**GATE Engineering Mathematics Syllabus**

**Section 1 : Linear Algebra **

Algebra of Matrices; Inverse, Rank of a matrix; System of linear equations, Symmetric, Skew – symmetric and Orthogonal matrices. Determinants; Eigenvalues and eigenvectors; Diagonalisation of matrices; Cayley – Hamilton Theorem.

**Section 2 : Calculus**

Functions of single variable : Limit, continuity and differentiability, Mean value theorems, Indeterminate forms and L’Hospital rule, Maxima and minima, Taylor’s series, Fundamental and mean value – theorems of integral calculus. Evaluation of definite and improper integrals, Applications of definite integrals to evaluate areas and volumes.

Functions of two variables : Limit, continuity and partial derivatives; Directional derivative; Total derivative; Tangent plane and normal line; Maxima, minima and saddle points; Method of Lagrange multipliers; Double and triple integrals, and their applications.

Sequence and series : Convergence of sequence and series; Tests for convergence; Power series; Taylor’s series; Fourier Series; Half range sine and cosine series.

**Section 3 : Vector Calculus **

Gradient, divergence and curl; Line and surface integrals; Green’s theorem, Stokes theorem and Gauss divergence theorem ( without proofs ).

**Complex variables**

Analytic functions, Cauchy – Riemann equations, Line integral, Cauchy’s integral theorem and integral formula ( without proof ); Taylor’s and Laurent’ series, Residue theorem ( without proof ) and its applications.

**Section 4 : Ordinary Differential Equations**

First order equation ( Linear and nonlinear ), Higher order linear differential equations with constant coefficients; Second order linear differential equations with variable coefficients, Method of Variation of parameters; Cauchy – Euler’s equations, power series solutions, Legendre polynomials and Bessel’s functions of the first kind and their properties.

**Section 5 : Partial Differential Equations **

Classification of second order linear partial differential equations; Method of separation of variables; Laplace equation; Solutions of one dimensional heat and wave equations.

**Section 6 : Probability and Statistics **

Axioms of probability; Conditional probability; Bayes’ Theorem; Discrete and continuous random variables : Binomial, Poisson and normal distributions; Correlation and linear regression.

**Section 7 : Numerical Methods **

Solution of a system of linear equations by LU decomposition, Gauss elimination and Gauss – Seidel methods; Lagrange and Newton’s interpolations, Solution of polynomial and transcendental equations by Newton – Raphson method; Numerical integration by trapezoidal rule, Simpson’s rule and Gaussian quadrature rule; Numerical solutions of first order differential equations by Euler’s method and 4th order Runge – Kutta method.