Fundamental Math and Physics for Scientists and Engineers

David Yevick, P. Eng. (Ontario) is Professor of Physics at the University of Waterloo, Canada. He received his A.B. and Ph.D. degrees respectively from Harvard University in Physics (1973) and Princeton University in Particle Physics (1977). Dr. Yevick is a leading scientist in the numerical simulation of optical communication systems, in particular electric field propagation in guided-wave optics, optical processes in semiconductors, and communication system modeling. Dr. Yevick is a fellow of the APS, OSA, and IEEE. Hannah Yevick holds a Ph.D. in Biological Physics from the Curie Institute, France, as well as a M.A. from Columbia University, and a B.A. from the University of Pennsylvania in Physics. Her experience with the Physics GRE and graduate comprehensive exams has enhanced the text.

1 Introduction 1

2 Problem Solving 3

2.1 Analysis 3

2.2 Test-Taking Techniques 4

2.2.1 Dimensional Analysis 5

3 Scientific Programming 6

3.1 Language Fundamentals 6

3.1.1 Octave Programming 7

4 Elementary Mathematics 12

4.1 Algebra 12

4.1.1 Equation Manipulation 12

4.1.2 Linear Equation Systems 13

4.1.3 Factoring 14

4.1.4 Inequalities 15

4.1.5 Sum Formulas 16

4.1.6 Binomial Theorem 17

4.2 Geometry 17

4.2.1 Angles 18

4.2.2 Triangles 18

4.2.3 Right Triangles 19

4.2.4 Polygons 20

4.2.5 Circles 20

4.3 Exponential, Logarithmic Functions, and Trigonometry 21

4.3.1 Exponential Functions 21

4.3.2 Inverse Functions and Logarithms 21

4.3.3 Hyperbolic Functions 22

4.3.4 Complex Numbers and Harmonic Functions 23

4.3.5 Inverse Harmonic and Hyperbolic Functions 25

4.3.6 Trigonometric Identities 26

4.4 Analytic Geometry 28

4.4.1 Lines and Planes 28

4.4.2 Conic Sections 29

4.4.3 Areas, Volumes, and Solid Angles 31

5 Vectors and Matrices 32

5.1 Matrices and Matrix Products 32

5.2 Equation Systems 34

5.3 Traces and Determinants 35

5.4 Vectors and Inner Products 38

5.5 Cross and Outer Products 40

5.6 Vector Identities 41

5.7 Rotations and Orthogonal Matrices 42

5.8 Groups and Matrix Generators 43

5.9 Eigenvalues and Eigenvectors 45

5.10 Similarity Transformations 48

6 Calculus of a Single Variable 50

6.1 Derivatives 50

6.2 Integrals 54

6.3 Series 60

7 Calculus of Several Variables 62

7.1 Partial Derivatives 62

7.2 Multidimensional Taylor Series and Extrema 66

7.3 Multiple Integration 67

7.4 Volumes and Surfaces of Revolution 69

7.5 Change of Variables and Jacobians 70

8 Calculus of Vector Functions 72

8.1 Generalized Coordinates 72

8.2 Vector Differential Operators 77

8.3 Vector Differential Identities 81

8.4 Gauss’s and Stokes’ Laws and Green’s Identities 82

8.5 Lagrange Multipliers 83

9 Probability Theory and Statistics 85

9.1 Random Variables, Probability Density, and Distributions 85

9.2 Mean, Variance, and Standard Deviation 86

9.3 Variable Transformations 86

9.4 Moments and Moment-Generating Function 86

9.5 Multivariate Probability Distributions, Covariance, and Correlation 87

9.6 Gaussian, Binomial, and Poisson Distributions 87

9.7 Least Squares Regression 91

9.8 Error Propagation 92

9.9 Numerical Models 93

10 Complex Analysis 94

10.1 Functions of a Complex Variable 94

10.2 Derivatives, Analyticity, and the Cauchy–Riemann Relations 95

10.3 Conformal Mapping 96

10.4 Cauchy’s Theorem and Taylor and Laurent Series 97

10.5 Residue Theorem 101

10.6 Dispersion Relations 105

10.7 Method of Steepest Decent 106

11 Differential Equations 108

11.1 Linearity, Superposition, and Initial and Boundary Values 108

11.2 Numerical Solutions 109

11.3 First-Order Differential Equations 112

11.4 Wronskian 114

11.5 Factorization 115

11.6 Method of Undetermined Coefficients 115

11.7 Variation of Parameters 116

11.8 Reduction of Order 118

11.9 Series Solution and Method of Frobenius 118

11.10 Systems of Equations, Eigenvalues, and Eigenvectors 119

12 Transform Theory 122

12.1 Eigenfunctions and Eigenvectors 122

12.2 Sturm–Liouville Theory 123

