GATE Mathematics Syllabus 2025

IIT Roorkee has released the GATE Mathematics Syllabus 2025 in its official portal, GOAPS, for the 2025 examination session. The student who selects this paper as the primary paper can choose from either PH (Physics), CS (Computer Science and Information Technology), or ST (Statistics) as the secondary paper. 

The first step in preparation is to grasp the GATE Mathematics syllabus. The Syllabus covers topics such as linear algebra, calculus, complex analysis, real analysis, algebra, ordinary differential equations, partial differential equations, numerical analysis, linear programming, and topology. 

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GATE Mathematics Syllabus sections

Below are the sections of the GATE Mathematics Syllabus:

The GATE Mathematics Syllabus has 10 sections, including linear algebra, calculus, real analysis, ordinary differential equations, complex analysis, functional analysis, numerical analysis, algebra, and partial differential analysis. Each section contains specific topics that students are expected to understand and know. By studying and mastering the topics in each section, aspirants can prepare themselves for success on the GATE Engineering Mathematics exam. 

Calculus Syllabus:

This section is one of the most renowned parts of the GATE Mathematics syllabus. Calculus covers key topics in maths, including differentiation, integration, functions, and limits. 

Below is the list of topics covered in the Calculus section of the GATE 2025 Mathematics syllabus:

Linear Algebra Syllabus 

This section of the GATE Mathematics Syllabus is focused on the study of linear sets of equations and the transform properties of equations. This section includes topics like: 

• Finite-dimensional vector spaces over real or complex fields 
• Systems of linear equations, characteristics of polynomials 
• Linear transformations and their matrix representation, rank, and nullity 
• Minimal Polynomial 
• Eigenvalues and eigenvectors, diagonalization 
• Caley Hamilton Theorem 
• Gram Schmidt orthonormalization process 
• Symmetric, Skew Symmetric 
• Diagonalization by a unitary matrix, Jordan Canonical form 
• Hermitian, Skew Hermitian, normal, orthogonal, and unitary matrices 
• Bilinear and Quadratic forms. 

Real Analysis Syllabus 

This section is a discipline of Mathematics that was made to formalize the study of numbers and functions. It covers important topics like 

1. Connectedness
2. Metric spaces
3.  Completeness
4. Compactness
5. Uniform convergence
6. Sequences and series of functions
7. Ascoli-Arzela theorem,
8. Contraction mapping principle
9. Weierstrass approximation theorem
10.  Contraction Mapping Principle 
11. Differentiation of functions of several variables 
12. Power Series 
13. Inverse and Implicit function theorems 
14. Lebesgue measure on the real line 
15. Measurable functions 
16. Fatou’s lemma 
17. Monotone convergence theorem
18.  Dominated convergence theorem. 

Complex Analysis Syllabus

This section discusses complex numbers as well as their properties, derivatives, and manipulations. It includes the following topics:

• Complex integration: Cauchy’s integral theorem and formula 
• Functions of a complex variable: Differentiability, Continuity, Harmonious functions, and analytic functions. 
• Zeroes and singularities 
• Liouville’s theorem, Morera’s theorem, and the maximum modulus principle 
• Power series, the radius of convergence 
• Taylor’s series and Laurent’s series 
• Residue theorem and applications for evaluating real integrals 
• Conformal mappings, Mobius transformations.
  
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Ordinary Differential Equations Syllabus 

This section consists of a differential equation that contains one or more functions of one independent variable, as well as their derivatives. It includes the following: 

• First-order ordinary differential equations
• Existence and uniqueness theorems for initial value problems
• Linear ordinary differential equations of higher order with constant coefficients
• Secondary-order linear ordinary differential equations with variable coefficients 
• Cauchy’s 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. 
• Strum’s oscillation and separation theorems 
• Sturm- Liouville eigenvalue problems
• Planar autonomous system of ordinary differential equations 
• Stability of stationary points for linear systems with constant coefficients 
• Linearized stability 
• Lyapunov Functions. 

Algebra Syllabus 

Algebra is a mathematical discipline that deals with symbols and rules for manipulating those symbols. This section includes important topics like:

• Groups 
• 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 
• Principal ideal domains 
• Euclidean domains 
• Polynomial rings 
• Eisenstein’s irreducibility criterion 
• Fields 
• Finite Fields 
• Field extensions 
• Algebraic Extensions 
• Algebraic closed fields 

Functional Analysis Syllabus 

This section is a study of mathematical analysis that deals with the functionals or functions of functions y = f(x). It includes a few topics in the GATE Engineering Mathematics syllabus, including:

1. Projection Theorem 
2. Hahn- Banach Theorem 
3. Normed linear spaces
4. Orthonormal bases 
5. Open Mapping 
6. Banach spaces
7. Hilbert spaces
8. Principle of uniform boundedness 
9. Inner product spaces
10. Riesz representation theorem 
11. Spectral theorem for compact self-adjoint operators 
12. Close graph theorems 

Numerical Analysis Syllabus 

GATE 2025 Syllabus of Engineering Mathematics includes various numerical methods for solving equations (roots), other techniques,  and bisection or half-interval as the foundation. 

