Details of MA3101 (Autumn 2016)

Level: 3 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
MA3101 Analysis III Sushil Gorai

MA3101: Analysis-III

Part I : Point Set Topology

Metric Spaces: Introduction to metric spaces, metric space topology, equivalent metrics; sequences, complete metric spaces, the Baire category theorem; limits and continuity, uniform continuity, extension of uniformly continuous functions, contractions and the Banach contraction mapping theorem.

Connectedness: Connected and disconnected metric spaces, equivalent formulations of connectedness, continuous functions on connected spaces, path-connectedness.

Compactness: Compact spaces, equivalent formulations of compactness, Heine-Borel and Bolzano-Weierstrass theorems, compactness and continuity, the Arzela-Ascoli theorem.

Part II: Differential Calculus

Differentiation Theory: Directional derivatives and its drawbacks, total derivative, mean value theorem, -functions, mixed derivatives, Taylors theorem.

Inverse Function Theorems: Local inverse function theorem, Hadamards global inverse function theorem (proof not required).

Implicit Function Theorem: Implicit function theorem and rank theorem; introduction to manifolds in Euclidean spaces.

Maxima and Minima: Necessity and sufficiency, critical points and the Hessian; constrained extrema and Lagranges multipliers.

Suggested Text/Reference Books:

1. Apostol, T. M., Mathematical Analysis (2nd Edition), Narosa Publishers, 1996.

2. Fleming, W. H., Functions of Several Variables (2nd Edition), Undergraduate Texts in Mathematics, Springer-Verlag, 1977.

3. Munkres, J. R., Analysis on Manifolds, Addison-Wesley, 1991.

4. Munkres, J. R., Topology (2nd Edition), Prentice Hall, 2000.

5. Rudin, W., Principles of Mathematical Analysis (3rd Edition), International Series in Pure and Applied Mathematics. McGraw-Hill, 1976.

6. Searcid, M. ., Metric Spaces, Springer Undergraduate Mathematics Series, Springer-Verlag, 2007.

7. Spivak, M., Calculus on Manifolds, Addison-Wesley, 1965.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 1 Core
2 IP 3 Not Allowed
3 IP 5 Not Allowed
4 MR 1 Not Allowed
5 MR 3 Not Allowed
6 MS 5 Core
7 RS 1 Not Allowed
8 RS 2 Not Allowed