Methods Of Theoretical Physics, Part I

Methods Of Theoretical Physics Part I......Page 1
Half-Title......Page 2
McGraw-Hill International Series In Pure And Applied Physics......Page 3
Title-Page......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 10
Chapter 1 Types Of Fields......Page 24
1.1 Scalar Fields......Page 27
Isotimic Surfaces......Page 28
The Laplacian......Page 29
1.2 Vector Fields......Page 31
Axial Vectors......Page 33
Lines Of Flow......Page 35
Potential Surfaces......Page 36
Source Point......Page 39
Line Integrals......Page 40
Vortex Line......Page 42
Singularities Of Fields......Page 43
1.3 Curvilinear Coordinates......Page 44
Direction Cosines......Page 45
Scale Factors......Page 47
Curvature Of Coordinate Lines......Page 48
The Volume Element And Other Formulas......Page 50
Rotation Of Axes......Page 51
Law Of Transformation Of Vectors......Page 52
Contravariant And Covariant Vectors......Page 53
The Gradient......Page 54
Directional Derivative......Page 55
Infinitesimal Rotation......Page 56
The Divergence......Page 57
Gauss' Theorem......Page 60
A Solution of Poisson's Equation......Page 61
The Curl......Page 62
Vorticity Lines......Page 65
Stokes' Theorem......Page 66
Covariant And Contravariant Vectors......Page 67
Axial Vectors......Page 69
Christoffel Symbols......Page 70
Covariant Derivative......Page 71
Other Differential Operators......Page 73
Vector As A Sum Of Gradient And Curl......Page 75
Dyadics......Page 77
Dyadics As Vector Operators......Page 78
Symmetric And Antisymmetric Dyadics......Page 81
Rotation Of Axes And Unitary Dyadics......Page 84
Dyadic Fields......Page 87
Deformation Of Elastic Bodies......Page 89
Types Of Strain......Page 91
Stresses In An Elastic Medium......Page 93
Static Stress-Strain Relations For An Isotropic Elastic Body......Page 94
Dyadic Operators......Page 95
Complex Numbers And Quaternions As Operators......Page 96
Abstract Vector Spaces......Page 99
Operators In Quantum Theory......Page 101
Probabilities And Uncertainties......Page 103
Complex Vector Space......Page 104
Generalized Dyadics......Page 105
Hermitian Operators......Page 106
Examples Of Unitary Operators......Page 107
Transformation Of Operators......Page 108
Quantum Mechanical Operators......Page 110
Spin Operators......Page 111
Quaternions......Page 113
Rotation Operators......Page 114
Proper Time......Page 116
The Lorentz Transformation......Page 117
Four-dimensional Invariants......Page 118
Four-vectors......Page 119
Stress-Energy Tensor......Page 121
Spin Space And Space-Time......Page 123
Spinors And Four-Vectors......Page 124
Space Rotation Of Spinors......Page 125
Spin Vectors And Tensors......Page 127
Rotation Operator In Spinor Form......Page 128
Problems For Chapter 1......Page 130
Table Of Useful Vector And Dyadic Equations......Page 137
Table of Properties Of Curvilinear Coordinates......Page 138
Bibliography......Page 140
Chapter 2 Equations Governing Fields......Page 142
Forces On An Element of String......Page 143
Poisson's Equation......Page 144
Concentrated Force, Delta Function......Page 145
The Wave Equation......Page 147
Simple Harmonic Motion, Helmholtz Equation......Page 148
Wave Energy......Page 149
Energy Flow......Page 150
Power And Wave Impedance......Page 151
Forced Motion Of The String......Page 152
Operator Equations For The String......Page 154
Eigenvectors For The Unit Shift Operator......Page 156
Limiting Case Of Continuous String......Page 157
Diffusion Equation......Page 160
Klein-Gordon Equation......Page 161
Forced Motion Of The Elastically Braced String......Page 163
Recapitulation......Page 164
Longitudinal Waves......Page 165
Transverse Waves......Page 166
Wave Motion in Three Dimensions......Page 167
Vector Waves......Page 170
Integral Representations......Page 171
Wave Energy And Impedance......Page 172
2.3 Motion Of Fluids......Page 174
Equation Of Continuity......Page 176
Solutions For Incompressible Fluids......Page 177
Examples......Page 180
Stresses In A Fluid......Page 181
Bernouilli's Equation......Page 184
The Wave Equation......Page 185
Irrotational Flow Of A Compressible Fluid......Page 186
Subsonic And Supersonic Flow......Page 188
Velocity Potential, Linear Approximation......Page 190
Mach Lines And Shock Waves......Page 191
