendi cular to
the ax is of the variab le x
Th e dis placement given by the equation
k
y
,
y
.
.
.
,
,
.
D1 = i A l
cos ( m ac
— n
(1)
t)
is likewise a plane wave perpendicular to the axis of a but
not stationar
Th e vib ration is harmonic and advances
along the direction i with a velocit equal to the uotient of
y
.
y
q
HAR M ON I C
n
by m
length
.
VIB RA TI ONS
be the velocity ;
If c
A N D B I VE C TORS
42 7
the pe ri o d ; and l the wav e
p
,
2 W
m
Th e
a
displacement
cos ( m a — n t )
differs from D1 in the partic ul ar that the displa cement ta kes
pl a ce in the direction not in the di rection i Th e wave as
b efore procee ds in the di rec t ion of x with the same veloci t y
This vibration is t rans verse instead of longitudi nal
By a
simple e x te nsion it is seen that
D2 = j A 2
j
.
,
.
.
D=A
c os
(m
ax—
n
t)
is a displacement in the di rection A Th e wave advances
along the dir ection of c Hence the vib ration is ob lique to
the wave -front A still more general form may be ob tained
b substituting m r for m 93 Then
.
.
y
.
.
.
D= A
This
c os
(m
o
r
is a displacement in the direction A Th e maximum
amoun t of that d splacement is the magni tude of A Th e
wave advances in t h e direction In ob lique to the displace
ment ; the velocity period and wave-length are as b efore
Elliptic ha r
So much for rectil inear harmonic motion
monic motion may be de fi ned b y the equation ( p
i
.
.
,
.
,
.
.
cl
2
d i
Th e
r
z
n
2
r.
general integral is ob tained a s
r
= A
c os n
t + B
s in n
t
.
discuss ion of waves may b e carried thr ough as pre
vi o us ly
Th e general wave of ellipti c harmonic motio n
adva ncing in the recti on m is seen to be
Th e
.
di
VECTOR ANA L YSIS
42 8
D = A c os
(I
D
— n
dt
(m
o
A s in
r — n
(m
o
t) — B
s in
(m
o
r — n
.
— n
r
y
4
( )
t)
t)
i
( 5)
y
is the velocit of the displ a ced point at an moment in t h e
ellipse in which it vib rates This is of course entirely differ
ent from the velocity of the wave
An interesting result is obtained by subtracting the dis
placement from the velocity multipli ed b the imaginar
unit V 1 and d vided by n
.
.
y
i
—
dt
n
+
V
— 1
y
.
A
cos ( m -r —
t) — B
n
sin ( m -r —
n
t)
6
( )
i
n
A
s
j
(m
9
“
d.
n
-r —
n
t ) + B s in
V—
+
l
n
(m
V:
.
-r
r— u
.
e xpression here obtained as far as its form is concern ed
is an imaginar vector It is the sum of two real vectors of
whi ch one has be en m ul tiplied by the imaginar sc a lar V 1
Such a vecto r is called a bi ve c to r or i m a gin a ry vector
Th e
ordinar imaginar scalars may be called bisc a la rs Th e use
of b ive c t o rs is foun d ver convenient in the discussion of
el lipti c ha rmonic motion Indeed any undamped ellip t ic har
m o n rc plane wave may be represented as above by the pro
duct of a b ivector and an e xponenti al fac tor Th e real part
of the product gives the displ a cement of any point and the
pure imaginary part gives the velocity of di splacement
reversed in Sign and divi ded b y n
1 59 ]
fe s from that of
Th e analyti c theory of b ive c t o rs di f
real vecto rs very m uch as the an a lyt ic theory of b is c a l a rs
differs from t hat of real scalars It is unnecessar to have
an dist nguis hi ng cha a cter for b ive c to rs jus t as it is need
Th e
y
y
,
,
y
.
y
.
.
y
.
.
.
