GATE 2025 syllabus for Mathematics (MA) gives you details of the latest GATE syllabus for the subject released by the official gate organizing institute IIT Roorkee 2025. We also created an easy-to-use ad-free mobile app for the GATE syllabus, previous year papers with keys, gate calculator, virtual calculator, and more. Download the iStudy App for all GATE preparation needs.
You will also find the IIT GATE college predictor, NIT GATE college predictor, GATE score calculator, and GATE Exam Info & Stats useful during GATE preparation. Without further due let’s jump into the GATE syllabus for Mathematics.
GATE 2025 Syllabus – Mathematics (MA)
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.
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.
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.
If you are preparing for GATE, download iStudy Mobile App for all GATE Syllabus, Previous Question Papers and Keys, preparation guide, Exam updates, Gate Virtual calculator, Gate Score calculator, and IITs & NITs cutoffs. It is a lightweight, easy to use, no images, and no ads platform to make students’ lives easier.
.
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.
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.
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.
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
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-Kutta method of order 2.
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, 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, connectedness, compactness, 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 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.
If you are preparing for GATE, download iStudy Mobile App for all GATE Syllabus, Previous Question Papers and Keys, preparation guide, Exam updates, Gate Virtual calculator, Gate Score calculator, and IITs & NITs cutoffs. It is a lightweight, easy to use, no images, and no ads platform to make students’ lives easier..
GENERAL APTITUDE (Common for all branches)
Verbal Aptitude
Basic English Grammar
Tenses, articles, adjectives, prepositions, conjunctions, verb-noun agreement, and other parts of speech
Basic Vocabulary
Words, idioms, and phrases in context
Reading and Comprehension
Narrative Sequencing
Quantitative Aptitude
Data Interpretation
Data graphs (bar graphs, pie charts, and other graphs representing data), 2- and 3-dimensional plots, maps, and tables
Numerical Computation and Estimation
Ratios, percentages, powers, exponents and logarithms, permutations and combinations, and series
Mensuration and Geometry
Elementary Statistics and Probability
Analytical Aptitude
Logic
Deduction and Induction
Analogy
Numerical Relations and Reasoning
Spatial Aptitude
Transformation of Shapes
Translation, rotation, scaling, mirroring, assembling, and grouping
Paper folding, cutting, and patterns in 2 and 3 dimensions
InI India has tons of information for M.Tech Admissions in IITs, NITs, IIITs, and other state-level institutes. Do benefit from the content and also share it with your friends. If you want to discuss things with friends do check our communities, All Q&A about GATE Preparation and M.Tech Admissions.
We wish you great luck with GATE preparations.