2.4 Diffusion And Other Percolative Fluid Motion......Page 194
Flow of Liquid Through A Porous Solid......Page 195
Diffusion......Page 196
Phase Space And The Distribution Function......Page 197
Pressure And The Equation Of State......Page 199
Mean Free Path And Scattering Cross Section......Page 201
Diffusion Of Light, Integral Equation......Page 203
Diffusion Of Light, Differential Equation......Page 205
Boundary Conditions......Page 208
Effect Of Nonuniform Scattering......Page 211
First-Order Approximation, The Diffusion Equation......Page 212
Unit Solutions......Page 214
Loss Of Energy On Collision......Page 215
Effect Of External Force......Page 217
Uniform Drift Due To Force Field......Page 218
Slowing Down Of Particles By Collisions......Page 219
Recapitulation......Page 222
The Electrostatic Field......Page 223
The Magnetostatic Field......Page 225
Maxwell's Equations......Page 227
Retardation And Relaxation......Page 229
Lorentz Transformation......Page 231
Gauge Transformations......Page 233
Field Of A Moving Charge......Page 235
Force And Energy......Page 238
Surfaces Of Conductors And Dielectrics......Page 240
Wave Transmission And Impedance......Page 241
Proca Equation......Page 244
2.6 Quantum Mechanics......Page 245
Photons And The Electromagnetic Field......Page 246
Conjugate Variables And Poisson Brackets......Page 252
The Fundamental Postulates Of Quantum Theory......Page 254
Independent Quantum Variables And Functions Of Operators......Page 256
Eigenvectors For Coordinates......Page 258
Transformation Functions......Page 259
Operator Equations For Transformation Functions......Page 261
Transformation To Momentum Space......Page 262
Hamiltonian Function And Schroedinger Equation......Page 265
The Harmonic Oscillator......Page 267
Dependence On Time......Page 270
Time As A Parameter......Page 271
Time-Dependent Hamiltonian......Page 274
Particle In Electromagnetic Field......Page 277
Relativity And Spin......Page 278
The Dirac Equation......Page 283
Total Angular Momentum......Page 286
Free-Field Wave Function......Page 287
Recapitulation......Page 288
Problems For Chapter 2......Page 290
Standard Forms For Some Of The Partial Differential Equations Of Theoretical Physics......Page 294
Bibliography......Page 296
Chapter 3 Fields And The Variational Principle......Page 298
The Euler Equations......Page 299
Auxillary Conditions......Page 301
3.2 Hamilton's Principle And Classical Dynamics......Page 303
Lagrange's Equations......Page 304
Energy And The Hamiltonian......Page 305
Impedance......Page 306
Canonical Transformations......Page 310
Poisson Brackets......Page 313
The Action Integral......Page 314
The Two-Dimensional Oscillator......Page 315
Charged Particles In Electromagnetic Field......Page 317
Relativistic Particle......Page 320
Dissipative Systems......Page 321
Impedance and Admittance For Dissipative Systems......Page 322
3.3 Scalar Fields......Page 324
The Flexible String......Page 325
The Wave Equation......Page 327
Helmholtz Equation......Page 329
Velocity Potential......Page 330
Compressional Waves......Page 331
Wave Impedance......Page 333
Plane-Wave Solution......Page 334
Diffusion Equation......Page 336
Schroedinger Equation......Page 337
Klein-Gordon Equation......Page 339
3.4 Vector Fields......Page 341
General Field Properties......Page 342
Isotropic Elastic Media......Page 345
Plane-Wave Solutions......Page 346
Impedance......Page 347
The Electromagnetic Field......Page 349
Stress-Energy Tensor......Page 350
Field Momentum......Page 353
Gauge Transformation......Page 355
Impedance Dyadic......Page 356
Plane-Wave Solution......Page 357
Dirac Equation......Page 358
Problems For Chapter 3......Page 360
Tabulation Of Variational Method......Page 364
Bibliography......Page 370
Chapter 4 Functions Of A Complex Variable......Page 371
4.1 Complex Numbers And Variables......Page 372
The Exponential Rotation Operator......Page 373
Vectors And Complex Numbers......Page 374
The Two-Dimensional Electrostatic Field......Page 375
Contour Integrals......Page 376
4.2 Analytic Functions......Page 379
Conformal Representation......Page 381
Integration In The Complex Plane......Page 385
Cauchy's Theorem......Page 386
Some Useful Corollaries Of Cauchy's Theorem......Page 388
