A substantial course in real analysis is an esential part of the preparation of any potential mathematician. Analysis on Manifolds is a thorough, class-tested approach that begins with the derivative and the Riemann integral for functions of several variables, followed by a treatment of differential forms and a proof of Stokes' theorem for manifolds in euclidean space.
The book includes careful treatment of both the inverse function theorem and the change of variables theorem for n-dimensional integrals, as well as a proof of the Poincare lemma.
Intended for students at the senior or first-year graduate level, this text includes more than 120 illustrations and exercises that range from the strightforward to the challenging . The book evolved from courses on real analysis taught by the author at the Massachusetts INstitute of Technology.
Author(s): James R. Munkres
Series: The Advanced Book Program
Publisher: Addison Wesley Publishing Company
Year: 1994
Language: English
Pages: 379
Tags: Mathematics Textbooks, Geometry, Nonfiction, Technology, Engineering
Title
Copyright Page
Preface
Contents
Chapter 1. The Algebra and Topology of Rn
1. Review of Linear Algebra
2. Matrix Inversion and Determinants
3. Review of Topology in Rn
4. Compact Subspaces and Connected Subspaces of Rn
Chapter 2. Differentiation
5. Derivative
6. Continuously Differentiable Functions
7. The Chain Rule
8. The Inverse Function Theorem
* 9. The Implicit Function Theorem
Chapter 3. Integration
10. The Integral over a Rectangle
11. Existence of the Integral
12. Evaluation of the Integral
13. The Integral over a Bounded Set
14. Rectifiable Sets
15. Improper Integrals
Chapter 4. Changes of Variables
16. Partitions of Unity
17. The Change of Variables Theorem
18. Diffeomorphisms in Rn
19. Proof of the Change of Variables Theorem
20. Application of Change of Variables
Chapter 5. Manifolds
21. The Volume of a Parallelopiped
22. The Volume of a Parametrized-Manifold
23. Manifolds in Rn
24. The Boundary of a Manifold 20
25. Integrating a Scalar Function over a Manifold
Chapter 6. Differential Forms
26. Multilinear Algebra
27. Alternating Tensors
28. The Wedge Product
29. Tangent Vectors and Differential Forms
30. The Differential Operator
* 31. Application to Vector and Scalar Fields
32. The Action of a Differentiable Map
Chapter 7. Stokes' Theorem
33. Integrating Forms over Parametrized-Manifold
34. Orientable Manifolds
35. Integrating Forms over Oriented Manifolds
* 36. A Geometric Interpretation of Forms and Integrals
37. The Generalized Stokes' Theorem
* 38. Applications to Vector Analysis
Chapter 8. Closed Forms and Exact Forms
39. The Poincaré Lemma
40. The deRham Groups of Punctured Euclidean Space
Chapter 9. Epilogue-Life Outside Rn
41. Differentiable Manifolds and Riemannian Manifolds
Bibliography
Index
