Mathematical Methods: For Students of Physics and Related Fields

Description

Intended to follow the usual introductory physics courses, this book has the unique feature of addressing the mathematical needs of sophomores and juniors in physics, engineering and other related fields. Many original, lucid, and relevant examples from the physical sciences, problems at the ends of chapters, and boxes to emphasize important concepts help guide the student through the material. Beginning with reviews of vector algebra and differential and integral calculus, the book continues with infinite series, vector analysis, complex algebra and analysis, ordinary and partial differential equations. Discussions of numerical analysis, nonlinear dynamics and chaos, and the Dirac delta function provide an introduction to modern topics in mathematical physics. This new edition has been made more user-friendly through organization into convenient, shorter chapters. Also, it includes an entirely new section on Probability and plenty of new material on tensors and integral transforms. Cover......Page 1 Preface to Second Edition......Page 6 Preface......Page 8 Note to the Reader......Page 12 Contents......Page 13 Part I Coordinates and Calculus......Page 22 1.1 Vectors in a Plane and in Space......Page 23 1.1.1 Dot Product......Page 25 1.1.2 Vector or Cross Product......Page 27 1.2 Coordinate Systems......Page 31 1.3 Vectors in Different Coordinate Systems......Page 36 1.3.1 Fields and Potentials......Page 41 1.3.2 Cross Product......Page 48 1.4 Relations Among Unit Vectors......Page 51 1.5 Problems......Page 57 2 Differentiation......Page 63 2.1 The Derivative......Page 64 2.2.1 Definition, Notation, and Basic Properties......Page 67 2.2.2 Differentials......Page 73 2.2.3 Chain Rule......Page 75 2.2.4 Homogeneous Functions......Page 77 2.3 Elements of Length, Area, and Volume......Page 79 2.3.1 Elements in a Cartesian Coordinate System......Page 80 2.3.2 Elements in a Spherical Coordinate System......Page 82 2.3.3 Elements in a Cylindrical Coordinate System......Page 85 2.4 Problems......Page 88 3.1 “∫” Means “∫um”......Page 96 3.2 Properties of Integral......Page 100 3.2.4 Partition of Range of Integration......Page 101 3.2.6 Small Region of Integration......Page 102 3.2.8 Symmetric Range of Integration......Page 103 3.2.9 Differentiating an Integral......Page 104 3.2.10 Fundamental Theorem of Calculus......Page 106 3.3 Guidelines for Calculating Integrals......Page 110 3.3.1 Reduction to Single Integrals......Page 111 3.3.2 Components of Integrals of Vector Functions......Page 114 3.4 Problems......Page 117 4.1.1 An Example from Mechanics......Page 120 4.1.2 Examples from Electrostatics and Gravity......Page 123 4.1.3 Examples from Magnetostatics......Page 128 4.2.1 Cartesian Coordinates......Page 134 4.2.2 Cylindrical Coordinates......Page 137 4.2.3 Spherical Coordinates......Page 139 4.3 Applications: Triple Integrals......Page 141 4.4 Problems......Page 147 5.1 One-Variable Case......Page 157 5.1.1 Linear Densities of Points......Page 161 5.1.2 Properties of the Delta Function......Page 163 5.1.3 The Step Function......Page 170 5.2 Two-Variable Case......Page 172 5.3 Three-Variable Case......Page 177 5.4 Problems......Page 184 Part II Algebra of Vectors......Page 189 6 Planar and Spatial Vectors......Page 190 6.1 Vectors in a Plane Revisited......Page 191 6.1.1 Transformation of Components......Page 193 6.1.2 Inner Product......Page 199 6.1.3 Orthogonal Transformation......Page 207 6.2 Vectors in Space......Page 209 6.2.1 Transformation of Vectors......Page 211 6.2.2 Inner Product......Page 215 6.3 Determinant......Page 219 6.4 The Jacobian......Page 224 6.5 Problems......Page 228 7 Finite-Dimensional Vector Spaces......Page 232 7.1 Linear Transformations......Page 233 7.2 Inner Product......Page 235 7.3 The Determinant......Page 239 7.4 Eigenvectors and Eigenvalues......Page 241 7.5 Orthogonal Polynomials......Page 244 7.6 Systems of Linear Equations......Page 247 7.7 Problems......Page 251 8 Vectors in Relativity......Page 254 8.1 Proper and Coordinate Time......Page 256 8.2 Spacetime Distance......Page 257 8.3 Lorentz Transformation......Page 260 8.4 Four-Velocity and Four-Momentum......Page 264 8.4.1 Relativistic Collisions......Page 267 8.4.2 Second Law of Motion......Page 270 8.5 Problems......Page 271 Part III Infinite Series......Page 273 9.1 Infinite Sequences......Page 274 9.2 Summations......Page 