ALGEBRA,
fgo
.l
;ri"i is nt"i week ) . (80
Mar*s)
Chapter 1l Thgory of Numbgq (!2 hrs)
Atgorithm - Divistbility;.frim^
^
'{ '\'\r''!^''tcita
nrtmbe*;oJ*:?:t:ri:tflrT.XH.illtHlffit"":
I,l'ffJ'th1."iH'1ilffiil'
uencss - rinear congruences
and Lagrange.
n-r,nr:Hil".::1':
of Euler'
,i"JffiiTi'Hfi'
aneous congruences - theorems
Ghapter 2 :Theory of Equatlons (16 hrs)
- polynomials. with integral coefficients
of equatiols - Euclid,s algorithm
(statement:ry]
theo
:'3l,Iffi
-
--.--^-'i 'i^^'s11'or at!6ora
rcrions
)ns
equations by cardon,s methoo
an coordinates
-
Remainder theorem
-
;'l':*H':i0"":"illil"i'$ffffi:
itj::,i-*'kn":$.,61"il1?'lllH':'ffffili[;JltTi"''
t*"r-"q'"tion'. ;;;!11t^t ..t*t^:i:15;y:t1'ti",t"ln'"-,n:::unn
- roruing quartic rqu"ilon. by Descartbs'and
Ferrari's Method'
Chapter 3: Analytlcal Geometry ( 3{ hrs)
cartesian co-ordinates and
in three dimensional space - Relation between
''
^- (ca rtes i an ani-vecto r
#ffi;-
fo
rm),:,
il?1'j* t:'T*i"(
:'31,il"o'
two lines
betwden
' Angle ?ffi
lff
Fio1""1on on a straight line
I
hedron.
plalg: - Equations of planes
ht lines (cartesian and vector form) coaxlal planes - parallel and
Annta between
hatween olanes
pr"n"d - coaxial
frJifi#i'ingr"
of angles between
perpendicutar frim a point to a plane - Bisectors
lines.
skew
two
Janb-pr"p. - Shortest distance between
(34 hrs)
Differentlal Catculus
,t
Absolu e value - Interval:
Chapter
4.
- Inequalities - ;*,el-itgtiry,*3,gyl#'fi
^,F:itiT::^gl*"- H#::::fri,",J;t':ht""i
;!il;H:"'?'l',',T-ffi
S--"fffl
tunctions
ffi ,Hfl"ffi
- Maxima
r r,*'"'- i/ronotone
Numbers
.;l'fi
l#"l'JgJi^:iljrl'i'lt";,ffi:'.;lltdJ#il-iliSnit
.!^- Fr-r-r
uacAit arac
annla heiweon thg fadiUS
ffi
,ffi1*l:
f, iilin:"i3"XT:f,t-#'J3'ff
':l:119:"r1,**"*g::lmil*:3gffi
'yrH,l;J;-,iiilli:i
j*.1,::y""ngmm;f"roefg
or
curuahrre
ll[i;?i,il3ll'I,?ffi
radius
p"i"i.n'oinates curvature -
S",'.,rTilj5:[t;llilif:ffi:ffift;;;;il;;;
evolutes'
of curvature
-
for reference
and Ganapathy - Algebra
Natarajan, Manicavachogam Pillay
of nnitytical Solid Geometry
Shanthinaraydn
:E6;;6
and 2
ral Calculu
calculus and Analytical Geometry'
courant and John - Introduction to
Shanthlnarayan''Differential Calculus
-topics from the Theory ol Numbers.
Theory
;M. Vd;gradov' Elements of Number
to Theory ol Numbers'
Introduction
'nn
H.d.
l. Niven and
S. e. Malik' Basic Number Theory'
dilnO
,ig.
'i;.10.
11.
i,itftit"n
3 Math.
-
SEMESTER
JI
ALGEBBn,
^nJlffiXfll?t"$frtl
ii?.o.culus
,
Chapter
-
rl
(96 hrs) (6 hrs/ week
)
.
