| | Contents | |
| | | |
| |
| 1 | Infinite Sequences and Series | 1 |
| 1.1 | Real and Complex Numbers | 3 |
| 1.1.1 | Arithmetic | 3 |
| 1.1.2 | Algebraic Equations | 4 |
| 1.1.3 | Infinite Sequences; Irrational Numbers | 5 |
| 1.1.4 | Sets of Real and Complex Numbers | 7 |
| 1.2 | Convergence of Infinite Series and Products | 8 |
| 1.2.1 | Convergence and Divergence; Absolute Convergence | 8 |
| 1.2.2 | Tests for Convergence of an Infinite Series of Positive Terms | 10 |
| 1.2.3 | Alternating Series and Rearrangements | 11 |
| 1.2.4 | Infinite Products | 13 |
| 1.3 | Sequences and Series of Functions | 14 |
| 1.3.1 | Pointwise Convergence and Uniform Convergence of Sequences of Functions | 14 |
| 1.3.2 | Weak Convergence; Generalized Functions | 15 |
| 1.3.3 | Infinite Series of Functions; Power Series | 16 |
| 1.4 | Asymptotic Series | 19 |
| 1.4.1 | The Exponential Integral | 19 |
| 1.4.2 | Asymptotic Expansions; Asymptotic Series | 20 |
| 1.4.3 | Laplace Integral; Watson’s Lemma | 22 |
| A | Iterated Maps, Period Doubling, and Chaos | 26 |
| | Bibliography and Notes | 30 |
| | Problems | 31 |
| 2 | Finite-Dimensional Vector Spaces | 37 |
| 2.1 | Linear Vector Spaces | 41 |
| 2.1.1 | Linear Vector Space Axioms | 41 |
| 2.1.2 | Vector Norm; Scalar Product | 43 |
| 2.1.3 | Sum and Product Spaces | 47 |
| 2.1.4 | Sequences of Vectors | 49 |
| 2.1.5 | Linear Functionals and Dual Spaces | 49 |
| 2.2 | Linear Operators | 51 |
| 2.2.1 | Linear Operators; Domain and Image; Bounded Operators | 51 |
| 2.2.2 | Matrix Representation; Multiplication of Linear Operators | 54 |
| 2.2.3 | The Adjoint Operator | 56 |
| 2.2.4 | Change of Basis; Rotations; Unitary Operators | 57 |
| 2.2.5 | Invariant Manifolds | 61 |
| 2.2.6 | Projection Operators | 63 |
| 2.3 | Eigenvectors and Eigenvalues | 64 |
| 2.3.1 | Eigenvalue Equation | 64 |
| 2.3.2 | Diagonalization of a Linear Operator | 65 |
| 2.3.3 | Spectral Representation of Normal Operators | 67 |
| 2.3.4 | Minimax Properties of Eigenvalues of Self-Adjoint Operators | 71 |
| 2.4 | Functions of Operators | 75 |
| 2.5 | Linear Dynamical Systems | 77 |
| A | Small Oscillations | 80 |
| | Bibliography and Notes | 83 |
| | Problems | 84 |
| 3 | Geometry in Physics | 93 |
| 3.1 | Manifolds and Coordinates | 97 |
| 3.1.1 | Coordinates on Manifolds | 97 |
| 3.1.2 | Some Elementary Manifolds | 98 |
| 3.1.3 | Elementary Properties of Manifolds | 101 |
| 3.2 | Vectors, Differential Forms, and Tensors | 104 |
| 3.2.1 | Smooth Curves and Tangent Vectors | 104 |
| 3.2.2 | Tangent Spaces and the Tangent Bundle T (M) | 105 |
| 3.2.3 | Differential Forms | 106 |
| 3.2.4 | Tensors | 109 |
| 3.2.5 | Vector and Tensor Fields | 110 |
| 3.2.6 | The Lie Derivative | 114 |
| 3.3 | Calculus on Manifolds | 116 |
| 3.3.1 | Wedge Product: p-Forms and p-Vectors | 116 |
| 3.3.2 | Exterior Derivative | 120 |
| 3.3.3 | Stokes’ Theorem andi ts Generalizations | 123 |
| 3.3.4 | Closed and Exact Forms | 128 |
| 3.4 | Metric Tensor and Distance | 130 |
