Gate 2018 Syllabus for Mathematics (MA)

Gate 2018 Syllabus for Mathematics (MA)

Linear Algebra: Finite Dimensional Vector Spaces; Linear Transformations And Their Matrix Representations, Rank; Systems Of Linear Equations, Eigen Values And Eigen Vectors, Minimal Polynomial, Cayley-Hamilton Theroem, Diagonalisation, Hermitian, Skew-Hermitian And Unitary Matrices; Finite Dimensional Inner Product Spaces, Gram-Schmidt Orthonormalization Process, Self-Adjoint Operators.

Complex Analysis: Analytic Functions, Conformal Mappings, Bilinear Transformations; Complex Integration: Cauchy’s Integral Theorem And Formula; Liouville’s Theorem, Maximum Modulus Principle; Taylor And Laurent’s Series; Residue Theorem And Applications For Evaluating Real Integrals.

Real Analysis: Sequences And Series Of Functions, Uniform Convergence, Power Series, Fourier Series, Functions Of Several Variables, Maxima, Minima; Riemann Integration, Multiple Integrals, Line, Surface And Volume Integrals, Theorems Of Green, Stokes And Gauss; Metric Spaces, Completeness, Weierstrass Approximation Theorem, Compactness; Lebesgue Measure, Measurable Functions; Lebesgue Integral, Fatou’s Lemma, Dominated Convergence Theorem.

Ordinary Differential Equations: First Order Ordinary Differential Equations, Existence And Uniqueness Theorems, Systems Of Linear First Order Ordinary Differential Equations, Linear Ordinary Differential Equations Of Higher Order With Constant Coefficients; Linear Second Order Ordinary Differential Equations With Variable Coefficients; Method Of Laplace Transforms For Solving Ordinary Differential Equations, Series Solutions; Legendre And Bessel Functions And Their Orthogonality.

Algebra: Normal Subgroups And Homomorphism Theorems, Automorphisms; Group Actions, Sylow’s Theorems And Their Applications; Euclidean Domains, Principle Ideal Domains And Unique Factorization Domains. Prime Ideals And Maximal Ideals In Commutative Rings; Fields, Finite Fields.

Functional Analysis: Banach Spaces, Hahn-Banach Extension Theorem, Open Mapping And Closed Graph Theorems, Principle Of Uniform Boundedness; Hilbert Spaces, Orthonormal Bases, Riesz Representation Theorem, Bounded Linear Operators.

Numerical Analysis: Numerical Solution Of Algebraic And Transcendental Equations: Bisection, Secant Method, Newton-Raphson Method, Fixed Point Iteration; Interpolation: Error Of Polynomial Interpolation, Lagrange, Newton Interpolations; Numerical Differentiation; Numerical Integration: Trapezoidal And Simpson Rules, Gauss Legendrequadrature, Method Of Undetermined Parameters; Least Square Polynomial Approximation; Numerical Solution Of Systems Of Linear Equations: Direct Methods (Gauss Elimination, LU Decomposition); Iterative Methods (Jacobi And Gauss-Seidel); Matrix Eigenvalue Problems: Power Method, Numerical Solution Of Ordinary Differential

Equations: Initial Value Problems: Taylor Series Methods, Euler’s Method, Runge-Kutta Methods.

Partial Differential Equations: Linear And Quasilinear 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, Wave And Diffusion Equations In Two Variables; Fourier Series And Fourier Transform And Laplace Transform Methods Of Solutions For The Above Equations.

Mechanics: Virtual Work, Lagrange’s Equations For Holonomic Systems, Hamiltonian Equations.

Topology: Basic Concepts Of Topology, Product Topology, Connectedness, Compactness, Countability And Separation Axioms, Urysohn’s Lemma.

Probability And Statistics: Probability Space, Conditional Probability, Bayes Theorem, Independence, Random Variables, Joint And Conditional Distributions, Standard Probability Distributions And Their Properties, Expectation, Conditional Expectation, Moments; Weak And Strong Law Of Large Numbers, Central Limit Theorem; Sampling Distributions, UMVU Estimators, Maximum Likelihood Estimators, Testing Of Hypotheses, Standard Parametric Tests Based On Normal, X2 , T, F – Distributions; Linear Regression; Interval Estimation.

Linear Programming: Linear Programming Problem And Its Formulation, Convex Sets And Their Properties, Graphical Method, Basic Feasible Solution, Simplex Method, Big-M And Two Phase Methods; Infeasible And Unbounded LPP’s, Alternate Optima; Dual Problem And Duality Theorems, Dual Simplex Method And Its Application In Post Optimality Analysis; Balanced And Unbalanced Transportation Problems, U -U Method For Solving Transportation Problems; Hungarian Method For Solving Assignment Problems.

Calculus Of Variation And Integral Equations: Variation Problems With Fixed Boundaries; Sufficient Conditions For Extremum, Linear Integral Equations Of Fredholm And Volterra Type, Their Iterative Solutions.