.
y i
r
.
r
y
VECTOR A NAL YSIS
43 0
r
1
‘
o
( rl
s
X S=
(
1
'
1
x
8
— l
s
a
r
'
5
1
X
'
3
i
2)
(
1
8
1
r
2
+
2
”
s
1
r
X
z
3
1
)
)
or b is c a la rs are said to b e conjugate when
their real parts are equal and their pur e imaginar parts
Th e conjugate of a real scalar or vector
difi e r only in Sign
is equal to the scalar or vector its elf Th e conjugate of an
sort of product of b ive c t o rs and b is c a l a rs is equal to the pro
duct of the conjugates taken in the same order A similar
statement may be made concerning sums and di fferences
Two b ive c t o rs
y
y
.
.
.
.
)
(
fr
+ i
r
2
)
1
r1
= 2 i
2
x r1 ,
i
r
e
“
r
'
l 1
"
l
'
1
y
i
If the
bivector r rl i r2 be multiplied by a root of unit
or cyclic factor as it is frequently called that is by an imag
nary sc al ar of the form
,
where
a
2
2
O
'
12
are unaltered
Thus if
I
X
'
1 1a
1
1,
:
the conjugate is multiplied by
products
3
,
i b,
a
1
2
4
"
e
and hence
ra rl
th e
four
"
m u ltiplying the bivector r by such a fa cto r
by
I
I
I
.
HA RIlI ON I C
VIB RA TI ONS AND
B I VE C TOR S
y
4 31
closer e xami nation of t h e e ffect of multipl ing a
bivector b y a cyclic fac t or ields inte resting and import ant
geometric results
et
A
.
y
L
( cos
i
sin
q)
(
i
r
1
cos
q
r
2
sin
q,
r, cos
q
r
1
sin
q
r
1
B y reference
du c e d in the
i
q
8
( )
.
to Art 1 29 it will be seen that the change pro
real and imaginary vector parts of a b ive c to r by
m u ltiplication with a cyclic fac t or is precisely the same as
would be produced upon those vectors by a cyclic dyadi c
.
,
,
O=
a a
’
+
c os
—
q
sin
g
(c
b
’
—
b
c
’
)
used as a prefac tor b and c are supposed to be t wo vectors
collinear respectively with r1 and r, a is any vector not in
their plane Consider the elli pse of whi ch rl and r2 are a
pa ir of conjugate semi -diameters It then appears that rl
and r2 are a lso a pair of conjugate semi-di ameters of that
ellips e They are rotated in the ellipse from r1 toward r2 b y
a sector of whi ch the area is to the area of the whole ellipse
as q is t o 2
Such a change of position has been called an
elliptic rotation through the sector q
Th e ellip s e of which r and r are a pair of conjugate semi
l
2
diameters is called t h e dire c tio n a l e llip s e of th e bi ve c to r r
When the b ivector has a real direction the di rectional ellipse
reduces to a right line in that direction When the b ivec t or
has a comple x di rection the ellipse is a true ellipse Th e
angular direction from the real part r1 to the complex part r2
is considered as the positive direction in t h e dir ec t ional
el lipse and m u st al ways b e known If th e real and imagi
nary parts of a b ivector turn in the positive dir ection in the
ellipse they are said to be advanced if in the negative dirc o
tion they are said to be retarded Hence mu ltip lic a tio n of a
.
.
.
’
.
’
,
.
.
.
.
.
,
.
.
VE C TOR ANAL YSIS
4 32
bi ve c to r by
a
a
s e c to r e q u a
c yc lic
re t a rds
f
a c to r
l t o th e
a n
l
e
g
i t i n i ts direc tio n a l
f the c y c li c fa c t o r
o
el
lip se by
y
.
is always possib le to multiply a bivector b such a cycli c
factor that the real and imaginar parts become coincident
with the ax es of the e llipse and are pe rp endi cular
It
y
.
To
accomplish the reducti on proceed as follo ws
r
Ifa
o
o
Form
r
b = 0,
r
-r
— b
r
o
tau
With this value of
b)
.