| 3.4.1 | Metric Tensor of a Linear Vector Space | 130 |
| 3.4.2 | Raising and Lowering Indices | 131 |
| 3.4.3 | Metric Tensor of a Manifold | 132 |
| 3.4.4 | Metric Tensor and Volume | 133 |
| 3.4.5 | The Laplacian Operator | 134 |
| 3.4.6 | Geodesic Curves on a Manifold | 135 |
| 3.5 | Dynamical Systems and Vector Fields | 139 |
| 3.5.1 | What is a Dynamical System? | 139 |
| 3.5.2 | A Model from Ecology | 140 |
| 3.5.3 | Lagrangian and Hamiltonian Systems | 142 |
| 3.6 | Fluid Mechanics | 148 |
| A | Calculus of Variations | 152 |
| B | Thermodynamics | 153 |
| | Bibliography and Notes | 158 |
| | Problems | 159 |
| 4 | Functions of a Complex Variable | 167 |
| 4.1 | Elementary Properties of Analytic Functions | 169 |
| 4.1.1 | Cauchy–Riemann Conditions | 169 |
| 4.1.2 | Conformal Mappings | 171 |
| 4.2 | Integration in the Complex Plane | 176 |
| 4.2.1 | Integration Along a Contour | 176 |
| 4.2.2 | Cauchy’s Theorem | 177 |
| 4.2.3 | Cauchy’s Integral Formula | 178 |
| 4.3 | Analytic Functions | 179 |
| 4.3.1 | Analytic Continuation | 179 |
| 4.3.2 | Singularities of an Analytic Function | 182 |
| 4.3.3 | Global Properties of Analytic Functions | 184 |
| 4.3.4 | Laurent Series | 186 |
| 4.3.5 | Infinite Product Representations | 188 |
| 4.4 | Calculus of Residues: Applications | 190 |
| 4.4.1 | Cauchy Residue Theorem | 190 |
| 4.4.2 | Evaluationof Real Integrals | 191 |
| 4.5 | Periodic Functions; Fourier Series | 195 |
| 4.5.1 | Periodic Functions | 195 |
| 4.5.2 | Doubly Periodic Functions | 197 |
| A | Gamma Function; Beta Function | 199 |
| A.1 | Gamma Function | 199 |
| A.2 | Beta Function | 203 |
| | Bibliography and Notes | 204 |
| | Problems | 205 |
| 5 | Differential Equations: Analytical Methods | 211 |
| 5.1 | Systems of Differential Equations | 213 |
| 5.1.1 | General Systems of First-Order Equations | 213 |
| 5.1.2 | Special Systems of Equations | 215 |
| 5.2 | First-Order Differential Equations | 216 |
| 5.2.1 | Linear First-Order Equations | 216 |
| 5.2.2 | Ricatti Equation | 218 |
| 5.2.3 | Exact Differentials | 220 |
| 5.3 | Linear Differential Equations | 221 |
| 5.3.1 | nth Order Linear Equations | 221 |
| 5.3.2 | Power Series Solutions | 222 |
| 5.3.3 | Linear Independence; General Solution | 223 |
| 5.3.4 | Linear Equation with Constant Coefficients | 225 |
| 5.4 | Linear Second-Order Equations | 226 |
| 5.4.1 | Classification of Singular Points | 226 |
| 5.4.2 | Exponents at a Regular Singular Point | 226 |
| 5.4.3 | One Regular Singular Point | 229 |
| 5.4.4 | Two Regular Singular Points | 229 |
| 5.5 | Legendre’s Equation | 231 |
| 5.5.1 | Legendre Polynomials | 231 |
| 5.5.2 | Legendre Functions of the Second Kind | 235 |
| 5.6 | Bessel’s Equation | 237 |
| 5.6.1 | Bessel Functions | 237 |
| 5.6.2 | Hankel Functions | 239 |
| 5.6.3 | Spherical Bessel Functions | 240 |
| A | Hypergeometric Equation | 241 |
| A.1 | Reduction to Standard Form | 241 |
| A.2 | Power Series Solutions | 242 |
| A.3 | Integral Representations | 244 |