(80 Marfuy
1. Matrfces (40 hrs)
ffim;,%m$ I,fr;$lTg#l*'ffi#'$g*:^":l^:::[^'v i:: Hermitian and
l.lH!%ffff
:lff 9,;?:fl #ff;['*ifmll,'*lj":$i:t"lf '".,^t":^:.:*"fi ri'::TT,,T$'ii:
:li;i:fffi
skew
-e
ff :_!nlTco
:5:i?:,*^ll;':?n:,'*gll':;:"i,":ld:i""'"'t::'ffi
of rank u noe i r'"r" ni.r'i=ip"l'.,irons Inverse
of a non-singuiar
:tr:1tiT:. op€rations.
efementary
system of m linear equations in n unknowns
5il*3'lHilf
- matrices
associated,with linear equations
t?'l"J*T,':1;':li:ri'*::g*;.*li,l',ilr,omof
-
ll
by
Diff
valt
infit
trivial and non-
Asl
matrix
"n.o-utanonon.homoseneous
Eigen values and eigen vectors of. a square
rnatrix - characteristic e_quation of
a square matrix - Eigen
values and eigen'vec-tors of a real symteiril
rari, - prop"rti.. -.ol"gon"rir"iion
of a reat symmetric
to deiermine rhe rio*ei, or square
llli?J";"?ofJr;,ffiil11"J.1ff::"''
matr'ices and
- 6;i;;;'ns
Chapter
ofs
Fun
limit
lhec
deri
2. euadric curves (20 hrs)
Jac,
I'i$?:trff:'!".1?*",1r"^1':ffi1i1:i'?j1:jiT?^. ly:,,'.:r..o11desree - Discriminant and trace
j5[?;":',il::':,',"ly"Tir;". Tfr :iTffiiJiile?'il'1,fl'TfJ'Til:
;jlr$:H33:i'jl1[f,il,i';i
)i
;l d; X' I ; ; "&?'i E J iT'j' ['"
g."g
I ilfili, *:.', :,,"" ff
lI
?*y,:9i^ : :il*;;;,
iy:i,
;
3fflT""J"in'J'i""::L::'^'x,ll-jfr i
"l ",,"IJf'.,"ilroli[,]'.;:"f,"i:ffilty'j
i;ffi:li?jl:?:f;?Lf"_.H:il:H[:,:^;trff;?."#:::::!i#,1".'B",jn";"?:Bfff.'H;:.il:
Hyperboloids
- paraboloids -'Ruled Surfaces.
Defi
adc
2l ir
Defi
Bob
Detinite integrals
-
properties.
;H:T
#:"i :ffi1"H.:'^:"1 ';"*ons . Reduction
4.:
Books for Reference
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
5.i
6.
ulus
1
and2
gral Calculus
calculus and analytical geometry
y.aF Ganapathy - Afgebra
alytical Solid Geometry
4 Math.
1
SEMESTER III
-
MATHEMATICS MAT 303
CALCULUS -l (96 hrs)(6 hrs /weeh ){8O,Marks)
Chapter
1.
LlmlB and Contlnulty (10 hrs)
Continuous functions - discontinuous functions - theorems on continuity
=l:
closed interval - Uniform continuity (explaining the idea).
-
/
i
-
Functions continuous
Chapter 2. Dlffercntlablllty and lts appllcatlons (26 hrs)
- Theorems - Rolle's theorem - Lagrange's Mean value theorem - Cauchy's mean
theorem
- Taylor's theorem - Maclaurin's theorern - Generalized mean value theorem - Taylor's
[tfr"
power series expansion - Maclaurin's infiniie series -lndeterminate forms ,
and
series
fifinite
points - Cusp, node and conjugate points - Tracing
Asymptotes - Envelopes - Singular points - Multiple
,..0f itandard curves with Cartesian and polar equations.
.{
Chapter
3.
Partlal Derlvatives (30 hrs)
Gtrapter 4. Llne and MulUple lntegrals (3(lhrs)
Definition of a line integral and basic properties - Examples on evaluation of line integrals - Definition of
a double integral - Conversion to iterated integrals - Evaluation of double integrals 1) under given limits
2) in regions bounded by given curves - Change of variables - Surface areas.
