\nTOPICS<\/strong><\/td>\nSUB-TOPICS\/CONCEPTS<\/strong><\/td>\n<\/tr>\n\nCalculus<\/strong><\/td>\n\n\n- Functions of two or more variables, continuity, directional derivatives, partial derivatives, total derivative, maxima and minima, saddle point, method of Lagrange\u2019s 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\u2019s theorem, Stokes\u2019 theorem, and Gauss divergence theorem.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nLinear Algebra<\/strong><\/td>\n\n\n- 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.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nReal Analysis<\/strong><\/td>\n\n\n- 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\u2019s lemma, monotone convergence theorem, dominated convergence theorem.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nComplex Analysis<\/strong><\/td>\n\n\n- Functions of a complex variable: continuity, differentiability, analytic functions, harmonic functions; Complex integration: Cauchy\u2019s integral theorem and formula; Liouville\u2019s theorem, maximum modulus principle, Morera\u2019s theorem; zeros and singularities; Power series, radius of convergence, Taylor\u2019s series and Laurent\u2019s series; Residue theorem and applications for evaluating real integrals; Rouche\u2019s theorem, Argument principle, Schwarz lemma; Conformal mappings, Mobius transformations.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nOrdinary Differential Equations<\/strong><\/td>\n\n\n- 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\u2019s 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.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nAlgebra<\/strong><\/td>\n\n\n- Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms; cyclic groups, permutation groups,Group action,Sylow\u2019s theorems and their applications; Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisenstein\u2019s irreducibility criterion; Fields, finite fields, field extensions,algebraic extensions, algebraically closed fields.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nFunctional Analysis<\/strong><\/td>\n\n\n- 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.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nNumerical Analysis<\/strong><\/td>\n\n\n- 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.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nPartial Differential Equations<\/strong><\/td>\n\n\n- 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\u2019Alembert formula, domains of dependence and influence, non-homogeneous wave equation; Heat equation: Cauchy problem; Laplace and Fourier transform methods.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nTopology<\/strong><\/td>\n\n\n- Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, quotient topology, metric topology, connectedness, compactness, countability and separation axioms, Urysohn\u2019s Lemma.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
\nLinear Programming<\/strong><\/td>\n| \n | | | | | | | | | | | | | | | | | | | | | | | |
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