**
Linear Algebra: **
Finite dimensional vector spaces; Linear transformations and
their matrix representations, rank; systems of linear
equations, eigen values and eigen vectors, minimal
polynomial, Cayley-Hamilton Theroem, diagonalisation,
Hermitian, Skew-Hermitian and unitary matrices; Finite
dimensional inner product spaces, Gram-Schmidt
orthonormalization process, self-adjoint operators.

**
Complex Analysis: **
Analytic functions, conformal mappings, bilinear
transformations; complex integration: Cauchy’s integral
theorem and formula; Liouville’s theorem, maximum modulus
principle; Taylor and Laurent’s series; residue theorem and
applications for evaluating real integrals.

**
Real Analysis: **
Sequences and series of functions, uniform convergence,
power series, Fourier series, functions of several
variables, maxima, minima; Riemann integration, multiple
integrals, line, surface and volume integrals, theorems of
Green, Stokes and Gauss; metric spaces, completeness,
Weierstrass approximation theorem, compactness; Lebesgue
measure, measurable functions; Lebesgue integral, Fatou’s
lemma, dominated convergence theorem.

**
Ordinary Differential Equations: **
First order ordinary differential equations, existence and
uniqueness theorems, systems of linear first order ordinary
differential equations, linear ordinary differential
equations of higher order with constant coefficients; linear
second order ordinary differential equations with variable
coefficients; method of Laplace transforms for solving
ordinary differential equations, series solutions; Legendre
and Bessel functions and their orthogonality.

**
Algebra: **
Normal subgroups and homomorphism theorems, automorphisms;
Group actions, Sylow’s theorems and their applications;
Euclidean domains, Principle ideal domains and unique
factorization domains. Prime ideals and maximal ideals in
commutative rings; Fields, finite fields.

**
Functional Analysis: **
Banach spaces, Hahn-Banach extension theorem, open mapping
and closed graph theorems, principle of uniform boundedness;
Hilbert spaces, orthonormal bases, Riesz representation
theorem, bounded linear operators.

**
Numerical Analysis: **
Numerical solution of algebraic and transcendental
equations: bisection, secant method, Newton-Raphson method,
fixed point iteration; interpolation: error of polynomial
interpolation, Lagrange, Newton interpolations; numerical
differentiation; numerical integration: Trapezoidal and
Simpson rules, Gauss Legendre quadrature, method of
undetermined
parameters; least square polynomial approximation; numerical
solution of systems of linear equations: direct methods
(Gauss elimination, LU decomposition); iterative methods (Jacobi
and Gauss-Seidel); matrix eigenvalue problems: power method,
numerical solution of ordinary differential equations:
initial value problems: Taylor series methods, Euler’s
method, Runge-Kutta methods.

**
Partial Differential Equations: **
Linear and quasilinear first order partial differential equations,
method of characteristics; second order linear equations in
two variables and their classification; Cauchy, Dirichlet
and Neumann problems; solutions of Laplace, wave and
diffusion equations in two variables; Fourier series and
Fourier transform and Laplace transform methods of solutions
for the above equations.

**
Mechanics: **
Virtual work, Lagrange’s equations for holonomic systems,
Hamiltonian equations.

**
Topology: **
Basic concepts of topology, product topology, connectedness,
compactness, countability and separation axioms, Urysohn’s
Lemma.

**
Probability and Statistics: **
Probability space, conditional probability, Bayes theorem,
independence, Random variables, joint and conditional
distributions, standard probability distributions and their
properties, expectation, conditional expectation, moments;
Weak and strong law of large numbers, central limit theorem;
Sampling distributions, UMVU estimators, maximum likelihood
estimators, Testing of hypotheses, standard parametric tests
based on normal*, X2 , t, F *– distributions; Linear
regression; Interval estimation.

**
Linear programming: **
Linear programming problem and its formulation, convex sets
and their properties, graphical method, basic feasible
solution, simplex method, big-M and two phase methods;
infeasible and unbounded LPP’s, alternate optima; Dual
problem and duality theorems, dual simplex method and its
application in post optimality analysis; Balanced and
unbalanced transportation problems, u -υ method for solving
transportation problems; Hungarian method for solving
assignment problems.

**
Calculus of Variation and Integral Equations: **
Variation problems with fixed boundaries; sufficient conditions for
extremum, linear integral equations of Fredholm and Volterra
type, their iterative solutions.