张量与黎曼几何 微分方程应用【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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Part Ⅰ Tensors and Riemannian spaces3

1 Preliminaries3

1.1 Vectors in linear spaces3

1.1.1 Three-dimensional vectors3

1.1.2 General case7

1.2 Index notation.Summation convention9

Exercises10

2 Conservation laws11

2.1 Conservation laws in classical mechanics11

2.1.1 Free fall of a body near the earth11

2.1.2 Fall ofa body in a viscous fluid13

2.1.3 Discussion of Kepler's laws16

2.2 General discussion of conservation laws20

2.2.1 Conservation laws for ODEs20

2.2.2 Conservation laws for PDEs21

2.3 Conserved vectors defined by symmetries27

2.3.1 Infinitesimal symmetries of differential equations27

2.3.2 Euler-Lagrange equations.Noether's theorem28

2.3.3 Method of nonlinear self-adjointness36

2.3.4 Short pulse equation40

2.3.5 Linear equations43

Exercises43

3 Introduction of tensors and Riemannian spaces45

3.1 Tensors45

3.1.1 Motivation45

3.1.2 Covariant and contravariant vectors46

3.1.3 Tensor algebra47

3.2 Riemannian spaces49

3.2.1 Differential metric form49

3.2.2 Geodesics.The Christoffel symbols52

3.2.3 Covariant differentiation.The Riemann tensor54

3.2.4 Flat spaces55

3.3 Application to ODEs56

Exercises59

4 Motions in Riemannian spaces61

4.1 Introduction61

4.2 Isometric motions62

4.2.1 Definition62

4.2.2 Killing equations62

4.2.3 Isometric motions on the plane63

4.2.4 Maximal group of isometric motions64

4.3 Conformal motions65

4.3.1 Definition65

4.3.2 Generalized Killing equations65

4.3.3 Conformally flat spaces66

4.4 Generalized motions67

4.4.1 Generalized motions,their invariants and defect68

4.4.2 Invariant family of spaces70

Exercises71

Part Ⅱ Riemannian spaces of second-order equations75

5 Riemannian spaces associated with linear PDEs75

5.1 Covariant form of second-order equations75

5.2 Conformally invariant equations78

Exercises78

6 Geometry of linear hyperbolic equations79

6.1 Generalities79

6.1.1 Covariant form of determining equations79

6.1.2 Equivalence transformations80

6.1.3 Existence of conformally invariant equations81

6.2 Spaces with nontrivial conformal group83

6.2.1 Definition of nontrivial conformal group83

6.2.2 Classification of four-dimensional spaces83

6.2.3 Uniqueness theorem86

6.2.4 On spaces with trivial conformal group87

6.3 Standard form of second-order equations88

6.3.1 Curved wave operator in V4 with nontrivial conformal group88

6.3.2 Standard form of hyperbolic equations with nontrivial conformal group90

Exercises90

7 Solution of the initial value problem93

7.1 The Cauchy problem93

7.1.1 Reduction to a particular Cauchy problem93

7.1.2 Fourier transform and solution of the particular Cauchy problem94

7.1.3 Simplification of the solution95

7.1.4 Verification of the solution97

7.1.5 Comparison with Poisson's formula99

7.1.6 Solution of the general Cauchy problem100

7.2 Geodesics in spaces with nontrivial conformal group100

7.2.1 Outline of the approach101

7.2.2 Equations of geodesics in spaces with nontrivial conformal group102

7.2.3 Solution of equations for geodesics102

7.2.4 Computation of the geodesic distance104

7.3 The Huygens principle105

7.3.1 Huygens'principle for classical wave equation106

7.3.2 Huygens'principle for the curved wave operator in V4 with nontrivial conformal group107

7.3.3 On spaces with trivial conformal group107

Exercises108

Part Ⅲ Theory of relativity111

8 Brief introduction to relativity111

8.1 Special relativity111

8.1.1 Space-time intervals111

8.1.2 The Lorentz group112

8.1.3 Relativistic principle of least action113

8.1.4 Relativistic Lagrangian114

8.1.5 Conservation laws in relativistic mechanics115

8.2 The Maxwell equations116

8.2.1 Introduction116

8.2.2 Symmetries of Maxwell's equations117

8.2.3 General discussion of conservation laws119

8.2.4 Evolutionary part of Maxwell's equations122

8.2.5 Conservation laws of Eqs.(8.2.1 )and(8.2.2 )129

8.3 The Dirac equation132

8.3.1 Lagrangian obtained from the formal Lagrangian133

8.3.2 Symmetries134

8.3.3 Conservation laws136

8.4 General relativity137

8.4.1 The Einstein equations137

8.4.2 The Schwarzschild space138

8.4.3 Discussion of Mercury's parallax138

8.4.4 Solutions based on generalized motions139

Exercises141

9 Relativity in de Sitter space143

9.1 The de Sitter space143

9.1.1 Introduction143

9.1.2 Reminder of the notation145

9.1.3 Spaces of constant Riemannian curvature147

9.1.4 Killing vectors in spaces of constant curvature148

9.1.5 Spaces with positive definite metric149

9.1.6 Geometric realization of the de Sitter metric152

9.2 The de Sitter group153

9.2.1 Generators of the de Sitter group153

9.2.2 Conformal transformations in R3154

9.2.3 Inversion156

9.2.4 Generalized translation in direction of x-axis158

9.3 Approximate de Sitter group158

9.3.1 Approximate groups158

9.3.2 Simple method of solution of Killing's equations161

9.3.3 Approximate representation of de Sitter group163

9.4 Motion of a particle in de Sitter space165

9.4.1 Introduction165

9.4.2 Conservation laws in Minkowski space166

9.4.3 Conservation laws in de Sitter space168

9.4.4 Kepler's problem in de Sitter space169

9.5 Curved wave operator171

9.6 Neutrinos in de Sitter space172

9.6.1 Two approximate representations of Dirac's equations in de Sitter space173

9.6.2 Splitting of neutrinos by curvature174

Exercises175

Bibliography177

Index181