# GATE Section-XE-A Engineering Mathematics Syllabus

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## 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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