Mathematical Physics - Eugene Butkov
Textbook information
- Text book title :Mathematical Physics
- Author : Eugene Butkov
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- File size : 6.81 Mb
- File format : DJVU File
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Chapter 1 Vectors, Matrices, and Coordinates
- 1.1 Introduction 1
- 1.2 Vectors in Cartesian Coordinate Systems 1
- 1.3 Changes of Axes. Rotation Matrices 4
- 1.4 Repeated Rotations. Matrix Multiplication 8
- 1.5 Skew Cartesian Systems. Matrices in General 11
- 1.6 Scalar and Vector Fields 14
- 1.7 Vector Fields in Plane 20
- 1.8 Vector Fields in Space 26
- 1.9 Curvilinear Coordinates 34
Chapter 2 Functions of a Complex Variable
- 2.1 Complex Numbers 44
- 2.2 Basic Algebra and Geometry of Complex Numbers 45
- 2.3 De Moivre Formula and the Calculation of Roots 48
- 2.4 Complex Functions. Euler's Formula 49
- 2.5 Applications of Euler's Formula 51
- 2.6 Multivalued Functions and Riemann Surfaces 54
- 2.7 Analytic Functions. Cauchy Theorem 58
- 2.8 Other Integral Theorems. Cauchy Integral Formula 62
- 2.9 Complex Sequences and Series 66
- 2.10 Taylor and Laurent Series 71
- 2.11 Zeros and Singularities 78
- 2.12 The Residue Theorem and its Applications 83
- 2.13 Conformal Mapping by Analytic Functions 97
- 2.14 Complex Sphere and Point at Infinity 102
- 2.15 Integral Representations 104
- 3.1 General Introduction. The Wronskian 123
- 3.2 General Solution of The Homogeneous Equation 125
- 3.3 The Nonhomogeneous Equation. Variation of Constants . . . . 126
- 3.4 Power Series Solutions 128
- 3.5 The Frobenius Method 130
- 3.6 Some other Methods of Solution 147
- 4.1 Trigonometric Series 154
- 4.2 Definition of Fourier Series 155
- 4.3 Examples of Fourier Series' 157
- 4.4 Parity Properties. Sine and Cosine Series 161
- 4.5 Complex Form of Fourier Series 165
- 4.6 Pointwise Convergence of Fourier Series 167
- 4.7 Convergence in the Mean 168
- 4.8 Applications of Fourier Series 172
Chapter 5 The Laplace Transformation
- 5.1 Operational Calculus 179
- 5.2 The Laplace Integral 180
- 5.3 Basic Properties of Laplace Transform 184
- 5.4 The Inversion Problem 187
- 5.5 The Rational Fraction Decomposition 189
- 5.6 The Convolution Theorem 194
- 5.7 Additional Properties of Laplace Transform 200
- 5.8 Periodic Functions. Rectification 204
- 5.9 The Mellin Inversion Integral *. 206
- 5.10 Applications of Laplace Transforms . . .• 210
Chapter 6 Concepts of the Theory of Distributions
- 6.1 Strongly Peaked Functions and The Dirac Delta Function . . . 221
- 6.2 Delta Sequences 223
- 6.3 The 6-Calculus 226
- 6.4 Representations of Delta Functions 229
- 6.5 Applications of The 6-Calculus 232
- 6.6 Weak Convergence 236
- 6.7 Correspondence of Functions and Distributions 240
- 6.8 Properties of Distributions 245
- 6.9 Sequences and Series of Distributions 250
- 6.10 Distributions in N dimensions 257
Chapter 7 Fourier Transforms
- 7.1 Representations of a Function 260
- 7.2 Examples of Fourier Transformations 262
- 7.3 Properties of Fourier Transforms 266
- 7.4 Fourier Integral Theorem 269
- 7.5 Fourier Transforms of Distributions 271
- 7.6 Fourier Sine and Cosine Transforms 273
- 7.7 Applications of Fourier Transforms. The Principle of Causality . . 276
Chapter 8 Partial Differential Equations
- 8.1 The Stretched String. Wave Equation 287
- 8.2 The Method of Separation of Variables 291
- 8.3 Laplace and Poisson Equations 295
- 8.4 The Diffusion Equation 297
- 8.5 Use of Fourier and Laplace Transforms 299
- 8.6 The Method of Eigenfunction Expansions and Finite Transforms . 304
- 8.7 Continuous Eigenvalue Spectrum 308
- 8.8 Vibrations of a Membrane. Degeneracy 313