Cauchy's Integral Formula......Page 390
Real And Imaginary Parts Of Analytic Functions......Page 393
Impedances......Page 395
Poisson's Formula......Page 396
4.3 Derivatives Of Analytic Functions, Taylor And Laurent Series......Page 397
The Taylor Series......Page 398
The Laurent Series......Page 401
Isolated Singularities......Page 403
Classification Of Functions, Liouville's Theorem......Page 404
Meromorphic Functions......Page 405
Behavior Of Power Series On The Circle Of Convergence......Page 408
Analytic Continuation......Page 412
Fundamental Theorems......Page 413
Branch Points......Page 414
Techniques Of Analytic Continuation......Page 415
4.4 Multivalued Functions......Page 421
Branch Points And Branch Lines......Page 422
Riemann Surfaces......Page 424
An Illustrative Example......Page 427
4.5 Calculus Of Residues; Gamma And Elliptic Functions......Page 431
Integrals Involving Branch Points......Page 433
Inversion Of Series......Page 434
Summation Of Series......Page 436
Integral Representation Of Functions......Page 437
Integrals Related To The Error Function......Page 439
Gamma Functions......Page 442
Contour Integrals For Gamma Functions......Page 443
Infinite Product Representation For Gamma Functions......Page 444
Derivatives Of The Gamma Funcion......Page 445
The Duplication Formula......Page 447
Periodic Functions......Page 448
Fundamental Properties Of Doubly Periodic Functions......Page 450
Elliptic Functions Of Second Order......Page 452
Integral Representations For Elliptic Functions......Page 455
An Example......Page 457
Averaging Successive Terms......Page 458
Integral Representations And Asymptotic Series......Page 460
Choosing The Contour......Page 461
First Term In The Expansion......Page 463
The Rest Of The Series......Page 464
4.7 Conformal Mapping......Page 466
General Properties Of The Transformation......Page 467
Schwarz-Christoffel Transformation......Page 468
Some Examples......Page 472
The Method Of Inversion......Page 474
4.8 Fourier Transforms......Page 476
Relation To Fourier Series......Page 477
Some Theorems On Integration......Page 478
The Fourier Integral Theorem......Page 481
Properties Of The Fourier Transform......Page 482
General Formulation......Page 485
Faltung......Page 487
Poisson Sum Formula......Page 489
The Laplace Transform......Page 490
The Mellin Transform......Page 492
Problems For Chapter 4......Page 494
Tabulation Of Properties Of Functions Of Complex Variable......Page 503
Bibliography......Page 513
Chapter 5 Ordinary Differential Equations......Page 515
5.1 Separable Coordinates......Page 517
Boundary Surfaces And Coordinate Systems......Page 518
Two-Dimensional Separable Coordinates......Page 521
Separation Of The Wave Equation......Page 522
Rectangular And Parabolic Coordinates......Page 524
Polar And Elliptic Coordinates......Page 525
Scale Factors And Coordinate Geometry......Page 527
Separation Constants And Boundary Conditions......Page 529
Separation In Three Dimensions......Page 531
The Stackel Determinant......Page 532
Confocal Quadric Surfaces......Page 534
Degenerate Forms Of Ellipsoidal Coordinates......Page 536
Confluence Of Singularities......Page 538
Separation Constants......Page 539
Laplace Equation In Three Dimensions, Modulation Factor......Page 541
Confocal Cyclides......Page 542
5.2 General Properties, Series Solutions......Page 546
The Wronskian......Page 547
Independent Solutions......Page 548
Integration Factors And Adjoint Equations......Page 549
Solution Of Inhomogeneous Equation......Page 552
Series Solutions About Ordinary Points......Page 553
Singular Points, Indicial Equation......Page 555
Classification Of Equations, Standard Forms......Page 557
Two Regular Singular Points......Page 559
Three Regular Singular Points......Page 560
Recursion Formulas......Page 562
The Hypergeometric Equation......Page 564
Functions Expressible By Hypergeometric Series......Page 566
Analytic Continuation Of Hypergeometric Series......Page 568
Gegenbauer Functions......Page 570
One Regular And One Irregular Singular Point......Page 573
Asymptotic Series......Page 577
Two Regular, One Irregular Singular Point......Page 578
Continued Fractions......Page 580
The Hill Determinant......Page 583
Mathieu Functions......Page 585
