National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : I
Paper:I
ALGEBRA
Unit I
Symmetric, Skew symmetric, Hermitian and Skew Hermitian matrices,
Elementary
operation on
matrices.
Inverse
of a matrix. Linear Independence of row and column matrices. Row rank
column rank and rank of a matrix.
Equivalence
of a column and row ranks. Eigenvalues,
eigenvectors and characteristic equation of a matrix: Cayley Hamilton theorem and
its use in
finding inverse of a matrix.
Unit II
Application of matrices to a system of linear (both homogenous and nonhomogeneous)
equations. Theorems on consistency of a system of linear equations.
Relations between the roots and coefficients of general
polynomial equation in one variable,
transformation of equations, Descarte’s rule of signs. Solution of cubic equations (
Cardon’s
method). Biquadra
tic equations.
Unit III
Mappings.
Equivalence
relations and partitions. Congruence modulo n. Definition of
a group
with examples and simple properties. Subgroup. Generation of groups. Cyclic groups. Coset
decomposition. Lagrange’s theorem and its consequences. Fermat’s and Euler’s theorem.
Homomorphism and isomorphism. Normal subgroups. Quotient groups.
Unit
I
V
The fundamental
theorem of Homomorphism and Isomorphism. Permutations groups. Even and
odd permutations. The alternating groups A
n
. Cayley’s theorem. Introductions to rings. Subrings,
Integral domains and fields. Characteristic of a ring.
National
P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : I
Paper: II
DIFFERENTIAL CALCULUS & VECTOR CALCULUS
Unit I
Definition
of the limit of a function, Basic properties
of a lim
it
.
Continuous
functions and
classification of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem.
Unit II
Maclaurin’s and Taylor’s series expansions. Tangents and Normals, Asymptotes.
Unit III
Curvature
. Tests for Concavity
and Convexity. Points of Inflexion. Multiple points. Tracing for
curves in Cartesian and polar coordinates.
Unit IV
Vector Differentiation
. Gradient, Divergence and Curl. Vector Integration. Theorems of Gauss,
Green,
Stokes
and problems based on these.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : II
Paper
:
I
INTEGRAL CALCULUS & TRIGNOMETRY
Unit I
Integration of irrational algebraic functions and transce
ndental functions. Reductions formulae.
Definite integrals. Quadrature. Rectification. Volumes and Surfaces of solid of revolution.
Unit II
Ordinary Differential Equations
Degree and order of a differential equation. Equation of first order and first degre
e. Equations in
which the variables are
separable
. Homogenous equation. Linear
equations
and equations
reducible to the linear form. Exact differential equations. First order higher degree equations
solvable for x, y, p. Clairaut’s form and singular soluti
ons. Geometric meaning of a differential
equation.
Orthogonal
trajectories
.
Unit III
Linear differential equation with constant coefficients. Homogenous linear ordinary differential
equations.
Linear differential equations of second order. Transformation o
f the equation by changing the
dependent variable/the independent variable. Method of variation of parameters.
Ordinary simultaneous differential equations.
Unit IV
Trignometry
DeMoivre’s theorem and its applications. Direct and Inverse circular and hyperb
olic functions.
Logarithm of a complex quantity. Expansion of trignometrical functions.
Gregory’s
series.
Summation of series
.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathemat
ics
Semester : II
Paper
:
I
I
Vector Analysis & Geometry
Unit I
Scalar and vector product of vectors. Product of four vectors. Reciprocal vectors.
Unit II
Geometry
General equation of second degree. Tracing of Conics. Systems of Conics. Confocal Conics.
Po
lar equation
of
conics.
Unit III
Plane, The straight line, Sphere,
Cone,
Cylinder
.
Unit IV
Central Conicoids. Parabolids. Plane
section
of Conicoids. Generating lines. Confocal
Conicoids
.
Reduction of second degree equations.
National P.G. College
Dep
artment of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : III
Paper I
Advanced Calculus
Unit I
Continuity, Sequential Continuity, properties of
continuous
functions, Uniform continuity, chain
rule
of differentiability. Mean value theorems and their geometrical interpretations.
Darboux’s
intermediate
value theorem for derivatives. Taylor’s theorem with various forms of remainder’s.
Unit II
Limit and Continuity of functions of two variables, Partial
differentiation, Change of variables,
Euler’s theorem on homogenous functions . Taylor’s theorem for functions of two variables,
Jacobians.
Unit III
Envelops, Evolutes,
Maxima,
Minima and Saddle points of functions of two variables,
Lagrange’s multiplier
method.
Indeterminate
form.
Unit IV
Beta
and Gamma functions
, Double and t
riple integrals, Dirichlet integral, change of order of
integration in double integrals
.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergrad
uate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : III
Paper
:
I
I
Mathematical Methods
Unit I
Calculus of variations

