Analysis on Manifolds

Preface......Page f005.djvu
Contents......Page f009.djvu
1 Review of Linear Algebra......Page p001.djvu
2 Matrix Inversion and Determinants......Page p011.djvu
3 Review of Topology in R^n......Page p025.djvu
4 Compact Subspaces and Connected Subspaces of R^n......Page p032.djvu
5 Derivative......Page p041.djvu
6 Continuously Differentiable Functions......Page p049.djvu
7 The Chain Rule......Page p056.djvu
8 The Inverse Function Theorem......Page p063.djvu
*9 The Implicit Function Theorem......Page p071.djvu
10 The Integral over a Rectangle......Page p081.djvu
11 Existence of the Integral......Page p091.djvu
12 Evaluation of the Integral......Page p098.djvu
13 The Integral over a Bounded Set......Page p104.djvu
14 Rectifiable Sets......Page p112.djvu
15 Improper Integrals......Page p121.djvu
4 Changes of Variables......Page p135.djvu
16 Partitions of Unity......Page p136.djvu
17 The Change of Variables Theorem......Page p144.djvu
18 Diffeomorphisms in R^n......Page p152.djvu
19 Proof of the Change of Variables Theorem......Page p160.djvu
20 Application of Change of Variables......Page p169.djvu
5 Manifolds......Page p179.djvu
21 The Volume of a Parallelopiped......Page p178.djvu
22 The Volume of a Parametrized-Manifold......Page p186.djvu
23 Manifolds in R^n......Page p194.djvu
24 The Boundary of a Manifold......Page p201.djvu
25 Integrating a Scalar Function over a Manifold......Page p207.djvu
6 Differential Forms......Page p219.djvu
26 Multilinear Algebra......Page p220.djvu
27 Alternating Tensors......Page p226.djvu
28 The Wedge Product......Page p236.djvu
29 Tangent Vectors and Differential Forms......Page p244.djvu
30 The Differential Operator......Page p252.djvu
*31 Application to Vector and Scalar Fields......Page p262.djvu
32 The Action of a Differentiable Map......Page p267.djvu
33 Integrating Forms over Parametrized-Manifold......Page p275.djvu
34 Orientable Manifolds......Page p281.djvu
35 Integrating Forms over Oriented Manifolds......Page p293.djvu
*36 A Geometric Interpretation of Forms and Integrals......Page p297.djvu
37 The Generalized Stokes\' Theorem......Page p301.djvu
*38 Applications to Vector Analysis......Page p310.djvu
8 Closed Forms and Exact Forms......Page p323.djvu
39 The Poincare Lemma......Page p324.djvu
40 The deRham Groups of Punctured Euclidean Space......Page p334.djvu
41 Differentiable Manifolds and Riemannian Manifolds......Page p345.djvu
Bibliography......Page p359.djvu
Index......Page p361.djvu