### GATE 2019 Syllabus for Mathematics

GATE 2019 –  It is conducted to offer admission into M.Tech/M.Sc in engineering/ technology/ architecture and P.hD. GATE 2019 Mock Tests have been released, GATE 2019 exam is managed by the IIT. Graduate Aptitude Test in Engineering (GATE) is a national level examination and in relevant branches of science.

The topics have been divided into two categories into each of the GATE 2019 subjects. On core topics, the corresponding sections (of the syllabus given below) of the question paper will contain 90% of their questions and the remaining 10% on Special Topics.

Engineering mathematics is a creative and exciting discipline, spanning traditional boundaries and it combines mathematical theory, practical engineering, and scientific computing.

## Syllabus of Mathematics for GATE 2019

Calculus
Riemann integration, Improper integrals; Functions of two or three variables, continuity, differentiability, mean value theorems; Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem, directional derivatives, partial derivatives, total derivative, maxima and minima, saddle point, method of Lagrange’s multipliers; Double and Triple integrals and their applications;  Finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property; Sequences and series, convergence; Limits, continuity, uniform continuity.

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

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

Complex Analysis
Analytic functions, harmonic functions; Complex integration: 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; Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle, Morera’s theorem.

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; Legendre and Bessel functions and their orthogonal properties; Systems of linear first order ordinary differential equations; Cauchy-Euler equation, method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method).

Algebra
Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria; Fields, finite fields, field extensions. 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

Functional Analysis
Hilbert spaces, orthonormal bases, Riesz representation theorem, Normed linear spaces, Banach spaces, Hahn-Banach theorem, open mapping and closed graph theorems, the principle of uniform boundedness; Inner-product spaces.

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 solution of initial value problems of ODEs: Euler’s method, Runge-Kutta methods of order 2; 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).

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

Partial Differential Equations
Fourier series and Fourier transform and Laplace transform methods of solutions for the equations mentioned above; 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.

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.

Exam Pattern for GATE 2019

 Exam Pattern for GATE 2019 Section Question No No of Questions Marks per Question Total Marks General Aptitude 1 to 5 5 1 5 6 to 10 5 2 10 Technical &  Engineering 1 to 25 25 1 25 Mathematics 26 to 55 30 2 60

Total Questions: 65
Total Marks: 100
Total Duration: 3 hours
Technical Section: 70 marks
General Aptitude: 15 marks
Engineering Mathematics: 15 marks

25 marks to 40 marks will be allotted to Numerical Answer Type Questions

Reference Books for Mathematics- GATE 2019

• Advanced Engineering Mathematics by RK Jain, SRK Iyengar
• Advanced Engineering Mathematics by HK Dass
• Advanced Engineering Mathematics by Erwin Kreyszig
• Engineering Mathematics solved papers by Made easy publications
• Engineering and Mathematics general aptitude by G.K Publications
• GATE Engineering and Mathematics by Nodia and company
• Higher Engineering Mathematics by Bandaru Ramana
• Higher Engineering Mathematics by B.S. Grewal

Other GATE 2019 Syllabus and Information