Table of Contents
Preface v
Introduction v
Organization vii
Notational Conventions xi
Acknowledgments xii
About the Authors xiii
Chapter 1 Multilinear Algebra 1
1.1 Background 1
1.2 Quotient and dual spaces 4
1.3 Tensors 9
1.4 Alternating k-tensors 13
1.5 The space Λk(V*) 19
1.6 The wedge product 23
1.7 The interior product 26
1.8 The pullback operation on Λk(V*) 29
1.9 Orientations 33
Chapter 2 The Concept of a Differential Form 37
2.1 Vector fields and 1-forms 37
2.2 Integral curves for vector fields 42
2.3 Differential k-forms 50
2.4 Exterior differentiation 53
2.5 The interior product operation 58
2.6 The pullback operation on forms 61
2.7 Divergence, curl, and gradient 68
2.8 Symplectic geometry and classical mechanics 72
Chapter 3 Integration of Forms 81
3.1 Introduction 81
3.2 The Poincaré lemma for compactly supported forms on rectangles 81
3.3 The Poincaré lemma for compactly supported forms on open subsets of Rn 86
3.4 The degree of a differentiable mapping 88
3.5 The change of variables formula 92
3.6 Techniques for computing the degree of a mapping 98
3.7 Appendix: Sard's theorem 106
Chapter 4 Manifolds and Forms on Manifolds 111
4.1 Manifolds 111
4.2 Tangent spaces 119
4.3 Vector fields and differential forms on manifolds 125
4.4 Orientations 133
4.5 Integration of forms on manifolds 142
4.6 Stokes' theorem and the divergence theorem 147
4.7 Degree theory on manifolds 153
4.8 Applications of degree theory 158
4.9 The index of a vector field 165
Chapter 5 Cohomology via Forms 171
5.1 The de Rham cohomology groups of a manifold 171
5.2 The Mayer-Vietoris sequence 182
5.3 Cohomology of good covers 190
5.4 Poincaré duality 197
5.5 Thom classes and intersection theory 203
5.6 The Lefschetz theorem 212
5.7 The Künneth theorem 221
5.8 Cech cohomology 225
Appendix A Bump Functions and Partitions of Unity 233
Appendix B The Implicit Function Theorem 237
Appendix C Good Covers and Convexity Theorems 245
Bibliography 249
Index of Notation 251
Glossary of Terminology 253