Details of MA3101 (Autumn 2012)
Level: 3 | Type: Theory | Credits: 3.0 |
Course Code | Course Name | Instructor(s) |
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MA3101 | Multivariable Analysis | Saugata Bandyopadhyay |
Syllabus |
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MA3101 Multivariable Analysis
Part I : Point set Topology : Metric spaces: Introduction to metric spaces, metric space topology, equivalent metrics; sequences, complete metric spaces, 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 in Rn: 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. |
References |
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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. | Programme | Semester No | Course Choice |
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1 | IP | 1 | Not Allowed |
2 | IP | 3 | Not Allowed |
3 | MS | 5 | Not Allowed |
4 | RS | 1 | Not Allowed |