Other topics of this section are: 

• Systems of linear equations: This section covers direct methods (Gaussian elimination, Cholesky factorization, and LU decomposition), iterative methods (Gauss-Seidel and Jacobi), and their convergence for diagonally dominant coefficient matrices.

• Numerical solutions of nonlinear equations: Secant method, bisection method, Newton-Raphson method, and fixed-point iteration. 
• Numerical integration: Trapezoidal and Simpson rules, Newton-Cotes integration formulas, mathematical errors involved in numerical integration formulae1. Lagrange and Newton forms of an interpolating polynomial, interpolation, error in the polynomial interpolation of a function, numerical differentiation, and error. 

• Numerical Integration: This section includes topics like trapezoidal and Simpson rules, composite rules, Newton-Cotes integration formulas, and mathematical errors involved in numerical integration formulae. 
• Numerical solution of initial value problems for ordinary differential equations: Methods of Euler and Runge-Kutta method of order 2.
  

Topology Syllabus

This section of the topology syllabus in mathematics analyses mathematical problems and proves convergence for partial differential equations using various numerical methods. The topics in this section comprise basic concepts of bases, topology, subbases, order topology, subbases, quotient topology, product topology, connectedness, metric topology, separation axiom, compactness, accountability, and Urysohn’s Lemm. 

Partial Differential Equations Syllabus
The partial differential equation is a differential equation that contains multiple unknown variables, as well as their partial derivatives. It is also known as a special case of an ordinary differential equation. The PDE section of the GATE Mathematics Syllabus consists of topics like 

1. Method of characteristics for first-order linear and quasilinear partial differential equations 
2. 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. 
3. Wave Equation: Cauchy problem and d’Alembert formula, non-homogenous wave equations, and domains of dependence and influence. 
4. Heat Equation: Cauchy problem 
5. Laplace and Fourier transform methods 
6. Numerical solution of initial value problems for ordinary differential equations: Methods of Euler, Runge-Kutta method of order 2. 

Linear Programming Syllabus 

Linear programming helps in determining the best outcome of a linear function. By taking a few simple assumptions, this is the best method for linear optimization.

The key topics in this section of the GATE Mathematics syllabus are:

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

GATE Maths Exam Pattern 

Knowing the GATE Mathematics exam pattern thoroughly will aid the students in preparing better for the exams. Below are the details of the GATE Maths Syllabus Exam pattern: 

• General Aptitude (GA) marks of Mathematics (MA): 15 marks 
• Subject marks: 85 marks 
• Total marks for MA: 100 marks 
• Total Time (in mins): 180 mins
  
  Read more: GATE Exam Pattern 2024 – Subject-Wise Question Paper Pattern, Marking Scheme

GATE Mathematics Syllabus: FAQs

1 . What are the important topics from the GATE Mathematics Syllabus?

The syllabus includes the key topics of linear algebra, calculus, complex analysis, real analysis, algebra, ordinary differential equations, numerical analysis, functional analysis, topology and linear programming, and partial differential equations.

2. Is the GATE Engineering Mathematics syllabus likely to change?

The GATE Engineering Mathematics Syllabus 2025 will remain the same. All questions will be taken from the prescribed syllabus only.

3. Which concepts are discussed under the main topic of the topology of the syllabus? 

The key concepts discussed under the topology section of the GATE Mathematics Syllabus 2025 are the bases, basic concepts of topology, subspace topology, bases, product topology, order topology, metric topology, quotient topology, connectedness, compactness, metric topology, countability and separation axioms, and Urysohn’s Lemma. 

4. Who releases the GATE Mathematics Syllabus? 

The GATE Mathematics Syllabus has been released by the exam conducting body. IISc Bangalore will release the syllabus for candidates to refer to. A downloadable PDF will soon be available. 

5. Does the GATE Mathematics syllabus differ from the Engineering Mathematics syllabus?

GATE Mathematics and Engineering Mathematics have different syllabuses. Compared to mathematics, the engineering mathematics syllabus isn’t as broad. All the topics in engineering mathematics are included in the mathematics syllabus. 

6. Is an MSc Mathematics student eligible to take the exam if they prepare with the GATE Mathematics Syllabus?

Yes, an MSc student can sit and take the GATE exam. They have to download the GATE Mathematics syllabus from the official website.