+ i b,
a
r
o
b
2 q
a
i
l p
the ax es of the d rectiona elli se
q
a re
given by the equation
a
+ i b =
(
cos
q
—
y
i
s in
q)
r
.
case the real and imaginar par t s a and b of a b ivector
are equal in magnitude and perpendic ul ar in direction b oth a
and b in the e x pression for r r vanish Hence the angle
A
is
indeterminate
rectional
ellipse
i
s
a
circle
T
h
di
e
q
b ivector whose di rec t ional ellipse is a circle is called a c i rc u
l a r bi ve c to r
Th e necessar and s u ffi cient con di tion that a
non-vanishing bivector r be circular is
In
.
.
.
y
.
r
r
condition r
is not su ffi cient
Th e
r
to
r
o
r
circular
.
r
which for real vecto rs implies
ens u re the vanishing of a bivector
O,
.
r
VECTOR ANA L YSIS
434
Th e fi rst
equation shows that r2 and 3 2 are perpendicular a n d
hence s 1 and 8 2 are perpen di cular M oreover the second
shows t hat the angular direc t ions from r1 t o r2 and from 3 1 t o
t h e a x es of the direc t ional ellipses
are
the
same
and
that
8
2
of r and s are propor t ional
Hence t h e conditions for perpendiculari t y of t wo b ive c t o rs
whose planes coincide are that their direc t ional el l ipses are
similar the angular direction in b oth is t h e s ame and t h e
1
major ax es of the ellipses are perpendicular Ifb o t h vectors
have real directions the condi t ions degenerate into the per
T
h
of
hose
directions
n di c u l a rit
t
e conditions t herefore
e
p
y
hold for real as well as for imaginar vectors
et r and s b e t wo perpendicular b ive c t o rs the planes of
which do not coincide Resolve r1 and r2 ea c h in t o t wo com
pone nte respectively parallel and perpendicular t o the plane
of s Th e components perpendicular to that plane contrib ute
nothing to t h e value of r 3 Hence the components of r1
and r2 parallel to t h e plane of 8 form a b ivector r whi c h is
perpendic u lar to 8 To thi s b ivector and s t h e conditions
stated ab ove apply Th e di rectional ellips e of t h e b ivector r
is eviden t ly the projection of the directional ellipse of r upon
the plane of 3
Hence if two b ive c t o rs are perpendicular the di rectional
ellipse of either b ivector and the dire c tional ellips e of the
other projected upon t h e plane of that one are similar have
the same angular direction and have their major ax es per
.
,
,
.
,
,
.
y
.
L
.
.
.
.
’
.
’
.
.
,
,
,
n
d
i
u
l
r
e
c
a
p
.
C onsider
a bivector of
n
th e
9
( )
A
where A and m are b ive c t o rs and
position vec t or of a point in space
.
1
t he
m
It s h o ul d be
sa
e
as
t he
type
is a b is c a la r r is the
It is t herefore to b e con
n
.
mm g
d t h a t t h e c o n d i t i o n o f p e rp e n di c u l a ri t y o f a j o r a x e s is n o t
c o n di t i o n o fp e rp e n di c ul a ri t y o f re a l p a rt s a n d i
a in a ry p a rt s
no te
.
HARM ONI C
VI B RA TI ONS AND
B I VE C TOR S
4 35
as real t is the scalar variable time and
be considered as real
et
si
de re d
.
.