Variational problems with fixed boundaries

Euler’s equation for
functionals containing first order derivative and one independent v
ariable. Extremals.
Functionals dependent on higher order derivatives. Functionals dependent on more than one
independent variable. Variational problems in parametric form. Invariance of Euler’s equation
under coordinates transformation
.
Unit II
Variationa
l problems with moving boundaries

Functionals dependent on one and two functions.
One sided variations.
Sufficient conditions for extremum

Jacobi and Legendre conditions. Second Variation.
Variational principle of least action
.
Unit III
Laplace Transform
ation

Linearity of the Laplace transformation. Existence theorem for
Laplace
transforms. Laplace transforms of derivatives and integrals. Shifting theorems. Differentiation
and Integration of transforms. Convolution theorem. Solution of Integral equations
and system of
differential equations using the Lap
lace transformation.
Unit IV
Definition
of a sequence. Theorems on limits of sequences. Bounded and Monotonic sequences.
Cauchy’s convergence criterion. Series of non

negative terms. Comparison tests. Cauc
hy’s
integral test. Ratio tests. Raabe’s logarithmic, de Morgan and Bertrand’s tests. Alternating series.
Leibnitz’s theorem. Absolute and conditional convergence
.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : IV
Paper
:
I
Differential Equations
Unit I
Series solution of differential equations

Power series method, Bessel, Legendre and
Hypergeometric equations. Bessel, Legendre and Hypergeometric functions and
their
proper6ties

convergence, recurrence and generating relations
.
Unit II
Orthogonality of functions
. Sturm

Liouville problem. Orthogonality of eigen functions,
Reality
of eigen values. Orthogonality of Bessel’s functions and Legendre polynomials
.
Un
it III
Partial differential equations of the first order. Lagrange’s solution. Some special types of
equations which can
solve
easily by methods other than the general methods. Charpit’s general
method of solution
.
Unit IV
Partial differential equations o
f the second and higher orders. Classification of linear partial
differential equations of second order. Homogenous and non

homogenous equations with
constant coefficients, Partial differential equations reducible to equations with constant
coefficients. M
onge’s method
.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester :
I
V
Paper
:I
I
Mechanics
Unit I
Vitual work, Catenary.
Unit II
Forces in three dimensions, Poins
ot’s central axis, Wrenches, Null lines and planes, Stable and
unstable equilibrium.
Unit III
Velocities
and accelerations along radial and transverse directions, and along tangential and
normal directions. Simple Harmonic motion, Elastic strings.
Uni
t IV
Motion on smooth and rough plane curves, Motion in resisting medium, Motion of particles of
varying mass.
Central orbit, Kepler’s law of motion.
Motion of particle in three dimensions. Acceleration in terms of different coordinates systems.
N
ation
al P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : V
Paper :I
Analysis
Unit I
Riemann integral. Integrability of con
tinuous and monotonic functions.
The fundamental
theo
rem of Integral calculus, Mean value theorem of integral calculus.
Improper integrals and their convergence, Comparison tests, Abel’s and Dirichl
et’s tests,
Frullani’s integral.
Unit II
I
ntegral as a function of parameter, Continuity, derivability and i
ntregrability of an integral of a
function of a parameter
.
Series of arbitrary terms.
Convergence
,
divergence and oscillation,
Abel’s and Dirichlet’s tests,
Multiplication of series
, Double series.
Unit III
Partial derivation and differentiability of real
valued functions of two variables, Schwarz and
Young’s theorem, Implicit function theorem.Fourier series, Fourier expansion of piece wise
monotonic functions
.
Complex numbers as ordered pairs, geometric representation of complex
numbers, Stereographic pro
jection
.
Unit IV
Continuity and Differentiability of complex functions, Analytic functions, Cauchy Riemann
equations, Harmonic functions. Elementary functions, mapping by elementary functions, Mobius
transformations, Fixed points, Cross ratio, Inverse poi
nts and critical mappings, Conformal
mappings
.
Books Recommended:
1.R.V.Churchil & J.W.Brown. Com
p
lex variable and a
pp
lications.
2.Shantinarayan, A course in Mathematical Analysis.
National P.G. College
Department of Mathematics
Semester wise syll
abus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : V
Paper :II
Abstract Algebra
and
Metric Space
s
Unit I
Group

