Sadri HASSANI - Mathematical Methods for Students of Physics and Related Fields, Second Edition


CONTENTS



I Coordinates and Calculus 1


 1 Coordinate Systems and Vectors 3

  1.1 Vectors in a Plane and in Space 3

   1.1.1 Dot Product 5

   1.1.2 Vector or Cross Product 7

  1.2 Coordinate Systems 11

  1.3 Vectors in Different Coordinate Systems 16

   1.3.1 Fields and Potentials 21

   1.3.2 Cross Product 28

  1.4 Relations Among Unit Vectors 31

  1.5 Problems 37


 2 Differentiation 43

  2.1 The Derivative 44

  2.2 Partial Derivatives 47

   2.2.1 Definition, Notation, and Basic Properties 47

   2.2.2 Differentials 53

   2.2.3 Chain Rule 55

   2.2.4 Homogeneous Functions 57

  2.3 Elements of Length, Area, and Volume 59

   2.3.1 Elements in a Cartesian Coordinate System 60

   2.3.2 Elements in a Spherical Coordinate System 62

   2.3.3 Elements in a Cylindrical Coordinate System 65

  2.4 Problems 68


 3 Integration: Formalism 77

  3.1 “∫” Means “∫um” 77

  3.2 Properties of Integral 81

   3.2.1 Change of Dummy Variable 82

   3.2.2 Linearity 82

   3.2.3 Interchange of Limits 82

   3.2.4 Partition of Range of Integration 82

   3.2.5 Transformation of Integration Variable 83

   3.2.6 Small Region of Integration 83

   3.2.7 Integral and Absolute Value 84

   3.2.8 Symmetric Range of Integration 84

   3.2.9 Differentiating an Integral 85

   3.2.10 Fundamental Theorem of Calculus 87

  3.3 Guidelines for Calculating Integrals 91

   3.3.1 Reduction to Single Integrals 92

   3.3.2 Components of Integrals of Vector Functions 95

  3.4 Problems 98


 4 Integration: Applications 101

  4.1 Single Integrals 101

   4.1.1 An Example from Mechanics 101

   4.1.2 Examples from Electrostatics and Gravity 104

   4.1.3 Examples from Magnetostatics 109

  4.2 Applications: Double Integrals 115

   4.2.1 Cartesian Coordinates 115

   4.2.2 Cylindrical Coordinates 118

   4.2.3 Spherical Coordinates 120

  4.3 Applications: Triple Integrals 122

  4.4 Problems 128


 5 Dirac Delta Function 139

  5.1 One-Variable Case 139

   5.1.1 Linear Densities of Points 143

   5.1.2 Properties of the Delta Function 145

   5.1.3 The Step Function 152

  5.2 Two-Variable Case 154

  5.3 Three-Variable Case 159

  5.4 Problems 166



II Algebra of Vectors 171


 6 Planar and Spatial Vectors 173

  6.1 Vectors in a Plane Revisited 174

   6.1.1 Transformation of Components 176

   6.1.2 Inner Product 182

   6.1.3 Orthogonal Transformation 190

  6.2 Vectors in Space 192

   6.2.1 Transformation of Vectors 194

   6.2.2 Inner Product 198

  6.3 Determinant 202

  6.4 The Jacobian 207

  6.5 Problems 211


 7 Finite-Dimensional Vector Spaces 215

  7.1 Linear Transformations 216

  7.2 Inner Product 218

  7.3 The Determinant 222

  7.4 Eigenvectors and Eigenvalues 224

  7.5 Orthogonal Polynomials 227

  7.6 Systems of Linear Equations 230

  7.7 Problems 234


 8 Vectors in Relativity 237

  8.1 Proper and Coordinate Time 239

  8.2 Spacetime Distance 240

  8.3 Lorentz Transformation 243

  8.4 Four-Velocity and Four-Momentum 247

   8.4.1 Relativistic Collisions 250

   8.4.2 Second Law of Motion 253

  8.5 Problems 254



III Infinite Series 257


 9 Infinite Series 259

  9.1 Infinite Sequences 259

  9.2 Summations 262

   9.2.1 Mathematical Induction 265

  9.3 Infinite Series 266

   9.3.1 Tests for Convergence 267

   9.3.2 Operations on Series 273

  9.4 Sequences and Series of Functions 274

   9.4.1 Properties of Uniformly Convergent Series 277

  9.5 Problems 279


 10 Application of Common Series 283

  10.1 Power Series 283

   10.1.1 Taylor Series 286

  10.2 Series for Some Familiar Functions 287

  10.3 Helmholtz Coil 291

  10.4 Indeterminate Forms and L’Hôpital’s Rule 294

  10.5 Multipole Expansion 297

  10.6 Fourier Series 299

  10.7 Multivariable Taylor Series 305

  10.8 Application to Differential Equations 307

  10.9 Problems 311


