# Gate Syllabus for Mathematics (MA)

Check here the detailed view of GATE Mathematics syllabus described in this article. Students are advised to collect information about syllabus in case they are going to appear in upcoming GATE entrance exam. Without having proper knowledge of syllabus one can’t plan appropriate preparation and as a result they can’t achieve the scores as expected. So, let’s have a look on the GATE syllabus for Mathematic (MA) branch.

Part 1: Linear Algebra

Finite dimensional vector spaces; systems of linear equations, Linear transformations and their matrix representations, rank; eigenvalues and eigenvectors, Cayley-Hamilton Theorem, minimal polynomial, diagonalization, Hermitian, Jordan-canonical form, SkewHermitian and unitary matrices; Finite dimensional inner product spaces, self-adjoint operators, definite forms, Gram-Schmidt orthonormalization process

Part 2: Complex Analysis

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

Part 3: Real Analysis

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

Part 4: Ordinary Differential Equations

First order ordinary differential equations, systems of linear first order ordinary differential equations, existence and uniqueness theorems for initial value problems, linear ordinary differential equations of higher order with constant coefficients; method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method); linear second order ordinary differential equations with variable coefficients; Legendre and Bessel functions and their orthogonal properties.

Part 5: Algebra

Groups, normal subgroups, subgroups, quotient groups and homomorphism theorems, Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria; automorphisms; cyclic groups and permutation groups, Sylow’s theorems and their applications;  Fields, finite fields, field extensions.

Part 6: Functional Analysis

Normed linear spaces, Hahn-Banach extension theorem, Banach spaces, open mapping and closed graph theorems,; Inner-product spaces, principle of uniform boundedness, Hilbert spaces, orthonormal bases, bounded linear operators, Riesz representation theorem.

Part 7: Numerical Analysis

Numerical solution of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration; numerical solution of ordinary differential equations: initial value problems: Euler’s method, Runge-Kutta methods of order 2, numerical differentiation; interpolation: error of polynomial interpolation, Lagrange, Newton interpolations; numerical integration: numerical solution of systems of linear equations: direct methods (Gauss elimination, LU decomposition); Trapezoidal and Simpson rules; iterative methods (Jacobi and Gauss-Seidel)

Part 8: Partial Differential Equations

Linear and quasilinear first order partial differential equations, second order linear equations in two variables and their classification; method of characteristics; Cauchy, Dirichlet and Neumann problems; wave in two dimensional Cartesian coordinates, solutions of Laplace, Interior and exterior Dirichlet problems in polar coordinates; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations; Separation of variables method for solving wave and diffusion equations in one space variable

Part 9: Topology

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

Part 10: Probability and Statistics

Probability space, Bayes theorem, conditional probability, independence, joint and conditional distributions, Random variables,  standard probability distributions and their properties (Discrete uniform, Geometric, Binomial, Poisson, Negative binomial, Exponential, Normal,  Gamma, Continuous uniform, Multinomial, Bivariate normal), expectation, conditional expectation, moments; Sampling distributions, Weak and strong law of large numbers, central limit theorem; UMVU estimators, maximum likelihood estimators;Testing of hypotheses, standard parametric tests based on normal, distributions; ; Interval estimation; Simple linear regression

Part 11: Linear programming

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