Linear Algebra

• Matrices: types, operations, rank, inverse, determinants.
• Systems of linear equations: Gaussian elimination, Cramer's rule, consistency.
• Eigenvalues & eigenvectors, diagonalization, Cayley–Hamilton theorem, spectral decomposition.

Calculus

• Functions of single variable: limits, continuity, differentiability.
• Taylor and Maclaurin series expansions.
• Mean value theorems (Rolle's, Lagrange, Cauchy).
• Definite and improper integrals: evaluation techniques, comparison test, convergence.
• Partial derivatives: higher order partials, Euler's theorem for homogeneous functions.
• Total derivative and chain rule.
• Maxima and minima of single and multivariable functions; Lagrange multipliers.
• Vector calculus: gradient, divergence, curl; directional derivatives.
• Vector identities and theorems: line, surface and volume integrals; Stokes', Gauss' (Divergence) and Green's theorems; applications.

Differential Equations

• First order equations: separable, linear, exact, integrating factors, homogeneous and Bernoulli.
• Higher order linear ODEs with constant coefficients, Cauchy–Euler equations, method of undetermined coefficients, variation of parameters.
• Initial and boundary value problems.
• Laplace transforms and inversion; use in solving linear ODEs with inputs.
• PDEs basics: 1-D heat equation, wave equation, Laplace equation and their solution methods (separation of variables).

Complex Variables

• Complex numbers, polar form, Euler's formula.
• Complex functions, analytic functions, Cauchy–Riemann equations (basic).
• Contour integrals concept (basics sufficient for GATE).
• Triangle inequality and modulus properties.

Probability & Statistics

• Probability axioms, conditional probability, Bayes' theorem.
• Random variables: discrete and continuous.
• Expectation, variance, moments, standard deviation.
• Distributions: Binomial, Poisson, Normal (Gaussian); properties and approximations.
• Sampling theorems (basic concepts), Central Limit Theorem (conceptual).
• Linear regression (least squares): simple linear regression, correlation coefficient.

Numerical Methods

• Numerical solution of algebraic and transcendental equations: bisection, Newton–Raphson, secant method.
• Numerical integration: Trapezoidal rule, Simpson's 1/3 rule.
• Numerical solution of ordinary differential equations: single-step (Euler, RK2, RK4), multi-step methods (Adams–Bashforth, Adams–Moulton — basics).
• Error analysis and convergence concepts.