| B | Confluent Hypergeometric Equation | 246 |
| B.1 | Reduction to Standard Form | 246 |
| B.2 | Integral Representations | 247 |
| C | Elliptic Integrals and Elliptic Functions | 249 |
| | Bibliography and Notes | 254 |
| | Problems | 255 |
| 6 | Hilbert Spaces | 261 |
| 6.1 | Infinite-Dimensional Vector Spaces | 264 |
| 6.1.1 | Hilbert Space Axioms | 264 |
| 6.1.2 | Convergence in Hilbert Space | 267 |
| 6.2 | Function Spaces; Measure Theory | 268 |
| 6.2.1 | Polynomial Approximation; Weierstrass Approximation Theorem | 268 |
| 6.2.2 | Convergence in the Mean | 270 |
| 6.2.3 | Measure Theory | 271 |
| 6.3 | Fourier Series | 273 |
| 6.3.1 | Periodic Functions and Trigonometric Polynomials | 273 |
| 6.3.2 | Classical Fourier Series | 274 |
| 6.3.3 | Convergence of Fourier Series | 275 |
| 6.3.4 | Fourier Cosine Series; Fourier Sine Series | 279 |
| 6.4 | Fourier Integral; Integral Transforms | 281 |
| 6.4.1 | Fourier Transform | 281 |
| 6.4.2 | Convolution Theorem; Correlation Functions | 284 |
| 6.4.3 | Laplace Transform | 286 |
| 6.4.4 | Multidimensional Fourier Transform | 287 |
| 6.4.5 | Fourier Transformin Quantum Mechanics | 288 |
| 6.5 | Orthogonal Polynomials | 289 |
| 6.5.1 | Weight Functions and Orthogonal Polynomials | 289 |
| 6.5.2 | Legendre Polynomials and Associated Legendre Functions | 290 |
| 6.5.3 | Spherical Harmonics | 292 |
| 6.6 | Haar Functions; Wavelets | 294 |
| A | Standard Families of Orthogonal Polynomials | 305 |
| | Bibliography and Notes | 310 |
| | Problems | 311 |
| 7 | Linear Operators on Hilbert Space | 319 |
| 7.1 | Some Hilbert Space Subtleties | 321 |
| 7.2 | General Properties of Linear Operators on Hilbert Space | 324 |
| 7.2.1 | Bounded, Continuous, and Closed Operators | 324 |
| 7.2.2 | Inverse Operator | 325 |
| 7.2.3 | Compact Operators; Hilbert–Schmidt Operators | 326 |
| 7.2.4 | Adjoint Operator | 327 |
| 7.2.5 | Unitary Operators; Isometric Operators | 329 |
| 7.2.6 | Convergence of Sequences of Operators in  | 329 |
| 7.3 | Spectrum of Linear Operators on Hilbert Space | 330 |
| 7.3.1 | Spectrum of a Compact Self-Adjoint Operator | 330 |
| 7.3.2 | Spectrum of Noncompact Normal Operators | 331 |
| 7.3.3 | Resolution of the Identity | 332 |
| 7.3.4 | Functions of a Self-Adjoint Operator | 335 |
| 7.4 | Linear Differential Operators | 336 |
| 7.4.1 | Differential Operators and Boundary Conditions | 336 |
| 7.4.2 | Second-Order Linear Differential Operators | 338 |
| 7.5 | Linear Integral Operators; Green Functions | 339 |
| 7.5.1 | Compact Integral Operators | 339 |
| 7.5.2 | Differential Operators and Green Functions | 341 |
| | Bibliography and Notes | 344 |
| | Problems | 345 |
| 8 | Partial Differential Equations | 353 |
| 8.1 | Linear First-Order Equations | 356 |
| 8.2 | The Laplacian and Linear Second-Order Equations | 359 |
| 8.2.1 | Laplacian and Boundary Conditions | 359 |
| 8.2.2 | Green Functions for Laplace’s Equation | 360 |
| 8.2.3 | Spectrum of the Laplacian | 363 |
