**GATE Syllabus for Mathematics**

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.