# 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 4

^{th}order Runge-Kutta method

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