Differential Forms

by Victor Guillemin, Peter Haine

Overview

'Guillemin and Haine's goal is to construct a well-documented road map that extends undergraduate understanding of multivariable calculus into the theory of differential forms. Throughout, the authors emphasize connections between differential forms and topology while making connections to single and multivariable calculus via the change of variables formula, vector space duals, physics; classical mechanisms, div, curl, grad, Brouwer's fixed-point theorem, divergence theorem, and Stokes's theorem … The exercises support, apply and justify the developing road map.'
CHOICEThere already exist a number of excellent graduate textbooks on the theory of differential forms as well as a handful of very good undergraduate textbooks on multivariable calculus in which this subject is briefly touched upon but not elaborated on enough.The goal of this textbook is to be readable and usable for undergraduates. It is entirely devoted to the subject of differential forms and explores a lot of its important ramifications.In particular, our book provides a detailed and lucid account of a fundamental result in the theory of differential forms which is, as a rule, not touched upon in undergraduate texts: the isomorphism between the Čech cohomology groups of a differential manifold and its de Rham cohomology groups.

Product Details

ISBN-13:	9789811213779
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/25/2019
Pages:	272
Product dimensions:	6.00(w) x 9.00(h) x 0.58(d)

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 Λ^k(V^*) 19

1.6 The wedge product 23

1.7 The interior product 26

1.8 The pullback operation on Λ^k(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