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Tables of Contents for Vector Analysis
Chapter/Section Title
Page #
Page Count
Preface to the English Edition
v
Preface to the First German Edition
vii
Differentiable Manifolds
1
24
The Concept of a Manifold
1
2
Differentiable Maps
3
2
The Rank
5
2
Submanifolds
7
2
Examples of Manifolds
9
3
Sums, Products, and Quotients of Manifolds
12
5
Will Submanifolds of Euclidean Spaces Do?
17
1
Test
18
3
Exercises
21
1
Hints for the Exercises
22
3
The Tangent Space
25
24
Tangent Spaces in Euclidean Space
25
2
Three Versions of the Concept of a Tangent Space
27
5
Equivalence of the Three Versions
32
4
Definition of the Tangent Space
36
1
The Differential
37
3
The Tangent Spaces to a Vector Space
40
1
Velocity Vectors of Curves
41
1
Another Look at the Ricci Calculus
42
3
Test
45
2
Exercises
47
1
Hints for the Exercises
48
1
Differential Forms
49
16
Alternating k-Forms
49
2
The Components of an Alternating k-Form
51
3
Alternating n-Forms and the Determinant
54
1
Differential Forms
55
2
One-Forms
57
2
Test
59
2
Exercises
61
1
Hints for the Exercises
62
3
The Concept of Orientation
65
14
Introduction
65
2
The Two Orientations of an n-Dimensional Real Vector Space
67
3
Oriented Manifolds
70
1
Construction of Orientations
71
3
Test
74
2
Exercises
76
1
Hints for the Exercises
77
2
Integration on Manifolds
79
22
What Are the Right Integrands?
79
4
The Idea behind the Integration Process
83
2
Lebesgue Background Package
85
3
Definition of Integration on Manifolds
88
5
Some Properties of the Integral
93
3
Test
96
2
Exercises
98
1
Hints for the Exercises
99
2
Manifolds-with-Boundary
101
16
Introduction
101
1
Differentiability in the Half-Space
102
1
The Boundary Behavior of Diffeomorphisms
103
2
The Concept of Manifolds-with-Boundary
105
1
Submanifolds
106
1
Construction of Manifolds-with-Boundary
107
2
Tangent Spaces to the Boundary
109
1
The Orientation Convention
110
1
Test
111
3
Exercises
114
1
Hints for the Exercises
115
2
The Intuitive Meaning of Stokes's Theorem
117
16
Comparison of the Responses to Cells and Spans
117
1
The Net Flux of an n-Form through an n-Cell
118
3
Source Strength and the Cartan Derivative
121
1
Stokes's Theorem
122
1
The de Rham Complex
123
1
Simplicial Complexes
124
3
The de Rham Theorem
127
6
The Wedge Product and the Definition of the Cartan Derivative
133
18
The Wedge Product of Alternating Forms
133
2
A Characterization of the Wedge Product
135
2
The Defining Theorem for the Cartan Derivative
137
2
Proof for a Chart Domain
139
1
Proof for the Whole Manifold
140
3
The Naturality of the Cartan Derivative
143
1
The de Rham Complex
144
1
Test
145
3
Exercises
148
1
Hints for the Exercises
148
3
Stokes's Theorem
151
16
The Theorem
151
1
Proof for the Half-Space
152
2
Proof for a Chart Domain
154
1
The General Case
155
1
Partitions of Unity
156
2
Integration via Partitions of Unity
158
2
Test
160
3
Exercises
163
1
Hints for the Exercises
163
4
Classical Vector Analysis
167
28
Introduction
167
1
The Translation Isomorphisms
168
2
Gradient, Curl, and Divergence
170
3
Line and Area Elements
173
2
The Classical Integral Theorems
175
3
The Mean-Value Property of Harmonic Functions
178
2
The Area Element in the Coordinates of the Surface
180
5
The Area Element of the Graph of a Function of Two Variables
185
1
The Concept of the Integral in Classical Vector Analysis
186
3
Test
189
2
Exercises
191
1
Hints for the Exercises
192
3
De Rham Cohomology
195
20
Definition of the de Rham Functor
195
2
A Few Properties
197
2
Homotopy Invariance: Looking for the Idea of the Proof
199
2
Carrying Out the Proof
201
2
The Poincare Lemma
203
3
The Hairy Ball Theorem
206
3
Test
209
2
Exercises
211
1
Hints for the Exercises
212
3
Differential Forms on Riemannian Manifolds
215
24
Semi-Riemannian Manifolds
215
3
The Scalar Product of Alternating k-Forms
218
3
The Star Operator
221
4
The Coderivative
225
2
Harmonic Forms and the Hodge Theorem
227
3
Poincare Duality
230
2
Test
232
3
Exercises
235
1
Hints for the Exercises
236
3
Calculations in Coordinates
239
30
The Star Operator and the Coderivative in Three-Dimensional Euclidean Space
239
2
Forms and Dual Forms on Manifolds without a Metric
241
2
Three Principles of the Ricci Calculus on Manifolds without a Metric
243
3
Tensor Fields
246
3
Raising and Lowering Indices in the Ricci Calculus
249
2
The Invariant Meaning of Raising and Lowering Indices
251
2
Scalar Products of Tensors in the Ricci Calculus
253
2
The Wedge Product and the Star Operator in the Ricci Calculus
255
2
The Divergence and the Laplacian in the Ricci Calculus
257
2
Concluding Remarks
259
2
Test
261
3
Exercises
264
2
Hints for the Exercises
266
3
Answers to the Test Questions
269
4
Bibliography
273
2
Index
275