- 8.9 Propagation of Sound. Helmholtz Equation 319
Chapter 9 Special Functions
- 9.1 Cylindrical and Spherical Coordinates 332
- 9.2 The Common Boundary-Value Problems 334
- 9.3 The Sturm-Liouville Problem 337
- 9.4 Self-Adjoint Operators 340
- 9.5 Legendre Polynomials 342
- 9.6 Fourier-Legendre Series 350
- 9.7 Bessel Functions 355
- 9.8 Associated Legendre Functions and Spherical Harmonics .... 372
- 9.9 Spherical Bessel Functions 381
- 9.10 Neumann Functions 388
- 9.11 Modified Bessel Functions 394
Chapter 10 Finite-Dimensional Linear Spaces
- 10.1 Oscillations of Systems with Two Degrees of Freedom .... 405
- 10.2 Normal Coordinates and Linear Transformations 411
- 10.3 Vector Spaces, Bases, Coordinates 419
- 10.4 Linear Operators, Matrices, Inverses . ■ 424
- 10.5 Changes of Basis 433
- 10.6 Inner Product. Orthogonality. Unitary Operators 437
- 10.7 The Metric. Generalized Orthogonality 441
- 10.8 Eigenvalue Problems. Diagonalization 443
- 10.9 Simultaneous Diagonalization 451
Chapter 11 Infinite-Dimensional Vector Spaces
- 11.1 Spaces of Functions 463
- 11.2 The Postulates of Quantum Mechanics 467
- 11.3 The Harmonic Oscillator 471
- 11.4 Matrix Representations of Linear Operators 476
- 11.5 Algebraic Methods of Solution 483
- 11.6 Bases with Generalized Orthogonality 488
- 11.7 Stretched String with a Discrete Mass in the Middle 492
- 11.8 Applications of Eigenfunctions 495
Chapter 12 Green's Functions
- 12.1 Introduction 503
- 12.2 Green's Function for the Sturm-Liouville Operator 508
- 12.3 Series Expansions for G(x\ £) 514
- 12.4 Green's Functions in Two Dimensions 520
- 12.5 Green's Functions for Initial Conditions 523
- 12.6 Green's Functions with Reflection Properties 527
- 12.7 Green's Functions for Boundary Conditions 531
- 12.8 The Green's Function Method 536
- 12.9 A Case of Continuous Spectrum 543
Chapter 13 Variational Methods
- 13.1 The Brachistochrone Problem 553
- 13.2 The Euler-Lagrange Equation 554
- 13.3 Hamilton's Principle 560
- 13.4 Problems involving Sturm-Liouville Operators 562
- 13.5 The Rayleigh-Ritz Method 565
- 13.6 Variational Problems with Constraints 567
- 13.7 Variational Formulation of Eigenvalue Problems 573
- 13.8 Variational Problems in Many Dimensions 577
- 13.9 Formulation of Eigenvalue Problems by The Ratio Method . . . 581
Chapter 14 Traveling Waves, Radiation, Scattering
- 14.1 Motion of Infinite Stretched String 589
- 14.2 Propagation of Initial Conditions 592
- 14.3 Semi-infinite String. Use of Symmetry Properties 595
- 14.4 Energy and Power Flow in a Stretched String 599
- 14.5 Generation of Waves in a Stretched String 603
- 14.6 Radiation of Sound from a Pulsating Sphere 611
- 14.7 The Retarded Potential 619
- 14.8 Traveling Waves in Nonhomogeneous Media 624
- 14.9 Scattering Amplitudes and Phase Shifts 628
- 14.10 Scattering in Three Dimensions. Partial Wave Analysis . . . 633
Chapter 15 Perturbation Methods
- 15.1 Introduction 644
- 15.2 The Born Approximation 647
- 15.3 Perturbation of Eigenvalue Problems 650
- 15.4 First-Order Rayleigh-Schrodinger Theory 653
- 15.5 The Second-Order Nondegenerate Theory 658
- 15.6 The Case of Degenerate Eigenvalues 665
Chapter 16 Tensors
- 16.1 Introduction 671
- 16.2 Two-Dimensional Stresses 672
- 16.3 Cartesian Tensors 676
- 16.4 Algebra of Cartesian Tensors 681
- 16.5 Kronecker and Levi-Civita Tensors. Pseudotensors 684
- 16.6 Derivatives of Tensors. Strain Tensor and Hooke's Law .... 687
- 16.7 Tensors in Skew Cartesian Frames. Covariant and
- Contravariant Representations 696
- 16.8 General Tensors 700
- 16.9 Algebra of General Tensors. Relative Tensors 705
- 16.10 The Covariant Derivative 711
- 16.11 Calculus of General Tensors 715
Index 727