Definition of a triple integral - Evaluation - Change of variables - Volutn€ as a triple integral.
Boolee
for reference:
1. Lipman Bers - Calculus, volumes 1 and 2
2. Asha Rani Singhal and M.K. Singhal - A first Course in Rear Analysis.
3. S.C. Malik - Real AnalYsis
4, Shanthinarayan' Differential Calculus
5. Shanthinarayan - Integral Calculus
6. N.Piskunov - Ditferential and Integral Calculus.
5 Math.
SEMESTER IV
ALG EBRA AND. DIFFER ENTIAL EOUATIONS
80 Marks (96 hrs) (6 hrsAreefty
Chapter 1 - Group Theory
(jS hrs)
.
Definition and examples of groups.-Some general propertles oftroups
Permutations group of permutatlons cycllc permutatlons Even and odd permutations.
Powers of an element of a group -Subgroups-Cyclic groups, Zn and Z
Cosets, Index of a group Lagrange's theorem consequences Normal subgroups
Quotient groups-Homomorphism lsomorphism Autornorphism
Fundamental theorem of homomorphism - lsomorphism -Direct product of groups
Cayley's theorem
Sequenct
sequenc€
Converge
sequenc€
.,
Chapter 2 - Dltterentlal Equatlons ( 48 hrs)
Definitions and examples of differential equations.
curves-Differential equations of first order, sepa
coefficients-Exact equations Linear equations of ord
Integrating factors found by inspection The determi
by the equation. Bersoulli's equation.. Coefficients
coordinates.
Equations of first ord_er and higher degee
Equations solvable for x, solvable
1or
y, solvabte for
p
lnfinite st
converge
Alembert'
test for at
Cluciraut's equation- Singular solutions and geometrical meaning.
Ordinary Linear differential equations with constant coefficients - complementary function - particular
integral - Inverse differential operators.
Summatir
Books for Referencesi
lmproper
following
integrals
Daniel A Murray - Introductory course to Differential equations.
Ranville and Bedient - A short course in Differential equations.
l.N..Herbtein - Topics in Algebra.
4. G.D. Birkhofi and S Maclane - A briei Survey oi Mociern Algebra.
5. J.B. Fraleigh - A first course in Abstract Algebra.
1.
2.
3.
6 Math.
_i
I
Books fr
1. Lipm.
2. N.Pis
3. Asha
4. S. C.
5. B.S.
6. E.Kn
SEMESTER V
iIATHEIIATICS MAT 505
80 Marks (48 hrs) (3 hrs /week)
CALCULUS ll
Chapter 1. Real Numbers (6 hrs)
-
- Order structure - Bounded and unhrlunded sets - Supremum and infimum
important
subsets of R - Archimedean Property of real numbers - Countable
Sorne
FieH stLcture
-
urpuntaHe sots.
Chapter 2. Real Sequences (16 hrs)
-
Infimum and supremum of a
Bounded and unbouncled sequences
Sequences of real numbers
quotients
theorems on lirnits
product
limits
Standard
of
and
sequence -Limit of a sequence Sum,
properties
Monotonic
Subsequences
Standard
sequences
Convergent, divergent and oscillatory
principle
general
of
convergence.
point
Cauchy's
sequence
properties
of
a
Limit
sequences and their
-
-
eous
r-Ber
rsted
rclar
orP
-
Chapter
)s of
-
-
3. Infinite Serles
-
-
-
-
(16 hrs)
-
Properties of
Convergence divergence and oscillation of series
lnfinite series of real numbers
test D'
root
Cauchy's
tests
Comparison
series
Geometric
term
series
Positive
convergence
Alombert's ratio test, Raabe's test, Integral test - Absolute and conditional convergence - D'Alembert's
test for absolute convergence - Leibnitz's test for alternating series.
-
-
-
'
-
Summation of Binomial, Exponential and logarithmic series.
Chapter 4. lmproper Integrals ( 10 hrs)
lmproper integrals of the first and second kinds - Convergence -Gamma and Beta functions, and results
foliowingi the definitions - Connection between Beta and gama functions - Applications to evaluation of
integrals - Duplication formula' Sterling formula.
