GATE MATHEMATICS 2021 Syllabus

## Calculus

Functions of two or more variables, continuity, directional derivatives, partial derivatives, total derivative, maxima and minima, saddle point, method of Lagranges multipliers

Double and Triple integrals and their applications to area,volume and surface area

gradient, divergence and curl, Line integrals and Surface integrals, Greens 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, Fatous lemma, monotone convergence theorem, dominated convergence theorem

## Complex Analysis

Functions of a complex variable: continuity, differentiability, analytic functions, harmonic functions

Complex integration: Cauchys integral theorem and formula

Liouvilles theorem, maximum modulus principle, Moreras theorem

zeros and singularities

Power series, radius of convergence, Taylors series and Laurents series

Residue theorem and applications for evaluating real integrals

Rouches 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,Sylows theorems and their applications

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisensteins 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, Urysohns 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, Vogels approximation method)

Optimal solution, modified distribution method

Solving assignment problems, Hungarian method.