Analysis on Manifolds

by Munkres,James R.

• ISBN13: 9780201315967
• ISBN10: 0201315963
• Format: Paperback
• Copyright: 1997-09-01
• Publisher: CRC Press

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
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