Calculus: Functions of two or more variables, continuity, directional derivatives, partial
derivatives, total derivative, maxima and minima, saddle point, method of Lagrange’s
multipliers; Double and Triple integrals and their applications to area, volume and surface area;
Vector Calculus: gradient, divergence and curl, Line integrals and Surface integrals, Green’s
theorem, Stokes’ theorem, and Gauss divergence theorem.

Linear Algebra: Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank and nullity; systems of linear equations,
characteristic polynomial, eigenvalues and eigenvectors, diagonalization, minimal polynomial,
Cayley-Hamilton Theorem, Finite dimensional inner product spaces, Gram-Schmidt
orthonormalization process, symmetric, skew-symmetric, Hermitian, skew-Hermitian, normal,
orthogonal and unitary matrices; diagonalization by a unitary matrix, Jordan canonical form;
bilinear and quadratic forms.

Real Analysis: Metric spaces, connectedness, compactness, completeness; Sequences and
series of functions, uniform convergence, Ascoli-Arzela theorem; Weierstrass approximation
theorem; contraction mapping principle, Power series; Differentiation of functions of several
variables, Inverse and Implicit function theorems; Lebesgue measure on the real line,
measurable functions; Lebesgue integral, Fatou’s lemma, monotone convergence theorem,
dominated convergence theorem.

Complex Analysis: Functions of a complex variable: continuity, differentiability, analytic
functions, harmonic functions; Complex integration: Cauchy’s integral theorem and formula;
Liouville’s theorem, maximum modulus principle, Morera’s theorem; zeros and singularities;
Power series, radius of convergence, Taylor’s series and Laurent’s series; Residue theorem
and applications for evaluating real integrals; Rouche’s theorem, Argument principle, Schwarz
lemma; Conformal mappings, Mobius transformations.

Ordinary Differential equations: First order ordinary differential equations, existence and
uniqueness theorems for initial value problems, linear ordinary differential equations of higher
order with constant coefficients; Second order linear ordinary differential equations with variable
coefficients; Cauchy-Euler equation, method of Laplace transforms for solving ordinary
differential equations, series solutions (power series, Frobenius method); Legendre and Bessel
functions and their orthogonal properties; Systems of linear first order ordinary differential
equations, Sturm's oscillation and separation theorems, Sturm-Liouville eigenvalue problems,
Planar autonomous systems of ordinary differential equations: Stability of stationary points for
linear systems with constant coefficients, Linearized stability, Lyapunov functions.

Algebra: Groups, subgroups, normal subgroups, quotient groups, homomorphisms,
automorphisms; cyclic groups, permutation groups, Group action, Sylow’s theorems and their
applications; Rings, ideals, prime and maximal ideals, quotient rings, unique factorization
domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisenstein’s
irreducibility criterion; Fields, finite fields, field extensions, algebraic extensions, algebraically
closed fields

Functional Analysis: Normed linear spaces, Banach spaces, Hahn-Banach theorem, open
mapping and closed graph theorems, principle of uniform boundedness; Inner-product spaces,
Hilbert spaces, orthonormal bases, projection theorem, Riesz representation theorem, spectral
theorem for compact self-adjoint operators.

Numerical Analysis: Systems of linear equations: Direct methods (Gaussian elimination, LU
decomposition, Cholesky factorization), Iterative methods (Gauss-Seidel and Jacobi) and their
convergence for diagonally dominant coefficient matrices; Numerical solutions of nonlinear
equations: bisection method, secant method, Newton-Raphson method, fixed point iteration;
Interpolation: Lagrange and Newton forms of interpolating polynomial, Error in polynomial
interpolation of a function; Numerical differentiation and error, Numerical integration:
Trapezoidal and Simpson rules, Newton-Cotes integration formulas, composite rules,
mathematical errors involved in numerical integration formulae; Numerical solution of initial
value problems for ordinary differential equations: Methods of Euler, Runge-Kutta method of
order 2.

Partial Differential Equations: Method of characteristics for first order linear and quasilinear
partial differential equations; Second order partial differential equations in two independent
variables: classification and canonical forms, method of separation of variables for Laplace
equation in Cartesian and polar coordinates, heat and wave equations in one space variable;
Wave equation: Cauchy problem and d'Alembert formula, domains of dependence and
influence, non-homogeneous wave equation; Heat equation: Cauchy problem; Laplace and
Fourier transform methods.

Topology: Basic concepts of topology, bases, subbases, subspace topology, order topology,
product topology, quotient topology, metric topology, connectedness, compactness, countability
and separation axioms, Urysohn’s Lemma.

Linear Programming: Linear programming models, convex sets, extreme points; Basic feasible
solution, graphical method, simplex method, two phase methods, revised simplex method ;
Infeasible and unbounded linear programming models, alternate optima; Duality theory, weak
duality and strong duality; Balanced and unbalanced transportation problems, Initial basic
feasible solution of balanced transportation problems (least cost method, north-west corner rule,
Vogel’s approximation method); Optimal solution, modified distribution method; Solving
assignment problems, Hungarian method.