Books for fieierence
1.
Liprnan Bers
- Calculus,
Volumes 1 and 2
2. N.Piskunov - Differential and Integral Calculus
g. Asha Rani Singhal and M.K. Singhal- A First Course in Real Analysis
4. S. C. Malik - Real Analysis
5. B.S. Grewal - Higher Engineering Mathematics
6. E.Kreyszig - Advanced Engineering Mathematics
.
7 Math.
SEMESTER V
nLceenA AND
"iltlTttot'*
TATn:?lr hrs) (3 hrsftveek)
Chapter 1. Rlngs and Flelds ( 32 hrs
-
)
Rings - Examples tntegral Domains - D
ring - Ordered integral domain
lmbeddi
-
Algebra of ldeals - Principal ideal ring Elements - Polynomial rings - Divisibili
Common Divisois - Euctidean Algorithm - Unique f
Homomorphism of rings - Kernel of a ring homomorg
Maximal ldeals - Prime ldeals - Properties - Unique Factorisation domain
irreducibility.
B
ft-
e'
-
Einstein's Ctiterion of
(r
B
Chapter 2. Riernann Integration (16 hrs)
The Riemann integral - Upper and lower sums - Criterion for integrability - Integrabitity of continuous
functions and monotonic functions Fundamental theorem of Calculus Chlange'of variables
integration by parts - First and second mean value theorems of integrat calculus.
4
6,
6
'i
Books for Reference
1. N. Herstein - Topics in Algebra
2. G.D. Birkhoff and S. Maclane - A Brief survey of Modern Algebra
3. T.K. Manicavachagom Pillai and K.S. Narayanan - Modern Atgebra Volume 2
4, J.B. Fraleigh - A First Course in Abstract Algebra
5. S.C. Malik - Real Analysis.
6. Leadership Project, Bombay University -1-ext Book of Mathematical Analysis.
cl
SEMESTER V
MATHEMATICS IIIIAT 507
APPLIED MATHEMATICS - B0 Marks 48 hrs (3 hrs/week)
Chapter 1. Laplace Transforms (14 hrs)
Definition and basic properties - Laptace transforms of exp kt, coskt, sinkt, t n, coshkt
and sinhkt Laplace transform of e"t f$Vz - problems - Theorems on the derivative of Laplace transform
and the
transform of derivatives - lnverse Laplace transforms problems - (alpha- funclion
theorem on the
Laplace transform of integrals - Laplace transform of F (t) A Convolution theorem - Simple initial value probtems - Special integral equations Solution
of first and
second order differential equations with constant coefficients by Laptace transform method
- Systems of
equations - Laplace transforms of periodic functions.
Chapter 2. Fourler series (10 hrs)'
Introduction
range series
- Periodic functions - Change of interval
Fourier series and Euler formulae
- Even and odd functions -
Hall
.
Chapter 3. Llnear dilferenilat Equations. ( 14 hrs)
Cauchy - Euler differential equations - Simultaneous differential equations (two variables
with constant
coefficients) - Solution of ordinary second order linear differentiai equations by the followinj
methods
8 Math.
t\J
'
changing the independerlf variable
!ii)
oparameter:.
cnansins
con
i) Reduction
iii)
;ufficient
;ffit*t
,'lQl
equations or the
fiil j.J\:*:ll*iJ:SiX^TTll'1ff;"X,'.H?;i:
. ,A -,--r?r, .r-rEr
dzlR'
torm dxlP - dy'o
=
(10 hrs)
Chapter 4. Partlal dltlercntlal equatlfie
g'
- Formation by e.rimination 91 1f'll"'i ^T::Pf;,1,gl*[:,H""j'i'tx'l?:^?t#iili3]
1,:3,T:",q5'':,1";":?gl"lff;i,l'13'j
Ii"Eff,L" $iJi]ii'ol'[iHi[i'!iiii;di^t:':t'."1^Y
:T'i' ti1;d_{1.,ry:1ft*,:l*#l^;"::ll,,A1',:'
i.['Jf#'?l,Tl3ii,.?iir;
:T,"i',?tr
-p;n,curar
/Fgts"'
concepts
/ Prime
eatest
ings
rism
-
ei:?:iT'J"30.-'il";[-"d[T1;1','fi;il;;.:f:::"::l',1ffi[::T
separation or variabres
inte$ar Method ol ;"31i#":lX'il3'[:
;3#,fii1,'-ffT{,*,,i31"n*T::"tlJ"il^?."i'ii"
""i':b?