 11 Integrals and Series as Functions 317

  11.1 Integrals as Functions 317

   11.1.1 Gamma Function 318

   11.1.2 The Beta Function 320

   11.1.3 The Error Function 322

   11.1.4 Elliptic Functions 322

  11.2 Power Series as Functions 327

   11.2.1 Hypergeometric Functions 328

   11.2.2 Confluent Hypergeometric Functions 332

   11.2.3 Bessel Functions 333

  11.3 Problems 336



IV Analysis of Vectors 341


 12 Vectors and Derivatives 343

  12.1 Solid Angle 344

   12.1.1 Ordinary Angle Revisited 344

   12.1.2 Solid Angle 347

  12.2 Time Derivative of Vectors 350

   12.2.1 Equations of Motion in a Central Force Field 352

  12.3 The Gradient 355

   12.3.1 Gradient and Extremum Problems 359

  12.4 Problems 362


 13 Flux and Divergence 365

  13.1 Flux of a Vector Field 365

   13.1.1 Flux Through an Arbitrary Surface 370

  13.2 Flux Density = Divergence 371

   13.2.1 Flux Density 371

   13.2.2 Divergence Theorem 374

   13.2.3 Continuity Equation 378

  13.3 Problems 383


 14 Line Integral and Curl 387

  14.1 The Line Integral 387

  14.2 Curl of a Vector Field and Stokes’ Theorem 391

  14.3 Conservative Vector Fields 398

  14.4 Problems 404


 15 Applied Vector Analysis 407

  15.1 Double Del Operations 407

  15.2 Magnetic Multipoles 409

  15.3 Laplacian 411

   15.3.1 A Primer of Fluid Dynamics 413

   15.4 Maxwell’s Equations 415

   15.4.1 Maxwell’s Contribution 416

   15.4.2 Electromagnetic Waves in Empty Space 417

  15.5 Problems 420


 16 Curvilinear Vector Analysis 423

  16.1 Elements of Length 423

  16.2 The Gradient 425

  16.3 The Divergence 427

  16.4 The Curl 431

   16.4.1 The Laplacian 435

  16.5 Problems 436


 17 Tensor Analysis 439

  17.1 Vectors and Indices 439

   17.1.1 Transformation Properties of Vectors 441

   17.1.2 Covariant and Contravariant Vectors 445

  17.2 From Vectors to Tensors 447

   17.2.1 Algebraic Properties of Tensors 450

   17.2.2 Numerical Tensors 452

  17.3 Metric Tensor 454

   17.3.1 Index Raising and Lowering 457

   17.3.2 Tensors and Electrodynamics 459

  17.4 Differentiation of Tensors 462

   17.4.1 Covariant Differential and Affine Connection 462

   17.4.2 Covariant Derivative 464

   17.4.3 Metric Connection 465

  17.5 Riemann Curvature Tensor 468

  17.6 Problems 471



V Complex Analysis 475


 18 Complex Arithmetic 477

  18.1 Cartesian Form of Complex Numbers 477

  18.2 Polar Form of Complex Numbers 482

  18.3 Fourier Series Revisited 488

  18.4 A Representation of Delta Function 491

  18.5 Problems 493


 1９ Complex Derivative and Integral 497

  1９.1 Complex Functions 497

   1９.1.1 Derivatives of Complex Functions 499

   1９.1.2 Integration of Complex Functions 503

   19.1.3 Cauchy Integral Formula 508

   19.1.4 Derivatives as Integrals 509

  19.2 Problems 511


 20 Complex Series 515

  20.1 Power Series 516

  20.2 Taylor and Laurent Series 518

  20.3 Problems 522


 21 Calculus of Residues 525

  21.1 The Residue 525

  21.2 Integrals of Rational Functions 529

  21.3 Products of Rational and Trigonometric Functions 532

  21.4 Functions of Trigonometric Functions 534

  21.5 Problems 536



VI Differential Equations 539


 22 From PDEs to ODEs 541

  22.1 Separation of Variables 542

  22.2 Separation in Cartesian Coordinates 544

  22.3 Separation in Cylindrical Coordinates 547

  22.4 Separation in Spherical Coordinates 548

  22.5 Problems 550


 23 First-Order Differential Equations 551

  23.1 Normal Form of a FODE 551

  23.2 Integrating Factors 553

  23.3 First-Order Linear Differential Equations 556

  23.4 Problems 561


 24 Second-Order Linear Differential Equations 563

  24.1 Linearity, Superposition, and Uniqueness 564

  24.2 The Wronskian 566

  24.3 A Second Solution to the HSOLDE 567

  24.4 The General Solution to an ISOLDE 569

  24.5 Sturm–Liouville Theory 570

   24.5.1 Adjoint Differential Operators 571

   24.5.2 Sturm–Liouville System 574

