GATE Section-XE-A Engineering Mathematics Syllabus


Course Structure

Units Topics
Unit 1 Linear Algebra
Unit 2 Calculus
Unit 3 Vector Calculus
Unit 4 Complex Variables
Unit 5 Ordinary Differential Equations
Unit 6 Partial Differential Equations
Unit 7 Probability and Statistics
Unit 8 Numerical Methods

Course Syllabus

Unit 1: Linear Algebra

  • Algebra of matrices
  • Inverse and rank of a matrix
  • System of linear equations
  • Symmetric, skew-symmetric and orthogonal matrices
  • Determinants
  • Eigenvalues and eigenvectors
  • Diagonalisation of matrices
  • Cayley-Hamilton Theorem

Unit 2: Calculus

Chapter 1: Functions of single variable

  • Limit, continuity and differentiability
  • Mean value theorems
  • Indeterminate forms and L'Hospital's rule
  • Maxima and minima
  • Taylor's theorem
  • Fundamental theorem and mean value-theorems of integral calculus
  • Evaluation of definite and improper integrals
  • Applications of definite integrals to evaluate areas and volumes

Chapter 2: 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

Chapter 3: Sequence and Series

  • Convergence of sequence and series
  • Tests for convergence
  • Power series
  • Taylor's series
  • Fourier Series
  • Half range sine and cosine series

Unit 3: Vector Calculus

  • Gradient, divergence and curl

  • Line and surface integrals

  • Green's theorem, Stokes theorem and Gauss divergence theorem (without proofs)

Unit 4: Complex Variables

  • Analytic functions
  • Cauchy-Riemann equations
  • Line integral, Cauchy's integral theorem and integral formula (without proof)
  • Taylor's series and Laurent series
  • Residue theorem (without proof) and its applications

Unit 5: Ordinary Differential Equations

  • First order equations (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 equation
  • Power series solutions
  • Legendre polynomials, Bessel functions of the first kind and their properties

Unit 6: 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

Unit 7: Probability and Statistics

  • Axioms of probability
  • Conditional probability
  • Bayes' Theorem
  • Discrete and continuous random variables −
    • Binomial
    • Poisson
    • Normal distributions
  • Correlation and linear regression

Unit 8: Numerical Methods

  • Solution of systems of linear equations using 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

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