ion ot
$
G.stephenson
1
- An Introduction
rms
hematies
Mathematlcs
hort Course in Ditlerential Equations
Differential Equations'
t{'
iflUOUS
rles -
to Partial Differential Equations'
1
{
$'
f,
s
aum Series)
ntial Equations
. SEMESTER VI
MATHEMATICS MAT 608
NUMEF|CAL,atrauvs|sANDvEcToRcALcuLUs
80 f,larks (48 h:e! {3 hrs lwaek}
Chapter
l. ( I
hre )
- Bisection method - The method of false
equations
ar sorutions of Argebraic and transcendentai
-|terationmetnoi-lt"*ton-Raphsonmethod-Secantmethod.'|
-
inhkt
rnd the
on the
Chapter2( B hrs)
equations
merical solutions of first order linear differential
Picard's
method
order
:
fourth
Xutta
illd Cung. -
- Euler - cauchy
method
method'
-
Eule/s modifierd
-:
and
ms of
Chaper3( 14 hrs)
il:f#l'iji,ffiil"jffi ffi ,i#:Wif
'fffi',[.{:tili:1il,h'}:"Jffi',:""i'iino]ig'Lrt
.l
Diffe rence oquations.
-gj"e:y
j:*:.-l*"0i;1ff
|i::formurae
ffi :fiffi
inierporation
usins10*:'[ffi
"iJJ*6nJ
Jerivatives
Chapter4. (8 hrs)
ronstanl
nethods
.Numericar integration - General quadrature formula
th rule - Weddlo's rule'
-
Trapezoidal Rule
9 Math.
- simpson's
1/3 rule
-
simpson's
-!
vectons
-
scalans
Clatrrr
- Vector
fleld'- scatar
6. V,ootor Getcufug (lO
rrr")
"..
- v-Tj9, diff.e.renfiqtion - The vestor differental operator
n" oiversance in*,,, del,
or
n"m*ni::%"#llnresrarion -
frelcl
;$',9':$;:l:"ffi?n,3;l;,,tff:H
gcpfi.lor_Retltrlna:
l.
2.
3.
4.
5.
6.
7.
lrgblems of V€c1or Analysis
A Tert Book of Vec.tor C6fculus
I
ance and
!,q. qh"stry -
flbigl
Intrcctuctory Methods
B.S. Grewal- r.trmerical Methods
E. Kreyszig - Advanced Engineering
gEF
rs
fo
il
neers
Egua
0omF
SEiIESTER VI
Lf
vector spaoes
Afgebra of
-
NEAR
Introduction
subspaces
o.oHf :T$l[t#itTl
(3 hrc/u,eek)
Chaptcr 1. Vector Specec (2a hm)
Examptes
- Vector subspaces - criterion
"?^_-!,1""i-rp"n.
and rinear
$1Ht;lTr:fof,T"fic
svstems
- Quotrent
spaces
-
tor a subset to be a subspace
,cependence
-
dependence and tinear
0"n." - Basis or a vector
Ihe (
Homomorphism
"t!lii!$r::3:l!n::transroJri$:131*:ffi$T
- Dlrecl sums - lnner p.o,r.irpaces
- Eucridegn vector space - Distance - iength Propertfes Normal orthogonal
,eciors --e"am-scrrmioi onhogonariz"fir--pr*"r.
complement.
