Mathematical Physics

Description

CONTENTS 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 . . . . . . . . 33 2.13 Conformal Mapping by Analytic Functions . . . . . . . . 97 2.14 Complex Sphere and Point at Infinity . . . . . . . . . 102 2.15 Integral Representations . . . . . . . . . . . . . 104 Chapter 3 Linear Differential Equations of Second Order 3.] General Introduction. The Wronskian . . . . . . . . . 123 3.2 General Solution of The Homogeneous Equation . . . . . . 125 3.3 The Non-homogeneous 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 Chapter 4 Fourier Series 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 Point-wise 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 The0rem . . . . . . . . . . . . . 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 δ-Calculus . . . . . . . . . . . . . . . . 226 6.4 Representations of Delta Functions . . . . . . . . . . 229 6.5 Applications of The δ-Calculus . . . . . . . . . . . 232 6.6 Weak Convergence . . . . . . . . . . . . . . . 236 6.? 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 . . . . . . . . . . 223 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 Transformation . . . . . . 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 Harmonics 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,E) . . . . . , . . . . . . 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 13.3 Variational Problems in Many Dimensions . . . . . . . . 577 13.9 Formulation of Eigenvalue Problems by The Ratio Method . . . 581 Chapter 14 Travelling 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 Travailing Waves in Inhomogeneous 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-Schrödinger Theory . . . . . . . . 653 15.5 The Second-Order Non-degenerate 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. Pseudo-tensors . . . . . 684 16.6 Derivatives of Tensors. Strain Tensor and Hooke’s Law . . . . 637 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