| 8.3 | Time-Dependent Partial Differential Equations | 366 |
| 8.3.1 | The Diffusion Equation | 367 |
| 8.3.2 | Inhomogeneous Wave Equation: Advanced and Retarded Green Functions | 369 |
| 8.3.3 | The Schrödinger Equation | 373 |
| 8.4 | Nonlinear Partial Differential Equations | 376 |
| 8.4.1 | Quasilinear First-Order Equations | 376 |
| 8.4.2 | KdV Equation | 378 |
| 8.4.3 | Scalar Field in 1 + 1 Dimensions | 380 |
| 8.4.4 | Sine-Gordon Equation | 383 |
| A | Lagrangian Field Theory | 384 |
| | Bibliography and Notes | 386 |
| | Problems | 387 |
| 9 | Finite Groups | 391 |
| 9.1 | General Properties of Groups | 393 |
| 9.1.1 | Group Axioms | 393 |
| 9.1.2 | Cosets and Classes | 395 |
| 9.1.3 | Algebras; Group Algebra | 397 |
| 9.2 | Some Finite Groups | 399 |
| 9.2.1 | Cyclic Groups | 399 |
| 9.2.2 | Dihedral Groups | 399 |
| 9.2.3 | Tetrahedral Group | 400 |
| 9.3 | The Symmetric Group SN | 401 |
| 9.3.1 | Permutations and the Symmetric Group SN | 401 |
| 9.3.2 | Permutations and Partitions | 404 |
| 9.4 | Group Representations | 406 |
| 9.4.1 | Group Representations by Linear Operators | 406 |
| 9.4.2 | Schur’s Lemmas and Orthogonality Relations | 410 |
| 9.4.3 | Kronecker Product of Representations | 417 |
| 9.4.4 | Permutation Representations | 418 |
| 9.4.5 | Representations of Groups and Subgroups | 422 |
| 9.5 | Representations of the Symmetric Group SN | 424 |
| 9.5.1 | Irreducible Representations of SN | 424 |
| 9.5.2 | Outer Products of Representations of Sm  Sn | 426 |
| 9.5.3 | Kronecker Products of Irreducible Representations of SN | 428 |
| 9.6 | Discrete Infinite Groups | 431 |
| A | Frobenius Reciprocity Theorem | 435 |
| B | S-Functions and Irreducible Representations of SN | 437 |
| B.1 | Frobenius Generating Function for the Simple Characters of SN | 437 |
| B.2 | Graphical Calculation of the Characters  (m) | 442 |
| B.3 | Outer Products of Representations of Sm Sn | 446 |
| | Bibliography and Notes | 451 |
| | Problems | 451 |
| 10 | Lie Groups and Lie Algebras | 457 |
| 10.1 | Lie Groups | 460 |
| 10.2 | Lie Algebras | 461 |
| 10.2.1 | The Generators of a Lie Group | 461 |
| 10.2.2 | The Lie Algebra of a Lie Group | 462 |
| 10.2.3 | Classification of Lie Algebras | 465 |
| 10.3 | Representations of Lie Algebras | 469 |
| 10.3.1 | Irreducible Representations of SU(2) | 469 |
| 10.3.2 | Addition of Angular Momenta | 471 |
| 10.3.3 | SN and the Irreducible Representations of SU (2) | 474 |
| 10.3.4 | Irreducible Representations of SU (3) | 476 |
| A | Tensor Representations of the Classical Lie Groups | 482 |
| A.1 | The Classical Lie Groups | 482 |
| A.2 | Tensor Representations of U (n) and SU (n) | 483 |
| A.3 | Irreducible Representations of SO (n) | 487 |
| B | Lorentz Group; Poincaré Group | 489 |
| B.1 | Lorentz Transformations | 489 |
| B.2 | SL (2, C) and the Homogeneous Lorentz Group | 493 |
| B.3 | Inhomogeneous Lorentz Transformations; Poincaré Group | 496 |
| | Bibliography and Notes | 498 |
| | Problems | 499 |
| | Index | 507 |
