IITs and IISc are institutions of national importance
IITs and IISc are institutions of national importance and are well known, the world over, for quality education in engineering, science & technology and research in frontier areas. The aim of IITs and IISc is to build a sound foundation of knowledge, pursue excellence and enhance creativity in an intellectually stimulating environment. The vibrant academic ambience and well-equipped research infrastructure of IISc & IITs motivate the students to pursue Research and Development careers in frontier areas of basic sciences as well as interdisciplinary areas of science and technology.
Joint Admission Test for M.Sc. (JAM) is being conducted from 2004 to provide admissions to M.Sc. (Four Semesters), Joint M.Sc.-Ph.D., M.Sc.-Ph.D. Dual Degree, M.Sc.-M.Tech., etc. Programmes at the IITs and Integrated Ph.D. Degree Programmes at IISc for consolidating Science as a career option for bright students. These postgraduate programmes at IITs and IISc offer high quality education in their respective disciplines, comparable to the best in the world. The curricula for these programmes are designed to provide opportunities to the students to develop academic talent leading to challenging and rewarding professional life.
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Sequence of real numbers, convergence of sequences, bounded and monotone sequences, convergence criteria for sequences of real numbers, Cauchy sequences, subsequences, Bolzano-Weierstrass theorem. Series of real numbers, absolute convergence, tests of convergence for series of positive terms – comparison test, ratio test, root test; Leibniz test for convergence of alternating series.
Limit, continuity, intermediate value property, differentiation, Rolle’s Theorem, mean value theorem, L'Hospital's rule, Taylor's theorem, maxima and minima.
Integration as the inverse process of differentiation, definite integrals and their properties, fundamental theorem of calculus. Double and triple integrals, change of order of integration, calculating surface areas and volumes using double integrals, calculating volumes using triple integrals.
Ordinary differential equations of the first order of the form y'=f(x,y), Bernoulli’s equation, exact differential equations, integrating factor, orthogonal trajectories, homogeneous differential equations, variable separable equations, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation.
Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.
Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups, normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups.
Finite dimensional vector spaces, linear independence of vectors, basis, dimension, linear transformations, matrix representation, range space, null space, rank-nullity theorem. Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions, eigenvalues and eigenvectors for matrices, Cayley-Hamilton theorem.
Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets, completeness of R. Power series (of real variable), Taylor’s series, radius and interval of convergence, term-wise differentiation and integration of power series.