Mathieu Functions Of The Second Kind......Page 590
More On Recursion Formulas......Page 591
Functional Series......Page 593
5.3 Integral Representations......Page 600
Some Simple Examples......Page 601
General Equations For The Integrand......Page 605
The Euler Transform......Page 608
Euler Transform For The Hypergeometric Function......Page 610
Analytic Continuation Of The Hypergeometric Series......Page 614
Legendre Functions......Page 616
Legendre Functions Of The Second Kind......Page 620
Gegenbauer Polynomials......Page 623
The Confluent Hypergeometric Function......Page 627
The Laplace Transform......Page 628
Asymptotic Expansion......Page 630
Stokes' Phenomenon......Page 632
Solutions Of The Third Kind......Page 634
The Solution Of The Second Kind......Page 638
Bessel Functions......Page 642
Hankel Functions......Page 646
Neumann Functions......Page 648
Approximate Formulas For Large Order......Page 650
The Coulomb Wave Function......Page 654
Mathieu Functions......Page 656
The Laplace Transform And The Seperated Wave Equation......Page 659
More On Mathieu Functions......Page 662
Spheroidal Wave Functions......Page 665
Kernels Which Are Functions Of zt......Page 667
Problems For Chapter 5......Page 669
Table Of Separable Coordinates In Three Dimensions......Page 678
Second-Order Differential Equations And Their Solutions......Page 690
Bibliography......Page 697
6.1 Types Of Equations And Of Boundary Conditions......Page 699
Types Of Boundary Conditions......Page 701
Cauchy's Problem And Characteristic Curves......Page 702
Hyperbolic Equations......Page 704
Cauchy Conditions And Hyperbolic Equations......Page 706
Waves In Several Space Dimensions......Page 709
Elliptic Equations And Complex Variables......Page 711
Parabolic Equations......Page 714
6.2 Difference Equations And Boundary Conditions......Page 715
First-Order Linear Difference Equations......Page 716
Difference Equations For Several Dimensions......Page 717
The Elliptic Equation And Dirichlet Conditions......Page 719
Eigenfunctions......Page 721
Green's Functions......Page 723
The Elliptic Equation And Cauchy Conditions......Page 725
The Hyperbolic Difference Equation......Page 726
The Parabolic Difference Equation......Page 728
6.3 Eigenfunctions And Their Uses......Page 729
Fourier Series......Page 730
The Green's Function......Page 733
Eigenfunctions......Page 734
Types Of Boundary Conditions......Page 735
Abstract Vector Space......Page 739
Sturm-Liouville Problem......Page 742
Degeneracy......Page 748
Series Of Eigenfunctions......Page 749
Factorization Of The Sturm-Liouville Equation......Page 752
Eigenfunctions And The Variational Principle......Page 759
The Completeness Of A Set Of Eigenfunctions......Page 761
Asymptotic Formulas......Page 762
Comparison With Fourier Series......Page 766
The Gibbs' Phenomenon......Page 768
Generating Functions, Legendre Polynomials......Page 771
Eigenfunctions In Several Dimensions......Page 776
Separability Of Separation Constants......Page 780
Density Of Eigenvalues......Page 782
Continuous Distribution Of Eigenvalues......Page 785
Eigenvalues For The Schroedinger Equation......Page 789
Discrete And Continuous Eigenvalues......Page 791
Differentiation And Integration As Operators......Page 792
The Eigenvalue Problem In Abstract Vector Space......Page 794
Problems For Chapter 6......Page 801
Table Of Useful Eigenfunctions And Their Properties......Page 804
Eigenfunctions By The Factorization Method......Page 811
Bibliography......Page 813
Chapter 7 Green's Functions......Page 814
Formulation In Abstract Vector Space......Page 816
Boundary Conditions And Surface Charges......Page 818
A Simple Example......Page 820
Relation Between Volume And Surface Green's Functions......Page 822
The General Solution......Page 824
Green's Functions And Generating Functions......Page 825
Green's Theorem......Page 826
Green's Function For The Helmholtz Equation......Page 827
Solution Of The Inhomogeneous Equation......Page 829
General Properties Of The Green's Function......Page 831
The Effect Of Boundary Conditions......Page 833
Method Of Images......Page 835
Series Of Images......Page 837
Other Expansions......Page 839
Expansion Of Green's Function In Eigenfunctions......Page 843
Expansion For The Infinite Domain......Page 845