Automorphism, I
nner automorphism. Automorphism groups and their computations.
Conjugacy relations
. Normaliser. Coun
ting principle
and the class equation of a finite group.
Center for group of prime order.
Unit II
Abelianizing of a group and its universal property. Sylow
’
s theorem.
p
–
Sylow subgroup.
Structure theorem for finite Abelian groups.
Ring theory
–
Ring hom
omorphism. Ideals and
Quotient rings. Field of Quotients of an integral Domain. Euclidean Rings.
Unit III
Polynomial Rings. Polynomial over the Rational Field
s
. The Eisenstein Criterion. Polynomial
Rings over Commutative Rings.
Unique factorization doma
in.
R unique factorization domain
implies so is
.
Unit IV
Definition and examples of metric spaces, Neighbourhoods, limit points, interior points. Open
and closed sets. Closure and interior. Boundary points, subspace of a metric space, Cauch
y
sequences, Completeness, Cantor’s interaction theorem.
Contraction principle, construction of real numbers as the completion of the incomplete metric
space of rationals, real numbers as a complete ordered field, dense subsets. Baire category
theorem. Sep
arable, second countable and first countable spaces. Continuous functions.
Books Recommended:
1.
.I.N.Herstein, Topics in Algebra.
2.
G.F.Simmons, Introduction to Topology and Modern Analysis.
National P.G. College
Department of Mat
hematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : V
Paper :III
Linear Algebra
Unit I
Definition and examples of vector spaces. Subspaces. Sum and direct sum of subspaces. Linear span.
Linear depende
nce, independence and their basic properties. Basis. Finite dimensional vector spaces.
Existence theorem for bases. Invariance of the number of elements of a basis set. Dimension, Existence of
complementary subspace of a subspace of a finite dimensional ve
ctor space, Dimension of sums of
subspaces, Quotient space and its dimension
Unit II
Linear transformations and their representation as matrices, The Algebra of linear transformations, The
rank nullity theorem, C
hange of basis, Dual space, Bidua
l space an
d natural isomorphism.
Unit III
Adjoint of a linear transformation, Eigenvalues and eigenvectors of a linear transformation.
Diagonalisation, Annihilator of a subspace, Bilinear, Quadratic and Hermitian forms.
Unit IV
Inner product spaces

Cauchy

Schwarz
inequality, orthogonal vectors. Orthogonal complements,
Orthonormal sets and bases, Bessel’s inequality for finite dimensional spaces, Gram

