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DIFFERENTIAL FORMS WITH APPLICATIONS TO THE PHYSICAL SCIENCES
  • HARLEY FLANDERS 著
  • 出版社： ACADEMIC PRESS
  • ISBN：
  • 出版时间：1963
  • 标注页数：203页
  • 文件大小：5MB
  • 文件页数：212页
  • 主题词：

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图书目录

Ⅰ．Introduction1

1．1．Exterior Differential Forms1

1．2．Comparison with Tensors2

Ⅱ．Exterior algebra5

2．1．The Space of p-vectors5

2．2．Determinants7

2．3．Exterior Products8

2．4．Linear Transformations10

2．5．Inner Product Spaces12

2．6．Inner Products of p-vectors14

2．7．The Star Operator15

2．8．Problems17

Ⅲ．The Exterior Derivative19

3．1．Differential Forms19

3．2．Exterior Derivative20

3．3．Mappings22

3．4．Change of Coordinates25

3．5．An Example from Mechanics26

3．6．Converse of the Poincaré Lemma27

3．7．An Example30

3．8．Further Remarks30

3．9．Problems31

Ⅳ．Applications32

4．1．Moving Frames in E332

4．2．Relation between Orthogonal and Skew-symmetric Matrices35

4．3．The 6-dimensional Frame Space37

4．4．The Laplacian，Orthogonal Coordinates38

4．5．Surfaces40

4．6．Maxwell＇s Field Equations44

4．7．Problems48

Ⅴ．Manifolds and Integration49

5．1．Introduction49

5．2．Manifolds49

5．3．Tangent Vectors53

5．4．Differential Forms55

5．5．Euclidean Simplices57

5．6．Chains and Boundaries61

6．7．Integration of Forms63

5．8．Stokes＇ Theorem64

5．9．Periods and De Rham＇s Theorems66

5．10．Surfaces； Some Examples69

5．11．Mappings of Chains71

5．12．Problems73

Ⅵ．Applications in Euclidean space74

6．1．Volumes in En74

6．2．Winding Numbers，Degree of a Mapping77

6．3．The Hopf Invariant79

6．4．Linking Numbers，the Gauss Integral，Ampere＇s Law79

Ⅶ．Applications to Differential Equations82

7．1．Potential Theory82

7．2．The Heat Equation90

7．3．The Frobenius Integration Theorem92

7．4．Applications of the Frobenius Theorem102

7．5．Systems of Ordinary Equations106

7．6．The Third Lie Theorem108

Ⅷ．Applications to Differential Geometry112

8．1．Surfaces(Continued)112

8．2．Hypersurfaces116

8．3．Riemannian Geometry，Local Theory127

8．4．Riemannian Geometry，Harmonic Integrals136

8．5．Affine Connection143

8．6．Problems148

Ⅸ．Application to Group Theory150

9．1．Lie Groups150

9．2．Examples of Lie Groups151

9．3．Matrix Groups153

9．4．Examples of Matrix Groups154

9．5．Bi-invariant Forms158

9．6．Problems161

Ⅹ．Applications to Physics163

10．1．Phase and State Space163

10．2．Hamiltonian Systems165

10．3．Integral-invariants171

10．4．Brackets179

10．5．Contact Transformations183

10．6．Fluid Mechanics188

10．7．Problems193

BIBLIOGRAPHY197

GLOSSARY OF NOTATION199

INDEX201

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