277 9.2.1 Mathematical Induction......Page 280 9.3 Infinite Series......Page 281 9.3.1 Tests for Convergence......Page 282 9.3.2 Operations on Series......Page 288 9.4 Sequences and Series of Functions......Page 289 9.4.1 Properties of Uniformly Convergent Series......Page 292 9.5 Problems......Page 294 10.1 Power Series......Page 297 10.1.1 Taylor Series......Page 300 10.2 Series for Some Familiar Functions......Page 301 10.3 Helmholtz Coil......Page 305 10.4 Indeterminate Forms and L'Hôpital's Rule......Page 308 10.5 Multipole Expansion......Page 311 10.6 Fourier Series......Page 313 10.7 Multivariable Taylor Series......Page 319 10.8 Application to Differential Equations......Page 321 10.9 Problems......Page 325 11.1 Integrals as Functions......Page 331 11.1.1 Gamma Function......Page 332 11.1.2 The Beta Function......Page 334 11.1.4 Elliptic Functions......Page 336 11.2 Power Series as Functions......Page 341 11.2.1 Hypergeometric Functions......Page 342 11.2.2 Confluent Hypergeometric Functions......Page 346 11.2.3 Bessel Functions......Page 347 11.3 Problems......Page 350 Part IV Analysis of Vectors......Page 354 12 Vectors and Derivatives......Page 355 12.1.1 Ordinary Angle Revisited......Page 356 12.1.2 Solid Angle......Page 359 12.2 Time Derivative of Vectors......Page 362 12.2.1 Equations of Motion in a Central Force Field......Page 364 12.3 The Gradient......Page 367 12.3.1 Gradient and Extremum Problems......Page 371 12.4 Problems......Page 374 13.1 Flux of a Vector Field......Page 376 13.1.1 Flux Through an Arbitrary Surface......Page 381 13.2.1 Flux Density......Page 382 13.2.2 Divergence Theorem......Page 385 13.2.3 Continuity Equation......Page 389 13.3 Problems......Page 394 14.1 The Line Integral......Page 397 14.2 Curl of a Vector Field and Stokes' Theorem......Page 401 14.3 Conservative Vector Fields......Page 408 14.4 Problems......Page 414 15.1 Double Del Operations......Page 417 15.2 Magnetic Multipoles......Page 419 15.3 Laplacian......Page 421 15.3.1 A Primer of Fluid Dynamics......Page 423 15.4 Maxwell's Equations......Page 425 15.4.1 Maxwell's Contribution......Page 426 15.4.2 Electromagnetic Waves in Empty Space......Page 427 15.5 Problems......Page 430 16.1 Elements of Length......Page 432 16.2 The Gradient......Page 434 16.3 The Divergence......Page 436 16.4 The Curl......Page 440 16.4.1 The Laplacian......Page 444 16.5 Problems......Page 445 17.1 Vectors and Indices......Page 448 17.1.1 Transformation Properties of Vectors......Page 450 17.1.2 Covariant and Contravariant Vectors......Page 454 17.2 From Vectors to Tensors......Page 456 17.2.1 Algebraic Properties of Tensors......Page 459 17.2.2 Numerical Tensors......Page 461 17.3 Metric Tensor......Page 463 17.3.1 Index Raising and Lowering......Page 466 17.3.2 Tensors and Electrodynamics......Page 468 17.4.1 Covariant Differential and Affine Connection......Page 471 17.4.2 Covariant Derivative......Page 473 17.4.3 Metric Connection......Page 474 17.5 Riemann Curvature Tensor......Page 477 17.6 Problems......Page 480 Part V Complex Analysis......Page 484 18.1 Cartesian Form of Complex Numbers......Page 485 18.2 Polar Form of Complex Numbers......Page 490 18.3 Fourier Series Revisited......Page 496 18.4 A Representation of Delta Function......Page 499 18.5 Problems......Page 501 19.1 Complex Functions......Page 505 19.1.1 Derivatives of Complex Functions......Page 507 19.1.2 Integration of Complex Functions......Page 511 19.1.3 Cauchy Integral Formula......Page 516 19.1.4 Derivatives as Integrals......Page 517 19.2 Problems......Page 519 20 Complex Series......Page 523 20.1 Power Series......Page 524 20.2 Taylor and Laurent Series......Page 526 20.3 Problems......Page 530 21.1 The Residue......Page 532 21.2 Integrals of Rational Functions......Page 536 21.3 Products of Rational and Trigonometric Functions......Page 539 21.4 Functions of Trigonometric Functions......Page 541 21.5 Problems......Page 543 Part VI Differential Equations......Page 545 22 From PDEs to ODEs......Page 546 22.1 Separation of Variables......Page 547 22.2 Separation in Cartesian Coordinates......Page 549 22.3 Separation in Cylindrical Coordinates......Page 552 22.4 Separation in Spherical Coordinates......Page 553 22.5 Problems......Page 555 23.1 Normal Form of a FODE......Page 556 23.2 Integrating Factors......Page 558 23.3 First-Order Linear Differential Equations......Page 561 23.4 Problems......Page 566 24 Second-Order