# Unit I

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National P.G. College

Department of Mathematics

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
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

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

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

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

Class: B.A/B.Sc.

Subject : Mathematics

Semester : III

Paper I

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

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

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

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

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

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

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

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
.

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

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

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.