Analysis On Manifolds |
Contents
PREFACE CHAPTER 1 The Algebra and Topology of R | 1 |
2 Matrix Inversion and Determinants | 11 |
3 Review of Topology in R | 25 |
4 Compact Subspaces and Connected Subspaces of R | 32 |
Differentiation | 41 |
6 Continuously Differentiable Functions | 53 |
7 The Chain Rule | 56 |
8 The Inverse Function Theorem | 63 |
19 Proof of the Change of Variables Theorem | 161 |
20 Applications of Change of Variables | 169 |
Manifolds | 179 |
21 The Volume of a Parallelopiped | 180 |
22 The Volume of a ParametrizedManifold | 188 |
23 Manifolds in R | 196 |
24 The Boundary of a Manifold | 203 |
25 Integrating a Scalar Function over a Manifold | 209 |
9 The Implicit Function Theorem | 71 |
Integration | 81 |
11 Existence of the Integral | 91 |
12 Evaluation of the Integral | 98 |
13 The Integral over a Bounded Set | 104 |
14 Rectifiable Sets | 112 |
15 Improper Integrals | 121 |
Change of Variables | 135 |
16 Partitions of Unity | 136 |
17 The Change of Variables Theorem | 144 |
18 Diffeomorphisms in R | 152 |
Differential Forms | 219 |
26 Multilinear Algebra | 220 |
27 Alternating Tensors | 226 |
28 The Wedge Product | 236 |
29 Tangent Vectors and Differential Forms | 244 |
30 The Differential Operator | 252 |
31 Application to Vector and Scalar Fields | 262 |
32 The Action of a Differentiable Map | 267 |
Stokes Theorem | 275 |
Closed Forms and Exact Forms | 323 |
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Common terms and phrases
a₁ alternating tensors arbitrary b₁ basis bounded function calculus called change of variables class Cr column compact rectifiable compute contained continuous function coordinate patch corresponding cover cube definition denote diffeomorphism differentiable equals equation f(x EXAMPLE EXERCISES exists extended integral Figure finite follows formula Fubini theorem function f ƒ and g ƒ is continuous ƒ is integrable given induced orientation interval inverse function k-form k-manifold k-tensor k-tuple lemma Let f Let g linear transformation manifold matrix measure zero metric space neighborhood non-singular notation one-to-one open set operations ordinary integral orthogonal partial derivatives partition of unity permutation Proof properties prove the theorem rectangle Q rectifiable sets scalar field sequence Step subset subspace Suppose tangent vector V₁ vanishes variables theorem vector field vector space wedge product θα