orthogonal
veclor spaces
Greer
lunctir
- Liou
ChrtrEr 2. UnrerTrenrfonnaflonr (24 hn)
urrar hansformations - Linear maps ag matri'.o-r-'-h^--,-r.^--- -r
L
-'L-KEmelanOimaoeof
and image of a
allnearlransfnnnqrr^nllnear transformation
lransformalions-- gnmeniaru'r"tri"es anrt
c r arlcte risuc por ynom i"
c Jr',l
' nct
nd_elfectof
effect of associated
associatedmatricesmatrices 5"t*
-lrancr^
- lingular and non-singular linear
I'8il"ffi1i:ff"t''Ig[9ffiffii;;glg-"9,T,".],:::.::,:Tl:n":E,3::'?:ft[1X#'J[XllTiX[il
r--,il6ilH ;:'ffi?."i' ff .f,lt"i' polynomial.
Automorphlsrn
r
,t
Definil
Slretc
of four
Oonlo
Boolrr for Reference
l.
2.
3.
4.
5.
6.
ffi:nm
pltay
-
Modem
Atgebra,
G.D.Birkhoff and s.Madane Briet
survey or Modem Argebra
Gopalakrlshna - University Algebra
Saymour Upschitr - Theory ario broofems
of Linear Atgebra.
Zeros,
Votume 2
Boolrr
l.
L.r
2- Br
&Se
dsl
5. s,
6. R.
l0 Math.
SEMEEIER VI
itATHEifiA:lrcgmfr6m:
GOIIPLEI( ANALYSIS
-
(S0 Madc)'(4&hrs); (3 hrs I weel$
Cha$rr l'. Introdustlott - Gomp|u' llumbett' (e hnq
of dtomplex number - Geometriel
- Absolute value andOeconfugate
- &rle/s formula - Dot and
theorem
Moivre's
numbe6
complex
of
POlar form
-
comptex number system
rsentation
product.
-
-
points Open sets
Enf,,Uoutt oods Umit points Interior, Exterior, lsolated and boundary
regions.
connected
Simply
Domain
sets
Connected
fiilg;";d"d sets - Compact sets -
-
-
line in complex form
&uation to a circle and a straight
fuRtet
-
Jordan arc
-
Closed contour
Plane.
Chapter 2. Functlone of
a Complex Varlable (1'2
hre)
-
-
Closed
The extended
'
of a function - Continuig and differentlabillty - Analytic functions
Onctlons ol a complex.variable Umit
points - baucny.Riemann equations in cartesian and polar forms Necessary and sufficient
Reat and imagiqw pals oL.an analytic function are
;,rdtid for f to ue Jnatytic - Harmonic lunctionsi)-Mllne
ii) using the concept of
Thomson Method
of anatytic lunctions
-
i6iil6
ilrrJni"
-
'
construaiori
lilarmonic function.
Chapter 3. Compler Integratlon ( 10 hrs )
Proof of Cauchy's Integ;al theorem using
Co
Grssn,s
function
fhe
em
- The Cauchy's Integral formula lor the
of simple line integrals
- Cauchy's
inequality
- Uouvi
'
Chapter 4. TransformaUons (10 hrs)
r,i..
-A-^-- 6-a:^ l - YsvYr
lltrv!
Jacobian of a transformation - ldentity_transformation - Reflection - . ilefinit6n
ttllll
-^? ^ -^
Srietcnins -rnveision -liiraiit".tformations -P.1119il^: T?iil,I:?:5::j:,T?H:" :ltff^Rn:
and circles
Ratio preserving propefi - Preservation ot the f"l,! oI straigh!.!in",:
p"i.sCross
v
;t"fil
v lvur
lrv.. ru
th(z+1lzl'
oli'nb trinsformations w =22, w = sinz, w = ez, w =
Translation
Rotation
-
tdf;fi;ir"pp,nir-Disrrsrion
*
.--^.--^
tr--lJ..^^,d
L--t
-, Resldueg (8 hre)
of
Ghapter 5. Calculus
T
ilt L.V.Ahlfors - ComPlex AnalYsis
p.pallca - tntroducfioh to the Theory of Func'tion of a Complex Variable
2. Bruce
$
0,
tions of a Complex Variable
Complex AnalYsis
6.
Analysis'
d
*
*
*
I
I Math.
tr