  24.6 SOLDEs with Constant Coefficients 575

   24.6.1 The Homogeneous Case 576

   24.6.2 Central Force Problem 579

   24.6.3 The Inhomogeneous Case 583

  24.7 Problems 587


 25 Laplace’s Equation: Cartesian Coordinates 591

  25.1 Uniqueness of Solutions 592

  25.2 Cartesian Coordinates 594

  25.3 Problems 603


 26 Laplace’s Equation: Spherical Coordinates 607

  26.1 Frobenius Method 608

  26.2 Legendre Polynomials 610

  26.3 Second Solution of the Legendre DE 617

  26.4 Complete Solution 619

  26.5 Properties of Legendre Polynomials 622

   26.5.1 Parity 622

   26.5.2 Recurrence Relation 622

   26.5.3 Orthogonality 624

   26.5.4 Rodrigues Formula 626

  26.6 Expansions in Legendre Polynomials 628

  26.7 Physical Examples 631

  26.8 Problems 635


 27 Laplace’s Equation: Cylindrical Coordinates 639

  27.1 The ODEs 639

  27.2 Solutions of the Bessel DE 642

  27.3 Second Solution of the Bessel DE 645

  27.4 Properties of the Bessel Functions 646

   27.4.1 Negative Integer Order 646

   27.4.2 Recurrence Relations 646

   27.4.3 Orthogonality 647

   27.4.4 Generating Function 649

  27.5 Expansions in Bessel Functions 653

  27.6 Physical Examples 654

  27.7 Problems 657


 28 Other PDEs of Mathematical Physics 661

  28.1 The Heat Equation 661

   28.1.1 Heat-Conducting Rod 662

   28.1.2 Heat Conduction in a Rectangular Plate 663

   28.1.3 Heat Conduction in a Circular Plate 664

  28.2 The Schrödinger Equation 666

   28.2.1 Quantum Harmonic Oscillator 667

   28.2.2 Quantum Particle in a Box 675

   28.2.3 Hydrogen Atom 677

  28.3 The Wave Equation 680

   28.3.1 Guided Waves 682

   28.3.2 Vibrating Membrane 686

  28.4 Problems 687



VII Special Topics 691


 29 Integral Transforms 693

  29.1 The Fourier Transform 693

   29.1.1 Properties of Fourier Transform 696

   29.1.2 Sine and Cosine Transforms 697

   29.1.3 Examples of Fourier Transform 698

   29.1.4 Application to Differential Equations 702

  29.2 Fourier Transform and Green’s Functions 705

   29.2.1 Green’s Function for the Laplacian 708

   29.2.2 Green’s Function for the Heat Equation 709

   29.2.3 Green’s Function for the Wave Equation 711

  29.3 The Laplace Transform 712

   29.3.1 Properties of Laplace Transform 713

   29.3.2 Derivative and Integral of the Laplace Transform 717

   29.3.3 Laplace Transform and Differential Equations 718

   29.3.4 Inverse of Laplace Transform 721

  29.4 Problems 723


 30 Calculus of Variations 727

  30.1 Variational Problem 728

   30.1.1 Euler-Lagrange Equation 729

   30.1.2 Beltrami identity 731

   30.1.3 Several Dependent Variables 734

   30.1.4 Several Independent Variables 734

   30.1.5 Second Variation 735

   30.1.6 Variational Problems with Constraints 738

  30.2 Lagrangian Dynamics 740

   30.2.1 From Newton to Lagrange 740

   30.2.2 Lagrangian Densities 744

  30.3 Hamiltonian Dynamics 747

  30.4 Problems 750


 31 Nonlinear Dynamics and Chaos 753

  31.1 Systems Obeying Iterated Maps 754

   31.1.1 Stable and Unstable Fixed Points 755

   31.1.2 Bifurcation 757

   31.1.3 Onset of Chaos 761

  31.2 Systems Obeying DEs 763

   31.2.1 The Phase Space 764

   31.2.2 Autonomous Systems 766

   31.2.3 Onset of Chaos 770

  31.3 Universality of Chaos 773

   31.3.1 Feigenbaum Numbers 773

   31.3.2 Fractal Dimension 775

  31.4 Problems 778


 32 Probability Theory 781

  32.1 Basic Concepts 781

   32.1.1 A Set Theory Primer 782

   32.1.2 Sample Space and Probability 784

   32.1.3 Conditional and Marginal Probabilities 786

   32.1.4 Average and Standard Deviation 789

   32.1.5 Counting: Permutations and Combinations 791

  32.2 Binomial Probability Distribution 792

  32.3 Poisson Distribution 797

  32.4 Continuous Random Variable 801

   32.4.1 Transformation of Variables 804

   32.4.2 Normal Distribution 806

  32.5 Problems 809


Bibliography 815


Index 817


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