# GATE Mathematics Syllabus 2021 and Exam Pattern GATE Mathematics Syllabus: Mathematics includes the study of topics like quantity, structure, space, and change. The syllabus of GATE 2021 Mathematics paper will be based on the topics studied at the graduation level. The question asked in paper will be based on formulas, equations, and graphs.

You must aware of the latest syllabus and marking scheme to prepare for the Mathematics exam. To prepare for the GATE Mathematics exam, you should know the latest ECE Syllabus and marking scheme.

In the latest GATE Syllabus for Mathematics, you will find the important chapters and concepts to be covered in all subjects.

Here we are providing you the complete guide on GATE Mathematics Syllabus 2021 and Marking Scheme.

## GATE Mathematics Syllabus 2021

GATE is conducted online and candidates need to solve the paper within 3 hours duration. After qualifying GATE 2021 candidates will be able to get admission in M.Tech courses offered at IITs, NITs, etc. Other than admission in M.tech courses candidates can also apply for various posts offered by PSUs.

You can start your exam preparation by creating a solid study plan and score a better mark in the exam with the latest Mathematics Syllabus. Based on the score in GATE Mathematics, you can shape your career in the proper way.

You must have Mathematics books & study materials, Previous years questions paper along with the latest Mathematics Syllabus to enhance your semester exam preparation.

The major topics of included in the syllabus are,

• Calculus,
• Linear Algebra,
• Real Analysis,
• Algebra,
• Functional Analysis,
• Numerical Analysis, etc.

Check the complete GATE Mathematics syllabus below.

### Chapter 1 – Linear Algebra

• Finite dimensional vector spaces over real or complex fields; Linear transformations and their matrix representations, rank and nullity; systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem, diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, definite forms.

### Chapter 2 – Complex Analysis

• 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 theorem and Laurent’s theorem; residue theorem and applications for evaluating real integrals; Rouche’s theorem, Argument principle, Schwarz lemma; conformal mappings, bilinear transformations

### Chapter 3 – Real Analysis

• Metric spaces, connectedness, compactness, completeness; Sequences and series of functions, uniform convergence; Weierstrass approximation theorem; Power series; Functions of several variables: Differentiation, contraction mapping principle, Inverse and Implicit function theorems; Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, monotone convergence theorem, dominated convergence theorem.

### Chapter 4 – 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.

### Chapter 5 – Algebra

• Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms; cyclic groups, permutation groups, Sylow’s theorems and their applications; Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria; Fields, finite fields, field extensions.

### Chapter 6 – 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, Riesz representation theorem.

### Chapter 7 – Numerical Analysis

• Numerical solutions of algebraic and transcendental equations: bisection, secant method, Newton-Raphson method, fixed point iteration; Interpolation: error of polynomial interpolation, Lagrange and Newton interpolations; Numerical differentiation; Numerical integration: Trapezoidal and Simpson’s rules; Numerical solution of a system of linear equations: direct methods (Gauss elimination, LU decomposition), iterative methods (Jacobi and Gauss-Seidel); Numerical solution of initial value problems of ODEs: Euler’s method, Runge-Kutta methods of order 2.

### Chapter 8 – Partial Differential Equations

• Linear and quasi-linear first order partial differential equations, method of characteristics; Second order linear equations in two variables and their classification; Cauchy, Dirichlet and Neumann problems; Solutions of Laplace and wave equations in two dimensional Cartesian coordinates, interior and exterior Dirichlet problems in polar coordinates; Separation of variables method for solving wave and diffusion equations in one space variable; Fourier series and Fourier transform and Laplace transform methods of solutions for the equations mentioned above.

### Chapter 9 – Topology

• Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, metric topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma.

### Chapter 10 – Calculus

• Finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property; Sequences and series, convergence; Limits, continuity, uniform continuity, differentiability, mean value theorems; Riemann integration, Improper integrals; Functions of two or three 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; Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem.

### Chapter 11 – Linear Programming

• Linear programming problem and its formulation, convex sets and their properties, graphical method, basic feasible solution, simplex method, two phase methods; infeasible and unbounded LPP’s, alternate optima; Dual problem and duality theorems; Balanced and unbalanced transportation problems, Vogel’s approximation method for solving transportation problems; Hungarian method for solving assignment problems.