DEAR STUDENTS
here we discuss the Syllabus of engineering math's for GATE 2024 Exam
The topics covered in the GATE Mathematics syllabus are as follows.
- Calculus,
- Linear Algebra,
- Real Analysis,
- Complex Analysis,
- Ordinary Differential Equations,
- Algebra,
- Functional Analysis,
- Numerical Analysis,
- Partial Differential Equations,
- Topology, and
- Linear Programming.
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.
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, nonhomogeneous 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.
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