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Ⅰ．Introduction 1

1．1．Exterior Differential Forms 1

1．2．Comparison with Tensors 2

Ⅱ．Exterior algebra 5

2．1．The Space of p-vectors 5

2．2．Determinants 7

2．3．Exterior Products 8

2．4．Linear Transformations 10

2．5．Inner Product Spaces 12

2．6．Inner Products of p-vectors 14

2．7．The Star Operator 15

2．8．Problems 17

Ⅲ．The Exterior Derivative 19

3．1．Differential Forms 19

3．2．Exterior Derivative 20

3．3．Mappings 22

3．4．Change of Coordinates 25

3．5．An Example from Mechanics 26

3．6．Converse of the Poincaré Lemma 27

3．7．An Example 30

3．8．Further Remarks 30

3．9．Problems 31

Ⅳ．Applications 32

4．1．Moving Frames in E3 32

4．2．Relation between Orthogonal and Skew-symmetric Matrices 35

4．3．The 6-dimensional Frame Space 37

4．4．The Laplacian，Orthogonal Coordinates 38

4．5．Surfaces 40

4．6．Maxwell＇s Field Equations 44

4．7．Problems 48

Ⅴ．Manifolds and Integration 49

5．1．Introduction 49

5．2．Manifolds 49

5．3．Tangent Vectors 53

5．4．Differential Forms 55

5．5．Euclidean Simplices 57

5．6．Chains and Boundaries 61

6．7．Integration of Forms 63

5．8．Stokes＇ Theorem 64

5．9．Periods and De Rham＇s Theorems 66

5．10．Surfaces； Some Examples 69

5．11．Mappings of Chains 71

5．12．Problems 73

Ⅵ．Applications in Euclidean space 74

6．1．Volumes in En 74

6．2．Winding Numbers，Degree of a Mapping 77

6．3．The Hopf Invariant 79

6．4．Linking Numbers，the Gauss Integral，Ampere＇s Law 79

Ⅶ．Applications to Differential Equations 82

7．1．Potential Theory 82

7．2．The Heat Equation 90

7．3．The Frobenius Integration Theorem 92

7．4．Applications of the Frobenius Theorem 102

7．5．Systems of Ordinary Equations 106

7．6．The Third Lie Theorem 108

Ⅷ．Applications to Differential Geometry 112

8．1．Surfaces(Continued) 112

8．2．Hypersurfaces 116

8．3．Riemannian Geometry，Local Theory 127

8．4．Riemannian Geometry，Harmonic Integrals 136

8．5．Affine Connection 143

8．6．Problems 148

Ⅸ．Application to Group Theory 150

9．1．Lie Groups 150

9．2．Examples of Lie Groups 151

9．3．Matrix Groups 153

9．4．Examples of Matrix Groups 154

9．5．Bi-invariant Forms 158

9．6．Problems 161

Ⅹ．Applications to Physics 163

10．1．Phase and State Space 163

10．2．Hamiltonian Systems 165

10．3．Integral-invariants 171

10．4．Brackets 179

10．5．Contact Transformations 183

10．6．Fluid Mechanics 188

10．7．Problems 193

BIBLIOGRAPHY 197

GLOSSARY OF NOTATION 199

INDEX 201