## Details of MA3101 (Autumn 2016)

Level: 3 |
Type: Theory |
Credits: 3.0 |

Course Code | Course Name | Instructor(s) |
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MA3101 |
Analysis III |
Sushil Gorai |

Syllabus |
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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. |

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 |
---|---|---|---|

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 |