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GATE Mathematics

Official 2025 syllabus extracted from https://gate2025.iitr.ac.in/doc/2025/GATE%20_MA_2025_Syllabus.pdf

Calculus

Functions of two or more variables

Continuity

Directional derivatives

Partial derivatives

Total derivative

Maxima and minima

Saddle point

Method of Lagrange's multipliers

Double and Triple integrals

Applications to area, volume, surface area

Vector Calculus: gradient, divergence and curl

Line integrals

Surface integrals

Green's theorem

Stokes' theorem

Gauss divergence theorem

Linear Algebra

Finite dimensional vector spaces over real or complex fields

Linear transformations and matrix representations

Rank and nullity

Systems of linear equations

Characteristic polynomial

Eigen values and eigen vectors

Diagonalization

Minimal polynomial

Cayley-Hamilton Theorem

Finite dimensional inner product spaces

Gram-Schmidt orthonormalization

Symmetric, skew-symmetric, Hermitian, skew-Hermitian, normal, orthogonal and unitary matrices

Diagonalization by unitary matrix

Jordan canonical form

Bilinear and quadratic forms

Real Analysis

Metric spaces

Connectedness

Compactness

Completeness

Sequences and series of functions

Uniform convergence

Ascoli-Arzela theorem

Weierstrass approximation theorem

Contraction mapping principle

Power series

Differentiation of functions of several variables

Inverse and implicit function theorems

Lebesgue measure on the real line

Measurable functions

Lebesgue integral

Fatou's lemma

Monotone convergence theorem

Dominated convergence theorem

Complex Analysis

Functions of a complex variable: continuity, differentiability

Analytic functions

Harmonic functions

Cauchy's integral theorem and formula

Liouville's theorem

Maximum modulus principle

Morera's theorem

Zeros and singularities

Power series, radius of convergence

Taylor's and Laurent's series

Residue theorem and applications

Rouche's theorem

Argument principle

Schwarz lemma

Conformal mappings

Mobius transformations

Ordinary Differential Equations

First order ODE

Existence and uniqueness theorems for IVPs

Linear ODEs of higher order with constant coefficients

Second order linear ODEs with variable coefficients

Cauchy-Euler equation

Laplace transforms for ODEs

Series solutions (power series, Frobenius method)

Legendre and Bessel functions and orthogonal properties

Systems of linear first order ODEs

Sturm's oscillation and separation theorems

Sturm-Liouville eigenvalue problems

Planar autonomous systems: stability of stationary points

Linearized stability

Lyapunov functions

Algebra

Groups, subgroups, normal subgroups

Quotient groups

Homomorphisms and automorphisms

Cyclic groups

Permutation groups

Group action

Sylow's theorems and applications

Rings

Ideals, prime and maximal ideals

Quotient rings

Unique factorization domains

Principal ideal domains

Euclidean domains

Polynomial rings

Eisenstein's irreducibility criterion

Fields, finite fields

Field extensions

Algebraic extensions

Algebraically closed fields

Functional Analysis

Normed linear spaces

Banach spaces

Hahn-Banach theorem

Open mapping and closed graph theorems

Principle of uniform boundedness

Inner-product spaces

Hilbert spaces

Orthonormal bases

Projection theorem

Riesz representation theorem

Spectral theorem for compact self-adjoint operators

Numerical Analysis

Direct methods for linear equations: Gaussian elimination, LU decomposition, Cholesky factorization

Iterative methods: Gauss-Seidel, Jacobi

Convergence for diagonally dominant matrices

Nonlinear equations: bisection, secant, Newton-Raphson, fixed point iteration

Interpolation: Lagrange and Newton forms

Error in polynomial interpolation

Numerical differentiation and error

Numerical integration: Trapezoidal and Simpson rules

Newton-Cotes integration formulas

Composite rules

Numerical solution of IVPs: Euler method

Runge-Kutta method of order 2

Partial Differential Equations

Method of characteristics for first order linear and quasilinear PDEs

Second order PDEs in two variables: classification and canonical forms

Separation of variables for Laplace equation (Cartesian and polar)

Heat and wave equations in one space variable

Wave equation: Cauchy problem and d'Alembert formula

Domains of dependence and influence

Non-homogeneous wave equation

Heat equation: Cauchy problem

Laplace and Fourier transform methods

Topology

Basic concepts of topology

Bases, subbases

Subspace topology

Order topology

Product topology

Quotient topology

Metric topology

Countability and separation axioms

Urysohn's Lemma

Linear Programming

Linear programming models

Convex sets, extreme points

Basic feasible solution

Graphical method

Simplex method

Two phase method

Revised simplex method

Infeasible and unbounded LP models

Alternate optima

Duality theory

Weak and strong duality

Balanced and unbalanced transportation problems

Least cost, north-west corner, Vogel's approximation methods

Modified distribution method

Assignment problems: Hungarian method

GATE Mathematics Syllabus 2025, 2026 and more: Subjects, Topics, and What to Study