\n| \n Unit 1<\/p>\n<\/td>\n | \n Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, Differentiability, Mean value theorem, Sequences, and series. Functions of several variables, Metric spaces, compactness, connectedness. Normed Linear Spaces.<\/p>\n Linear Algebra: Vector spaces, algebra of linear transformations. Algebra of matrices, determinant of matrices, linear equations. Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.Quadratic forms, reduction, and classification of quadratic forms<\/p>\n<\/td>\n<\/tr>\n |
\n| \n Unit 2<\/p>\n<\/td>\n | \n Complex Analysis: Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric, and hyperbolic functions. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy\u2019s theorem, Cauchy\u2019s integral formula, Liouville\u2019s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations.<\/p>\n Algebra: Permutations, combinations, Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler\u2019s \u00d8- function, primitive roots, Cayley\u2019s theorem, Sylow theorems. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, Polynomial rings, and irreducibility criteria. Fields, finite fields, field extensions, Galois Theory.<\/p>\n Topology: basis, dense sets, subspace and product topology, separation axioms, connectedness, and compactness. <\/p>\n<\/td>\n<\/tr>\n |
\n| \n Unit 3<\/p>\n<\/td>\n | \n Ordinary Differential Equations (ODEs): Existence and uniqueness of solutions of initial value problems for first-order ordinary differential equations, singular solutions of first-order ODEs, the system of first-order ODEs.<\/p>\n Partial Differential Equations (PDEs): Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first-order PDEs. Classification of second-order PDEs, General solution of higher-order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations.<\/p>\n Numerical Analysis: Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, <\/p>\n Calculus of Variations: Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations.<\/p>\n <\/p>\n Linear Integral Equations: Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.<\/p>\n Classical Mechanics: Generalized coordinates, Lagrange\u2019s equations, Hamilton\u2019s canonical equations, Hamilton\u2019s principle and the principle of least action, Two-dimensional motion of rigid bodies, Euler\u2019s dynamical equations for the motion of a rigid body about an axis, the theory of small oscillations. <\/p>\n<\/td>\n<\/tr>\n |
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