Analysis on Manifolds
by Munkres,James R.Format: Paperback
Pub. Date: 1997-09-01
Publisher(s): CRC Press
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Summary
A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.
Table of Contents
| Preface | p. v |
| The Algebra and Topology of R[superscript n] | p. 1 |
| Review of Linear Algebra | p. 1 |
| Matrix Inversion and Determinants | p. 11 |
| Review of Topology in R[superscript n] | p. 25 |
| Compact Subspaces and Connected Subspaces of R[superscript n] | p. 32 |
| Differentiation | p. 41 |
| The Derivative | p. 41 |
| Continuously Differentiable Functions | p. 49 |
| The Chain Rule | p. 56 |
| The Inverse Function Theorem | p. 63 |
| The Implicit Function Theorem | p. 71 |
| Integration | p. 81 |
| The Integral over a Rectangle | p. 81 |
| Existence of the Integral | p. 91 |
| Evaluation of the Integral | p. 98 |
| The Integral over a Bounded Set | p. 104 |
| Rectifiable Sets | p. 112 |
| Improper Integrals | p. 121 |
| Change of Variables | p. 135 |
| Partitions of Unity | p. 136 |
| The Change of Variables Theorem | p. 144 |
| Diffeomorphisms in R[superscript n] | p. 152 |
| Proof of the Change of Variables Theorem | p. 161 |
| Applications of Change of Variables | p. 169 |
| Manifolds | p. 179 |
| The Volume of a Parallelopiped | p. 180 |
| The Volume of a Parametrized-Manifold | p. 188 |
| Manifolds in R[superscript n] | p. 196 |
| The Boundary of a Manifold | p. 203 |
| Integrating a Scalar Function over a Manifold | p. 209 |
| Differential Forms | p. 219 |
| Multilinear Algebra | p. 220 |
| Alternating Tensors | p. 226 |
| The Wedge Product | p. 236 |
| Tangent Vectors and Differential Forms | p. 244 |
| The Differential Operator | p. 252 |
| Application to Vector and Scalar Fields | p. 262 |
| The Action of a Differentiable Map | p. 267 |
| Stokes' Theorem | p. 275 |
| Integrating Forms over Parametrized-Manifolds | p. 275 |
| Orientable Manifolds | p. 281 |
| Integrating Forms over Oriented Manifolds | p. 293 |
| A Geometric Interpretation of Forms and Integrals | p. 297 |
| The Generalized Stokes' Theorem | p. 301 |
| Applications to Vector Analysis | p. 310 |
| Closed Forms and Exact Forms | p. 323 |
| The Poincare Lemma | p. 324 |
| The deRham Groups of Punctured Euclidean Space | p. 334 |
| Epilogue--Life Outside R[superscript n] | p. 345 |
| Differentiable Manifolds and Riemannian Manifolds | p. 345 |
| Bibliography | p. 359 |
| Index | p. 361 |
| Table of Contents provided by Rittenhouse. All Rights Reserved. |
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