**GATE Mathematics Syllabus:**

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.

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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.

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.

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.

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.

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.

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.

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.

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.

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

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