物理学家用的数学方法 第6版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Vector Analysis1

1.1 Definitions,Elementary Approach1

1.2 Rotation ofthe Coordinate Axes7

1.3 Scalar or Dot Product12

1.4 Vector or Cross Product18

1.5 Triple Scalar Product,Triple Vector Product25

1.6 Gradient,？32

1.7 Divergence,？38

1.8 Curl,？×43

1.9 SuccessiveApplications of？49

1.10 VectorIntegration54

1.11 Gauss'Theorem60

1.12 Stokes'Theorem64

1.13 Potential Theory68

1.14 Gauss'Law, Poisson's Equation79

1.15 Dirac Delta Function83

1.16 Helmholtz's Theorem95

Additional Readings101

2 Vector Analysis in Curved Coordinates and Tensors103

2.1 Orthogonal Coordinates in R3103

2.2 Differential Vector Operators110

2.3 Special Coordinate Systems：Introduction114

2.4 Circular Cylinder Coordinates115

2.5 Spherical Polar Coordinates123

2.6 Tensor Analysis133

2.7 Contraction,Direct Product139

2.8 Quotient Rule141

2.9 Pseudotensors,Dual Tensors142

2.10 General Tensors151

2.11 Tensor Derivative Operators160

Additional Readings163

3 Determinants and Matrices165

3.1 Determinants165

3.2 Matrices176

3.3 Orthogonal Matrices195

3.4 Hermitian Matrices,Unitary Matrices208

3.5 Diagonalization of Matrices215

3.6 Normal Matrices231

Additional Readings239

4 Group Theory241

4.1 Introduction to Group Theory241

4.2 Generators of Continuous Groups246

4.3 Orbital Angular Momentum261

4.4 Angular Momentum Coupling266

4.5 Homogeneous Lorentz Group278

4.6 Lorentz Covariance ofMaxwell's Equations283

4.7 Discrete Groups291

4.8 Differential Forms304

Additional Readings319

5 Infinite Series321

5.1 Fundamental Concepts321

5.2 Convergence Tests325

5.3 Alternating Series339

5.4 Algebra ofSeries342

5.5 Series ofFunctions348

5.6 Taylor's Expansion352

5.7 Power Series363

5.8 Elliptic Integrals370

5.9 Bernoulli Numbers,Euler-Maclaurin Formula376

5.10 Asymptotic Series389

5.11 Infinite Products396

Additional Readings401

6 Functions of a Complex Variable I Analytic Properties,Mapping403

6.1 Complex Algebra404

6.2 Cauchy-Riemann Conditions413

6.3 Cauchy's Integral Theorem418

6.4 Cauchy's Integral Formula425

6.5 Laurent Expansion430

6.6 Singularities438

6.7 Mapping443

6.8 Conformal Mapping451

Additional Readings453

7 Functions of a Complex Variable II455

7.1 Calculus of Residues455

7.2 Dispersion Relations482

7.3 Method of Steepest Descents489

Additional Readings497

8 The Gamma Function（Factorial Function）499

8.1 Definitions,Simple Properties499

8.2 Digamma and Polygamma Functions510

8.3 Stirling's Series516

8.4 The Beta Function520

8.5 Incomplete Gamma Function527

Additional Readings533

9 Differential Equations535

9.1 Partial Differential Equations535

9.2 First-Order Differential Equations543

9.3 Separation of Variables554

9.4 Singular Points562

9.5 Series Solutions—Frobenius'Method565

9.6 A Second Solution578

9.7 Nonhomogeneous Equation—Green's Function592

9.8 Heat Flow,or Diffusion,PDE611

Additional Readings618

10 Sturm-Liouville Theory—Orthogonal Functions621

10.1 Self-Adjoint ODEs622

10.2 Hermitian Operators634

10.3 Gram-Schmidt Orthogonalization642

10.4 Completehess of Eigenfunctions649

10.5 Green's Function—Eigenfunction Expansion662

Additional Readings674

11 Bessel Functions675

11.1 Bessel Functions of the First Kind,Jv(x)675

11.2 Orthogonality694

11.3 Neumann Functions699

11.4 Hankel Functions707

11.5 Modified Bessel Functions,Iv(x)and Kv(x)713

11.6 Asymptotic Expansions719

11.7 Spherical Bessel Functions725

Additional Readings739

12 Legendre Functions741

12.1 Generating Function741

12.2 Recurrence Relations749

12.3 Orthogonality756

12.4 Alternate Definitions767

12.5 Associated Legendre Functions771

12.6 Spherical Harmonics786

12.7 Orbital Angular Momentum Operators793

12.8 Addition Theoremfor Spherical Harmonics797

12.9 Integrals of Three Y's803

12.10 Legendre Functions ofthe SecondKind806

12.11 Vector Spherical Harmonics813

Additional Readings816

13 More Special Functions817

13.1Hermite Functions817

13.2 Laguerre Functions837

13.3 Chebyshev Polynomials848

13.4 Hypergeometric Functions859

13.5 Confluent Hypergeometric Functions863

13.6 Mathieu Functions869

Additional Readings879

14 Fourier Series881

14.1 General Properties881

142 Advantages,Uses of Fourier Series888

14.3 Applications of Fourier Series892

14.4 Properties of Fourier Series903

14.5 Gibbs Phenomenon910

14.6 Discrete Fourier Transform914

14.7 Fourier Expansions of Mathieu Functions919

Additional Readings929

15 Integral Transforms931

15.1 Integral Transforms931

15.2 Development of the Fourier Integral936

15.3 Fourier Transforms—Inversion Theorem938

15.4 Fourier Transform of Derivatives946

15.5 Convolution Theorem951

15.6 Momentum Representation955

15.7 Transfer Functions961

15.8 Laplace Transforms965

15.9 Laplace Transform of Derivatives971

15.10 Other Properties979

15.11 Convolution（Faltungs）Theorem990

15.12 Inverse Laplace Transform994

Additional Readings1003

16 Integral Equations1005

16.1 Introduction1005

16.2 Integral Transforms,Generating Functions1012

16.3 Neumann Series,Separable（Degenerate）Kernels1018

16.4 Hilbert-Schmidt Theory1029

Additional Readings1036

17 Calculus of Variations1037

17.1 A Dependent and an Independent Variable1038

17.2 Applications of the Euler Equation1044

17.3 Several Dependent Variables1052

17.4 Several Independent Variables1056

17.5 Several Dependent and Independent Variables1058

17.6 Lagrangian Multipliers1060

17.7 Variation with Constraints1065

17.8 Rayleigh-Ritz Variational Technique1072

Additional Readings1076

18 Nonlinear Methods and Chaos1079

18.1 Introduction1079

18.2 The Logistic Map1080

18.3 Sensitivity to Initial Conditions and Parameters1085

18.4 NonlinearDifferentialEquations1088

Additional Readings1107

19 Probability1109

19.1 Definitions,Simple Properties1109

19.2 Random Variables1116

19.3 Binomial Distribution1128

19.4 Poisson Distribution1130

19.5 Gauss'Normal Distribution1134

19.6 Statistics1138

Additional Readings1150

General References1150

Index1153