Engineering Mathematics

The Graduate Aptitude Test in Engineering (GATE) Mathematics syllabus covers a wide range of topics from various branches of mathematics. Below is a comprehensive list of topics typically included in the GATE Mathematics syllabus:

1. Calculus:

   • Functions of single and multiple variables
   • Limit, continuity, and differentiability
   • Mean value theorems
   • Indeterminate forms and L’Hospital’s rule
   • Maxima and minima
   • Taylor’s theorem and Taylor series
   • Partial derivatives and Jacobian
   • Double and triple integrals
   • Line, surface, and volume integrals
   • Vector calculus: Gradient, divergence, and curl
   • Green’s, Gauss’s, and Stokes’ theorems

2. Linear Algebra:

   • Vector spaces and subspaces
   • Linear dependence and independence
   • Basis, dimension, and rank of matrices
   • Eigenvalues and eigenvectors
   • Diagonalization of matrices
   • Orthogonalization and orthonormal bases
   • Inner products, norms, and orthogonal complements
   • Canonical forms: Jordan, rational, and Smith normal forms

3. Ordinary Differential Equations (ODEs):

   • First-order ordinary differential equations (ODEs)
   • Higher-order linear ordinary differential equations with constant coefficients
   • Variation of parameters
   • Cauchy-Euler equations
   • Laplace transforms and inverse transforms
   • Initial and boundary value problems

4. Partial Differential Equations (PDEs):

   • First-order linear and nonlinear partial differential equations (PDEs)
   • Method of characteristics
   • Classification of second-order linear partial differential equations
   • Separation of variables
   • Fourier series and boundary value problems
   • Solutions of heat, wave, and Laplace’s equations

5. Probability and Statistics:

   • Sample spaces, events, and probability
   • Conditional probability and Bayes’ theorem
   • Random variables and probability distributions (discrete and continuous)
   • Expectation and variance
   • Moment-generating functions
   • Standard distributions: Binomial, Poisson, exponential, normal distributions
   • Joint distributions and correlation
   • Limit theorems: Law of large numbers, central limit theorem
   • Estimation theory and hypothesis testing

6. Numerical Methods:

   • Solutions of nonlinear equations
   • Interpolation and approximation
   • Numerical integration and differentiation
   • Numerical solutions of ordinary and partial differential equations
   • Finite difference and finite element methods

7. Complex Analysis:

   • Analytic functions
   • Contour integration
   • Cauchy’s integral theorem and Cauchy’s integral formula
   • Taylor and Laurent series
   • Residue theorem
   • Conformal mapping