GATE Mathematics Syllabus 2027 with Topic Wise Weightage

GATE Mathematics Syllabus for 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 and their applications to area
• Volume and surface area
• Vector Calculus: gradient, divergence and curl, Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem

GATE Mathematics Syllabus for Linear Algebra

• Finite dimensional vector spaces over real or complex fields
• Linear transformations and their matrix representations
• Rank and nullity
• Systems of linear equations
• Characteristic polynomial
• Eigenvalues and eigenvectors
• Diagonalization
• Minimal polynomial
• Cayley-Hamilton Theorem
• Finite dimensional inner product spaces
• Gram-Schmidt orthonormalization process
• Symmetric
• Skew-symmetric
• Hermitian
• Skew-Hermitian
• Normal
• Orthogonal and unitary matrices
• Diagonalization by a unitary matrix
• Jordan canonical form
• Bilinear and quadratic forms

GATE Mathematics Syllabus for 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

GATE Mathematics Syllabus for Complex Analysis

• Functions of a complex variable: continuity, differentiability, analytic functions, harmonic functions
• Complex integration: 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 series and Laurent’s series
  • Residue theorem and applications for evaluating real integrals
  • Rouche’s theorem, Argument principle, Schwarz lemma
  • Conformal mappings, Mobius transformations.

GATE Mathematics Syllabus for Ordinary Differential Equations

• First-order ordinary differential equations, existence and uniqueness theorems for initial value problems, linear ordinary differential equations of higher order with constant coefficients
• Second-order linear ordinary differential equations with variable coefficients
• Cauchy-Euler equation, method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method)
• Legendre and Bessel functions and their orthogonal properties
• Systems of linear first order ordinary differential equations, Sturm's oscillation and separation theorems, Sturm-Liouville eigenvalue problems
• Planar autonomous systems of ordinary differential equations: Stability of stationary points for linear systems with constant coefficients, Linearized stability, Lyapunov functions

GATE Mathematics Syllabus for Algebra

• Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms
• Cyclic groups, permutation groups, Group action, Sylow’s theorems and their applications
• Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisenstein’s irreducibility criterion
• Fields, finite fields, field extensions, algebraic extensions, algebraically closed fields

GATE Mathematics Syllabus for 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

GATE Mathematics Syllabus for Numerical Analysis

• Systems of linear equations: Direct methods (Gaussian elimination, LU decomposition, Cholesky factorization), Iterative methods (Gauss-Seidel and Jacobi) and their convergence for diagonally dominant coefficient matrices
• Numerical solutions of nonlinear equations: bisection method, secant method, Newton-Raphson method, fixed point iteration
• Interpolation: Lagrange and Newton forms of interpolating polynomial, Error in polynomial interpolation of a function; Numerical differentiation and error,
• Numerical integration: Trapezoidal and Simpson rules, Newton-Cotes integration formulas, composite rules, mathematical errors involved in numerical integration formulae
• Numerical solution of initial value problems for ordinary differential equations: Methods of Euler, Runge- method of order 2

GATE Mathematics Syllabus for Partial Differential Equations

• Method of characteristics for first order linear and quasilinear partial differential equations
• Second order partial differential equations in two independent variables: classification and canonical forms, method of separation of variables for Laplace equation in Cartesian and polar coordinates, heat and wave equations in one space variable;
• Wave equation: Cauchy problem and d'Alembert formula, domains of dependence and influence, nonhomogeneous wave equation
• Heat equation: Cauchy problem; Laplace and Fourier transform methods.

GATE Mathematics Syllabus for Topology

• Basic concepts of topology
• Bases
• Subbases
• Subspace topology
• Order topology
• Product topology
• Quotient topology
• Metric topology
• Connectedness
• Compactness
• Countability and separation axioms
• Urysohn’s Lemma

GATE Mathematics Syllabus for Linear Programming

• Linear programming models, convex sets, extreme points
• Basic feasible solution, graphical method, simplex method, two phase methods, revised simplex method
• Infeasible and unbounded linear programming models, alternate optima
• Duality theory, weak duality and strong duality
• Balanced and unbalanced transportation problems, Initial basic feasible solution of balanced transportation problems (least cost method, north-west corner rule, Vogel’s approximation method)
• Optimal solution, modified distribution method; Solving assignment problems, Hungarian method

Vector Calculus

20%

Probability & Statistics

20%

Numerical Methods

20%

Differential Equation

10%

Calculus

10%

Linear Algebra

10%

Complex Variables

10%