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, uniform continuity, differentiability, mean value theorem.
Sequences and series of functions, uniform convergence.
Riemann sums and Riemann integral, Improper Integrals.
Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral.
Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.
Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.
Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations.
Algebra of matrices, rank and 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.
Inner product spaces, orthonormal basis.
Quadratic forms, reduction and classification of quadratic forms.
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’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem.
Taylor series, Laurent series, calculus of residues.
Conformal mappings, Mobius transformations.
Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle, derangements.
Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s function, primitive roots.
Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems.
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain.
Polynomial rings and irreducibility criteria.
Fields, finite fields, field extensions, Galois Theory.
Basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.
Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs.
General theory of homogeneous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function.
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.
Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, Gauss elimination and Gauss-Seidel methods, interpolation, numerical differentiation and integration, numerical solutions of ODEs.
Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema.
Fredholm and Volterra integral equations, separable kernels, characteristic numbers, eigenfunctions and resolvent kernel.
Generalized coordinates, Lagrange’s equations, Hamilton’s equations, principle of least action, rigid body motion and oscillations.
Descriptive statistics, probability, Bayes theorem, random variables, probability distributions, expectation and moments.
Markov chains, Poisson process, sampling distributions, estimation and hypothesis testing.
Regression analysis, ANOVA, multivariate analysis, sampling methods, experimental designs, queuing models and linear programming.
All students are expected to answer questions from Unit I. Students in Mathematics are expected to answer additional questions from Unit II and III. Students in Statistics are expected to answer additional questions from Unit IV.