Real Analysis - 1

Code: MA642 | L-T-P-C: 3-1-0-8

MA642 Real Analysis - 1 [3-1-0-8] Prerequisite: Nil

Completeness properties of real numbers, countable and uncountable sets, cardinality. Norms and metrics: Metric spaces, convergence of sequences, completeness, connectedness and sequential compactness; Continuity and uniform continuity; sequences and series of functions, uniform convergence, equicontinuity, Ascoli's theorem, Weierstrass approximation theorem, power series. Calculus of functions of a real variable: Differentiability, Mean value theorems, Taylor's theorem. Calculus of functions of several real variables: Partial and directional derivatives, differentiability, Chain Rule, Taylor's theorem, Maxima and Minima, Lagrange multipliers, Inverse function theorem, Implicit function theorem. Multiple Integration: Fubini's Theorem, Line integrals, Surface integrals, Green, Gauss and Stokes theorems.

Texts/References:

1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, 1993.
2. P. M. Fitzpatrick, Advanced Calculus, 2nd Edition, AMS, Indian Edition, 2010.
3. N. L. Carothers, Real Analysis, Cambridge University Press, Indian Edition, 2009.
4. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw Hill, 1976.