Functional Differential Geometry

Functional Differential Geometry

Hardcover

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Product Details

ISBN-13: 9780262019347
Publisher: MIT Press
Publication date: 07/05/2013
Series: The MIT Press
Pages: 248
Product dimensions: 9.00(w) x 6.10(h) x 0.60(d)
Age Range: 18 Years

About the Author

Gerald Jay Sussman is Panasonic Professor of Electrical Engineering at MIT.

Jack Wisdom is Professor of Planetary Science at MIT.

Will Farr is a CIERA Fellow at Northwestern University.

Table of Contents

Preface xi

Prologue xv

1 Introduction 1

2 Manifolds 11

2.1 Coordinate Functions 12

2.2 Manifold Functions 14

3 Vector Fields and One-Form Fields 21

3.1 Vector Fields 21

3.2 Coordinate-Basis Vector Fields 26

3.3 Integral Curves 29

3.4 One-Form Fields 32

3.5 Coordinate-Basis One-Form Fields 34

4 Basis Fields 41

4.1 Change of Basis 44

4.2 Rotation Basis 47

4.3 Commutators 48

5 Integration 55

5.1 Higher Dimensions 57

5.2 Exterior Derivative 62

5.3 Stokes's Theorem 65

5.4 Vector Integral Theorems 67

6 Over a Map 71

6.1 Vector Fields Over a Map 71

6.2 One-Form Fields Over a Map 73

6.3 Basis Fields Over a Map 74

6.4 Pullbacks and Pushforwards 76

7 Directional Derivatives 83

7.1 Lie Derivative 85

7.2 Covariant Derivative 93

7.3 Parallel Transport 104

7.4 Geodesic Motion 111

8 Curvature 115

8.1 Explicit Transport 116

8.2 Torsion 124

8.3 Geodesic Deviation 125

8.4 Bianchi Identities 129

9 Metrics 133

9.1 Metric Compatibility 135

9.2 Metrics and Lagrange Equations 137

9.3 General Relativity 144

10 Hodge Star and Electrodynamics 153

10.1 The Wave Equation 159

10.2 Electrodynamics 160

11 Special Relativity 167

11.1 Lorentz Transformations 172

11.2 Special Relativity Frames 179

11.3 Twin Paradox 181

A Scheme 185

B Our Notation 195

C Tensors 211

References 217

Index 219

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Endorsement

Another gem in the tradition of Structure and Interpretation of Computer Programs and Structure and Interpretation of Classical Mechanics, providing for applied mathematics what the previous two books did for computer science and physics.

Piet Hut, Institute for Advanced Study, Princeton, New Jersey

Piet Hut

Another gem in the tradition of Structure and Interpretation of Computer Programs and Structure and Interpretation of Classical Mechanics, providing for applied mathematics what the previous two books did for computer science and physics.

From the Publisher

Another gem in the tradition of Structure and Interpretation of Computer Programs and Structure and Interpretation of Classical Mechanics, providing for applied mathematics what the previous two books did for computer science and physics.

Piet Hut, Institute for Advanced Study, Princeton, New Jersey

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