GATE SYLLABUS FOR MATHEMATICS (MA)

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 -u 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.


Source: http://gate.iitd.ac.in/
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