L
A
AI
i A
m
m1
i m2 ,
n
I)
D
A
( I
A
( I
=
n
i Aa )
i A2 )
1
e
e
i
'
( mx o
- m
e
'
n
r
g,
+ t mg
m
e
r
a
g
r
— n1
ua
t
rd
—mt)
has been seen b efore the factor ( AI + i A2 )
represents a t rain of plane waves of ell ip t ic harmonic vi b ra
tions Th e vib rations ta ke place in t h e plane of AI and A2
in an ellips e of which AI and A2 are conjuga t e semi di am
Th e di s placement of t h e vi b ra t ing poin t from t h e
e t e rs
cente r of t h e ellipse is given b y the real part of t h e fac t or
Th e veloci t y of t h e poin t reversed in di rec t ion and divided
b y u 1 is given b y the pure imaginary par t
Th e wave a d
vances in t h e direction m l Th e other fac t ors in t h e e xpre s
m
sion are dampers Th e factor e
is a damper in t h e
di rec t ion m2 A s the wav e pro c eeds in t h e direc t ion 1112 it
dies a way Th e fac t or e is a damper in time If n 2 is
negative the wave dies away as time goes on If n 2 is posi
tive t h e wave incre a ses in energy as time increas e s Th e
presence ( for un limi t ed t ime ) of any such factor in an e x
pre s sion which represents an ac t ual vi b ra t ion is c learly inad
missib le It con t radic t s t h e law of cons erva t ion of energy
In any physical vi b ra t ion of a con s erva t ive s y stem n 2 is n e
c e s s a ril y nega t ive or z ero
Th e general e x pression ( 9 ) t herefore represents a train of
plane waves of ellipt ic harmonic vib ra t ions da mped in a
de fin ite direc t ion and in t ime Two such waves may b e com
pounded b y addi ng t h e b ive c t o rs which represen t t hem If
the e x ponent m r n t is t h e same for b o t h the resul t ing
train of waves advances in the same direction and has t h e
As
,
.
,
-
.
.
.
.
‘
"
r
.
.
"
"
.
.
.
.
.
.
.
.
.
VECTOR A NAL YSIS
43 6
same period and wave -length as the individual waves
vib rations however take place In a di fferent ellipse
waves are
—
-—
,
,
A
g
.
um
the resultant is
r
n
t)
a nd
B)
(A
e
B
g
flm
r
a
—nc
If the
a t)
t
un
Th e
.
.
comb i n i ng two t rains of waves which advance in opposite
direc t ions b u t wh ich are in other respects equal a system of
sta t ionary waves is ob tained
By
.
A
Ae
— mz
o
r
e
— mc
.
f
6
{ ml
e
011 1
3
o
.
r — n t)
A
e
— mg
'
r
e
i ( — m1
-—
(m 1
r
r
.
t)
n
)
r
e
- mz
d
'
6
theory of b ive c t o rs and their applications will not be
carried furt her Th e ob ject in entering a t all upon t his very
shor t and condensed dis cus sion of b ive c t o rs was fi rs t to show
t h e reader h o w t h e s imple idea of a di e c ti o n has to give way
to t h e more complicat ed b u t no less useful idea of a dire c tio n a l
e l li s e when t h e generali z ation from real to imagi n ar
ve c tors
p
is made and second t o set fort h the manner in which a single
b ivector D may be employed to represent a train of plane
waves of elliptic harmonic vi b ra t ions
This application of b i
vec t ors may b e used to gi ve the Theory of ight a wonderfully
1
simple and elegant trea t ment
Th e
.
r
y
,
L
.
m dL by
.
P ro fe s s o r Gibb s in h is c o u rs e o f l e c t u re s o n
Th E l t m g t i Th o y of ight d e l i ve re d bi a n u a ll y a t Ya l e Un i ve rs i t y
B ive c t o s we re n o t u s e d in t h e s e c o n d p a rt o f t h i s c h a p t e
b e c a u s e in t h e O pi i o n
o f t h e p re s e n t a u t h o r t h e y p o s s e s s n o e s s e n t i a l a dva n t a
e o ve r re a l ve c t o rs u n t il
t he
o re a d va n c e d p a rt s o f t h e t h e o y
ro t a t i o n o f t h e pl a n e o f p o la ri a t i o n by
a n e t s a n d c ry s t a ls t o t a l a n d
a re re a c h e d
e t a ll ic re fle c t i o n e t c
1
Su c h
e
us e o
e c ro
f bive c t o rs is
a
ne
c
e
r
a
e
n
,
.
g
r
m
mg
n
r,
,
m
r
z
,
,
.
,
.