Linear Differential Equations......Page 568 24.1 Linearity, Superposition, and Uniqueness......Page 569 24.2 The Wronskian......Page 571 24.3 A Second Solution to the HSOLDE......Page 572 24.4 The General Solution to an ISOLDE......Page 574 24.5 Sturm Liouville Theory......Page 575 24.5.1 Adjoint Differential Operators......Page 576 24.5.2 Sturm Liouville System......Page 579 24.6 SOLDEs with Constant Coefficients......Page 580 24.6.1 The Homogeneous Case......Page 581 24.6.2 Central Force Problem......Page 584 24.6.3 The Inhomogeneous Case......Page 588 24.7 Problems......Page 592 25 Laplace's Equation: Cartesian Coordinates......Page 596 25.1 Uniqueness of Solutions......Page 597 25.2 Cartesian Coordinates......Page 599 25.3 Problems......Page 608 26 Laplace's Equation: Spherical Coordinates......Page 611 26.1 Frobenius Method......Page 612 26.2 Legendre Polynomials......Page 614 26.3 Second Solution of the Legendre DE......Page 621 26.4 Complete Solution......Page 623 26.5.2 Recurrence Relation......Page 626 26.5.3 Orthogonality......Page 628 26.5.4 Rodrigues Formula......Page 630 26.6 Expansions in Legendre Polynomials......Page 632 26.7 Physical Examples......Page 635 26.8 Problems......Page 639 27.1 The ODEs......Page 642 27.2 Solutions of the Bessel DE......Page 645 27.3 Second Solution of the Bessel DE......Page 648 27.4.2 Recurrence Relations......Page 649 27.4.3 Orthogonality......Page 650 27.4.4 Generating Function......Page 652 27.5 Expansions in Bessel Functions......Page 656 27.6 Physical Examples......Page 657 27.7 Problems......Page 660 28.1 The Heat Equation......Page 663 28.1.1 Heat-Conducting Rod......Page 664 28.1.2 Heat Conduction in a Rectangular Plate......Page 665 28.1.3 Heat Conduction in a Circular Plate......Page 666 28.2 The Schrödinger Equation......Page 668 28.2.1 Quantum Harmonic Oscillator......Page 669 28.2.2 Quantum Particle in a Box......Page 677 28.2.3 Hydrogen Atom......Page 679 28.3 The Wave Equation......Page 682 28.3.1 Guided Waves......Page 684 28.3.2 Vibrating Membrane......Page 688 28.4 Problems......Page 689 Part VII Special Topics......Page 692 29.1 The Fourier Transform......Page 693 29.1.1 Properties of Fourier Transform......Page 696 29.1.2 Sine and Cosine Transforms......Page 697 29.1.3 Examples of Fourier Transform......Page 698 29.1.4 Application to Differential Equations......Page 702 29.2 Fourier Transform and Green's Functions......Page 705 29.2.1 Green's Function for the Laplacian......Page 708 29.2.2 Green's Function for the Heat Equation......Page 709 29.2.3 Green's Function for the Wave Equation......Page 711 29.3 The Laplace Transform......Page 712 29.3.1 Properties of Laplace Transform......Page 713 29.3.2 Derivative and Integral of the Laplace Transform......Page 717 29.3.3 Laplace Transform and Differential Equations......Page 718 29.3.4 Inverse of Laplace Transform......Page 721 29.4 Problems......Page 723 30 Calculus of Variations......Page 727 30.1 Variational Problem......Page 728 30.1.1 Euler-Lagrange Equation......Page 729 30.1.2 Beltrami identity......Page 731 30.1.4 Several Independent Variables......Page 734 30.1.5 Second Variation......Page 735 30.1.6 Variational Problems with Constraints......Page 738 30.2.1 From Newton to Lagrange......Page 740 30.2.2 Lagrangian Densities......Page 744 30.3 Hamiltonian Dynamics......Page 747 30.4 Problems......Page 750 31 Nonlinear Dynamics and Chaos......Page 752 31.1 Systems Obeying Iterated Maps......Page 753 31.1.1 Stable and Unstable Fixed Points......Page 754 31.1.2 Bifurcation......Page 756 31.1.3 Onset of Chaos......Page 760 31.2 Systems Obeying DEs......Page 762 31.2.1 The Phase Space......Page 763 31.2.2 Autonomous Systems......Page 765 31.2.3 Onset of Chaos......Page 769 31.3.1 Feigenbaum Numbers......Page 772 31.3.2 Fractal Dimension......Page 774 31.4 Problems......Page 777 32.1 Basic Concepts......Page 779 32.1.1 A Set Theory Primer......Page 780 32.1.2 Sample Space and Probability......Page 782 32.1.3 Conditional and Marginal Probabilities......Page 784 32.1.4 Average and Standard Deviation......Page 787 32.1.5 Counting: Permutations and Combinations......Page 789 32.2 Binomial Probability Distribution......Page 790 32.3 Poisson Distribution......Page 795 32.4 Continuous Random Variable......Page 799 32.4.1 Transformation of Variables......Page 802 32.4.2 Normal Distribution......Page 804 32.5 Problems......Page 807 Bibliography......Page 812 Index......Page 814