目錄
Preface
1: Prologue: What This Book Is About
1.1 Introduction
1.2 Where We Begin
1.3 What Vectors Are Not
1.4 What Traditional Vectors Are
1.5 Do Vectors Have a Location?
1.6 Coordinate Transformations vs Distortions of System
1.7 Why Is Topological Invariance Important?
Problems
2: Vector Types and Vector Operations
2.1 What Is the Problem?
2.2 Stacks
2.3 The Size of Mathematical Objects
2.4 Orientation of Lines and of Planes
2.5 Algebra of Arrows and of Stacks
2.6 The Dot Product
Problems
3: More Operations, More Vectors
3.1 A New Kind of Product
3.2 The Cross Product of Arrows
3.3 Polar and Axial Sense Genders
3.4 Algebra of Polar and Axial Senses
3.5 Polar and Axial Scalars
3.6 The Fourth and Last Vector: The Sheaf
3.7 Algebra of Thumbtacks and Sheaves
Problems
4: Completion of the Menagerie
4.1 Reprise: The Need for Invariance
4.2 What Is Still Missing
4.3 The Remaining Cross Products
4.4 More Dot Products: Scalar Densities and Capacities
4.5 Formalization of Nomenclature
4.6 The Geometrical Meaning of "Co-" and 「Contra
4.7 Multiplication by Scalar Densities and Capacities
Problems
5: Fields and the Geometrical Calculus
5.1 Fields
5.2 The Gradient
5.3 The Curl
5.4 The Divergence
5.5 The Inverse Operations
5.6 The Meaning of Differential Operations
Problems
6: Coordinates and Components
6.1 Coordinate Systems
6.2 The Bases of Scalar Capacities and Densities
6.3 The Arrow and Stack Bases
6.4 Stack-Arrow Dot Product in Terms of Components
6.5 How Coordinate Systems Differ
6.6 But Off the Record, How Does It Look?
Problems
7: The Grand Algebraization Rule
7.1 Statement of the Rule
7.2 The Remaining Bases
7.3 Cross Products in Terms of Components
7.4 Hermaphrodite Sense Gender of the Bases
7.5 Computation of the Gradient
7.6 Computation of the Curl
7.7 And Finally, the Divergence
Problems
8: Goodbye to the Rubber Universe
8.1 The Need for Measurement
8.2 Example: The Electromagnetic Field
8.3 Underlying Cartesian System
8.4 Legalization of Illegal Operations; the Laplacian
8.5 The "Del" Operator
8.6 Orthogonal Systems
8.7 The Metric
Problems
9: Epilogue: Where This Book Leaves Us
9.1 Some Remaining Problems
9.2 Number of Dimensions
9.3 Curved Spaces
9.4 Indefinite Metric
9.5 The Nature of Tensor Analysis
9.6 Conclusion
Index
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