Schmidt
orthogonalization process.
Books Recommended:
K. Hoffman and Kunze, Linear Algebra.
Nation
al P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : VI
Paper :I
Numerical Analysis
Unit I
Solution of equations: bisection, Secant, Regula Falsi, Newton’s method, Roots o
f polynomials.
Interpolation, Lagrange and Hermite interpolation, Dividend differences, Difference schemes,
Interpolation formula using differences.
Unit II
Numerical differentiation
.
Numerical Quadrature: Newton’s Cote’s Formulas, Gauss Quadrature
Formul
as, Chebyehev’s Formulas.
Unit III
Linear equations: Direct method for solving systems of linear equations( Gauss elimination, LU
Decomposition, Cholesky Decomposition), Iterative methods(Jacobi, Gauss Seidel, Relaxation
methods).
The Algebraic Eigenvalue
problem: Jacobi’s method, Givens method’ Householder’s
method, Power method, QR method, Lanczos’ method
.
Unit IV
Approximation: Different types of approximation, Least square polynomial approximation,
Polynomial approximation using Orthogonal Polynomials
. Approximation with Trignometrical
functions, Exponential functions, Chebychev Polynomials, Rational Functions.
Books Recommended:
M.K.Jain, S.R.K.Iyenger, R.K.Jain, Numerical Methods

Problems and solutions, New Age
International (P) Ltd.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Mathematics
Semester : VI
Paper :II
Differential Geometry
Unit I
Local theory of curve
s

Space curves. Examples. Planar
curves, tang
ent and normal and
binormal, Osculating plane, normal plane and rectifying plane, Helices, Serret

Frenet apparatus,
contact between curve and surfaces, tangent surfaces, involutes and evolutes of curves, Intrinsic
equations, fundamental existence theorem f
or space curves.
Unit II
Local theory of surfaces

Parametric patches on surface curve of a surface, surfaces of
revolutions, Helicoids, metric

first fundamental form and arc length.
Local theory of surfaces (
Cont.). Direction coefficients, families of cu
rves, intrinsic properties.
Unit III
Geodesics, canonical geodesic equations, normal properties of geodesics, geodesics curvature,
geodesics polars, Gauss

Bonnet theorem, Gauss
ian curvature,
space of constant curvature.
Unit IV
Second fundamental form of
a space, principal curvature, normal curvature, Meusneir’s theorem,
mean curvature, Gaussian curvature, umbilic points, lines of curvature, Rodrigue’s formula,
Euler’s theorem. The fundamental equation of surface theory

The equation of Gauss, the
equatio
n of Weingarten, the Mainardi

Codazzi equation.
Books Recommended:
An Introduction to Differential Geometry, T.J.Willmore.
National P.G. College
Department of Mathematics
Semester wise syllabus for undergraduate
Class: B.A/B.Sc.
Subject : Ma
thematics
Semester : VI
Paper :III
Numerical Solution of Differential Equations and Tensor
Unit I
Ordinary differential equations: Euler
method, S
ingle step methods, Runge

Kutta method, Multi

step methods, Milne

Simpson method, Methods based on Numeric
al integration, Methods based
on numerical differentiation, boundary value problems, Eigenvalue problems.
Unit II
Finite difference method for linear second order differential equations. Local truncation error,
derivative boundary conditions, solution of
tridiagonal systems.
Finite difference method for nonlinear second order differential equations, Local truncation
error, Newton Raphson method for system of algebraic equations. Linearization through Newton
Raphson method, Irreducible monotonic matrices, c
onvergence of finite difference scheme using
exact inverse and also using bounds for the inverse
.
Unit III
Tensor algebra: Vector spaces, the dual spaces, tensor product of vector spaces, transformation
formulae, contraction, special tensor, inner produc
t, associated tensor.
Differential Manifold

examples, tangent vectors, connexions, covariant differentiation.
Unit IV
Elements of general Riemannian geometry

Riemannian metric, the fundamental theorem of
local Riemannian Geometry, Differential parameter
s,
curvature tensors, Geodesics, G
eodesics
curvature, geometrical interpretation of the curvature tensor and special Riemannian spaces
.
Books Recommended:
1.
M.K.Jain, S.R.K.Iyenger, R.K.Jain, Numerical Methods

Problems and solutions, New
Age Internationa
l (P) Ltd.
2.
An Introduction to Differential Geometry, T.J.Willmore.
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