Metrics and norms - metric spaces, normed vector spaces, convergence in metric spaces, completeness, pointwise and uniform convergence of sequence of functions; Functions of several variables - differentiability, chain rule, Taylor's theorem, inverse function theorem, implicit function theorem; Introduction to Lebesgue measure and integral - measureable sets, measurable functions, Lebesgue integral, dominated convergence theorem, monotone convergence theorem.

1. N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
2. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Ed., W. H. Freeman, 1993.

1. M. Capinski and E. Kopp, Measure, Integral and Probability, 2nd Ed., Springer, 2007.

Probability - probability spaces, random variables and random vectors, functions of random vectors, univariate and multivariate distributions, mathematical expectations, moment generating functions, convergence in probability and in distribution and related results; Sampling distributions; Point estimation - estimators, sufficiency, completeness, minimum variance unbiased estimation, maximum likelihood estimation, method of moments, Cramer-Rao inequality, consistency; Interval estimation; Testing of hypotheses - tests and critical regions, Neymann-Pearson lemma, uniformly most powerful tests, likelihood ratio tests; Correlation and linear regression.

1. R. V. Hogg, J. W. McKean and A. Craig, Introduction to Mathematical Statistics, 6th Ed., Pearson Education India, 2006.
2. B. L. S. Prakasa Rao, A First Course in Probability and Statistics, World Scientific/Cambridge University Press India, 2009.

1. V. K. Rohatgi and A. K. Md. E. Saleh, An Introduction to Probability and Statistics, 2nd Ed., Wiley India, 2000.
2. G. Casella and R. L. Berger, Statistical Inference, 2nd Ed., Duxbury Press/Thompson Learning India, 2002.

Errors; Iterative methods for nonlinear equations; Polynomial interpolation, spline interpolations; Numerical integration based on interpolation, quadrature methods, Gaussian quadrature; Initial value problems for ordinary differential equations - Euler method, Runge-Kutta methods, multi-step methods, predictor-corrector method, stability and convergence analysis; Finite difference schemes for partial differential equations - Explicit and implicit schemes; Consistency, stability and convergence; Stability analysis (matrix method and von Neumann method), Lax equivalence theorem; Finite difference schemes for initial and boundary value problems (FTCS, Backward Euler and Crank-Nicolson schemes, ADI methods, Lax Wendroff method, upwind scheme).

1. D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Ed., AMS, 2002.
2. G. D. Smith, Numerical Solutions of Partial Differential Equations, 3rd Ed., Calrendorn Press, 1985.

1. K. E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989.
2. S. D. Conte and C. de Boor, Elementary Numerical Analysis - An Algorithmic Approach, McGraw-Hill, 1981.
3. R. Mitchell and S. D. F. Griffiths, The Finite Difference Methods in Partial Differential Equations, Wiley, 1980.

Formal properties of integers, equivalence relations, congruences, rings homomorphisms, ideals, integral domains, fields; Groups, homomorphisms, subgroups, cosets, Lagrange's theorem , normal subgroups, quotient groups, permutation groups; Groups actions, orbits, stabilizers, Cayley's theorem, conjugacy, class equation, Sylow's theorems and applications; Principal ideal domains, Euclidean domains, unique factorization domains, polynomial rings; Characteristic of a field, field extensions, algebraic extensions, finite fields.

1. N. H. McCoy and G. J. Janusz, Introduction to Abstract Algebra, 6th Ed., Elsevier, 2005.
2. J. A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1998.

1. I. N. Herstein, Topics in Algebra, Wiley, 2004.
2. J. B. Fraleigh, A First Course in Abstract Algebra, Addison Wesley, 2002.

Local theory of plane and space curves, curvature and torsion formulas, Serret-Frenet formulas, fundamental Theorem of space curves; Regular surfaces, change of parameters, differentiable functions, tangent plane, differential of a map; First and second fundamental form; Orientation, Gauss map and its properties, Euler's theorem on principal curvatures; Isometries, Gauss's Theorema Egregium; Parallel transport, geodesics, Gauss-Bonnet theorem and its applications to surfaces of constant curvature.

1. J. McCleary, Geometry from a Differentiable Viewpoint, Cambridge University Press, 1994.
2. A. Pressley, Elementary Differential Geometry, Springer, 2002.

1. M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.