Polar Coordinates......Page 847
A General Technique......Page 848
A General Formula......Page 851
Green's Functions And Eigenfunctions......Page 855
7.3 Green's Functions For The Scalar Wave Equation......Page 857
The Reciprocity Relation......Page 858
Form Of The Green's Function......Page 861
Field Of A Moving Source......Page 864
Two-Dimensional Solution......Page 865
Initial Conditions......Page 866
Huygen's Principle......Page 870
Boundaries In The Finite Region......Page 871
Eigenfunction Expansions......Page 872
Transient Motion Of Circular Membrane......Page 874
Klein-Gordon Equation......Page 877
7.4 Green's Function For Diffusion......Page 880
Causality And Reciprocity......Page 881
Inhomogeneous Boundary Conditions......Page 882
Green's Function For Infinite Domain......Page 883
Finite Boundaries......Page 885
Eigenfunction Solutions......Page 887
Maximum Velocity Of Heat Ransmission......Page 888
7.5 Green's Function In Abstract Operator Form......Page 892
Generalization Of Green's Theorem, Adjoint Operators......Page 893
Effect Of Boundary Conditions......Page 894
More On Adjoint Differential Operators......Page 897
Adjoint Integral Operators......Page 900
Generalization To Abstract vector Space......Page 901
Adjoint, Conjugate, And Hermitian Operators......Page 903
Green's Function And Green's Operator......Page 904
Reciprocity Relation......Page 905
Expansion Of Green's Operator For Hermitian Case......Page 906
Non-Hermitian Operators; Biorthogonal Functions......Page 907
Problems For Chapter 7......Page 909
Table Of Green's Functions......Page 913
Bibliography......Page 918
8.1 Integral Equations Of Physics, Their Classification......Page 919
Examples From Acoustics......Page 921
An Example From Wave Mechanics......Page 922
Boundary Conditions And Integral Equations......Page 923
Equations For Eigenfunctions......Page 925
Eigenfunctions And Their Integral Equations......Page 926
Types Of Integral Equantion; Fredholm Equations......Page 927
Volterra Equations......Page 928
8.2 General Properties Of Integral Equations......Page 930
Kernels Of Integral Equations......Page 931
Transformation To Definite Kernels......Page 933
Properties Of The Symmetric, Definite Kernel......Page 935
Kernels And Green's Functions For The Inhomogeneous Equation......Page 937
Semidefinite And Indefinite Kernels......Page 939
Kernel Not Real Or Definite......Page 942
Volterra Integral Equation......Page 943
Singular Kernels......Page 944
8.3 Solution Of Fredholm Equations Of The First Kind......Page 948
Series Solutions For Fedholm Equations......Page 949
Determining The Coefficients......Page 950
Orthogonalization......Page 951
Biorthogonal Series......Page 954
Integral Equations Of The First Kind And Generating Functions......Page 958
Use Of Gegenbauer Polynomials......Page 961
Integral Equations Of The First Kind And Green's Functions......Page 962
Transforms And Integral Equations Of The First Kind......Page 965
Differential Equations And Integral Equations Of The First Kind......Page 968
The Moment Problem......Page 970
8.4 Solution Of Integral Equations Of The Second Kind......Page 972
Expansion Of The First Class......Page 974
Expansion Of The Second Class......Page 975
Expansion Of The Third Class......Page 979
Other Classes......Page 980
Inhomogeneous Fredholm Integral Equation Of The Second Kind......Page 982
The Fourier Transform And Kernels Of The Form v(x-minus-x-naught)......Page 983
The Hankel Transform......Page 985
The Kernel v(x-minus-x-naught) In The Infinite Domain......Page 986
The Homogeneous Equation......Page 989
An Example......Page 990
The Kernel v(x-plus-x-naught) In The Infinite Domain......Page 992
An Example......Page 994
Applications Of The Laplace Transform......Page 995
Volterra Integral Equation, Limits (x, lemiscate)......Page 996
Mellin Transform......Page 999
The Method Of Weiner And Hopf......Page 1001
Illustration Of The Method......Page 1004
The Milne Problem......Page 1008
A General Method For Factorization......Page 1010
Milne Problem, Continued......Page 1012
Inhomogeneous Weiner-Hopf Equation......Page 1013
Table Of Integral Equations And Their Solutions......Page 1015
Bibliography......Page 1019
Index......Page 1022
Back Cover......Page 1063