Topics | Sub Topics |
GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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.
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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- method of order 2
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GATE Mathematics Syllabus for 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.
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GATE Mathematics Syllabus for 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
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GATE Mathematics Syllabus for 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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