Functional differential geometry

Physics is naturally expressed in mathematical language. Students new to the subject must simultaneously learn an idiomatic mathematical language and the content that is expressed in that language. It is as if they were asked to read Les Miserables while struggling with French grammar. This book offers an innovative way to learn the differential geometry needed as a foundation for a deep understanding of general relativity or quantum field theory as taught at the college level.

The approach taken by the authors (and used in their classes at MIT for many years) differs from the conventional one in several ways, including an emphasis on the development of the covariant derivative and an avoidance of the use of traditional index notation for tensors in favor of a semantically richer language of vector fields and differential forms. But the biggest single difference is the authors' integration of computer programming into their explanations. By programming a computer to interpret a formula, the student soon learns whether or not a formula is correct. Students are led to improve their program, and as a result improve their understanding.

Author Notes

Gerald Jay Sussman is Panasonic (formerly Matsushita) Professor of Electrical Engineering in the Department of Electrical Engineering and Computer Science at MIT and the coauthor (with Hal Abelson) of Structure and Interpretation of Computer Programs (MIT Press). Jack Wisdom is Professor of Planetary Science in the Department of Earth, Atmospheric, and Planetary Sciences at MIT. Sussman and Wisdom are the coauthors of Structure and Interpretation of Classical Mechanics (MIT Press). Will Farr is a CIERA Fellow at Northwestern University.

Preface	p. xi
Prologue	p. xv
1 Introduction	p. 1
2 Manifolds	p. 11
2.1 Coordinate Functions	p. 12
2.2 Manifold Functions	p. 14
3 Vector Fields and One-Form Fields	p. 21
3.1 Vector Fields	p. 21
3.2 Coordinate-Basis Vector Fields	p. 26
3.3 Integral Curves	p. 29
3.4 One-Form Fields	p. 32
3.5 Coordinate-Basis One-Form Fields	p. 34
4 Basis Fields	p. 41
4.1 Change of Basis	p. 44
4.2 Rotation Basis	p. 47
4.3 Commutators	p. 48
5 Integration	p. 55
5.1 Higher Dimensions	p. 57
5.2 Exterior Derivative	p. 62
5.3 Stokes's Theorem	p. 65
5.4 Vector Integral Theorems	p. 67
6 Over a Map	p. 71
6.1 Vector Fields Over a Map	p. 71
6.2 One-Form Fields Over a Map	p. 73
6.3 Basis Fields Over a Map	p. 74
6.4 Pullbacks and Pushforwards	p. 76
7 Directional Derivatives	p. 83
7.1 Lie Derivative	p. 85
7.2 Covariant Derivative	p. 93
7.3 Parallel Transport	p. 104
7.4 Geodesic Motion	p. 111
8 Curvature	p. 115
8.1 Explicit Transport	p. 116
8.2 Torsion	p. 124
8.3 Geodesic Deviation	p. 125
8.4 Bianchi Identities	p. 129
9 Metrics	p. 133
9.1 Metric Compatibility	p. 135
9.2 Metrics and Lagrange Equations	p. 137
9.3 General Relativity	p. 144
10 Hodge Star and Electrodynamics	p. 153
10.1 The Wave Equation	p. 159
10.2 Electrodynamics	p. 160
11 Special Relativity	p. 167
11.1 Lorentz Transformations	p. 172
11.2 Special Relativity Frames	p. 179
11.3 Twin Paradox	p. 181
A Scheme	p. 185
B Our Notation	p. 195
C Tensors	p. 211
References	p. 217
Index	p. 219

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Table of Contents