12.3 Fourier Series 125

12.4 Fourier Transforms 127

12.5 Delta Functions 128

12.6 Green’s Functions 131

12.7 Laplace Transforms 135

12.8 z-Transforms 137

13 Partial Differential Equations and Special Functions 138

13.1 Separation of Variables and Rectangular Coordinates 138

13.2 Legendre Polynomials 145

13.3 Spherical Harmonics 150

13.4 Bessel Functions 156

13.5 Spherical Bessel Functions 162

14 Integral Equations and the Calculus of Variations 166

14.1 Volterra and Fredholm Equations 166

14.2 Calculus of Variations the Euler-Lagrange Equation 168

15 Particle Mechanics 170

15.1 Newton’s Laws 170

15.2 Forces 171

15.3 Numerical Methods 173

15.4 Work and Energy 174

15.5 Lagrange Equations 176

15.6 Three-Dimensional Particle Motion 180

15.7 Impulse 181

15.8 Oscillatory Motion 181

15.9 Rotational Motion About a Fixed Axis 185

15.10 Torque and Angular Momentum 187

15.11 Motion in Accelerating Reference Systems 188

15.12 Gravitational Forces and Fields 189

15.13 Celestial Mechanics 191

15.14 Dynamics of Systems of Particles 193

15.15 Two-Particle Collisions and Scattering 197

15.16 Mechanics of Rigid Bodies 199

15.17 Hamilton’s Equation and Kinematics 206

16 Fluid Mechanics 210

16.1 Continuity Equation 210

16.2 Euler’s Equation 212

16.3 Bernoulli’s Equation 213

17 Special Relativity 215

17.1 Four-Vectors and Lorentz Transformation 215

17.2 Length Contraction, Time Dilation, and Simultaneity 217

17.3 Covariant Notation 219

17.4 Casuality and Minkowski Diagrams 221

17.5 Velocity Addition and Doppler Shift 222

17.6 Energy and Momentum 223

18 Electromagnetism 227

18.1 Maxwell’s Equations 227

18.2 Gauss’s Law 233

18.3 Electric Potential 235

18.4 Current and Resistivity 238

18.5 Dipoles and Polarization 241

18.6 Boundary Conditions and Green’s Functions 244

18.7 Multipole Expansion 248

18.8 Relativistic Formulation of Electromagnetism, Gauge Transformations, and Magnetic Fields 249

18.9 Magnetostatics 256

18.10 Magnetic Dipoles 259

18.11 Magnetization 260

18.12 Induction and Faraday’s Law 262

18.13 Circuit Theory and Kirchoff’s Laws 266

18.14 Conservation Laws and the Stress Tensor 270

18.15 Lienard–Wiechert Potentials 274

18.16 Radiation from Moving Charges 275

19 Wave Motion 282

19.1 Wave Equation 282

19.2 Propagation of Waves 284

19.3 Planar Electromagnetic Waves 286

19.4 Polarization 287

19.5 Superposition and Interference 288

19.6 Multipole Expansion for Radiating Fields 292

19.7 Phase and Group Velocity 295

19.8 Minimum Time Principle and Ray Optics 296

19.9 Refraction and Snell’s Law 297

19.10 Lenses 299

19.11 Mechanical Reflection 301

19.12 Doppler Effect and Shock Waves 302

19.13 Waves in Periodic Media 303

19.14 Conducting Media 304

19.15 Dielectric Media 306

19.16 Reflection and Transmission 307

19.17 Diffraction 311

19.18 Waveguides and Cavities 313

20 Quantum Mechanics 318

20.1 Fundamental Principles 318

20.2 Particle–Wave Duality 319

20.3 Interference of Quantum Waves 320

20.4 Schrödinger Equation 321

20.5 Particle Flux and Reflection 322

20.6 Wave Packet Propagation 324

20.7 Numerical Solutions 326

20.8 Quantum Mechanical Operators 328

20.9 Heisenberg Uncertainty Relation 331

20.10 Hilbert Space Representation 334

20.11 Square Well and Delta Function Potentials 336

20.12 WKB Method 339

20.13 Harmonic Oscillators 342

20.14 Heisenberg Representation 343

20.15 Translation Operators 344

20.16 Perturbation Theory 345

20.17 Adiabatic Theorem 351

21 Atomic Physics 353

21.1 Properties of Fermions 353

21.2 Bohr Model 354

21.3 Atomic Spectra and X-Rays 356

21.4 Atomic Units 356

21.5 Angular Momentum 357

21.6 Spin 358

21.7 Interaction of Spins 359

21.8 Hydrogenic Atoms 360

21.9 Atomic Structure 362

21.10 Spin–Orbit Coupling 362

21.11 Atoms in Static Electric and Magnetic Fields 364

21.12 Helium Atom and the H+2 Molecule 368

21.13 Interaction of Atoms with Radiation 371

21.14 Selection Rules 373

21.15 Scattering Theory 374

22 Nuclear and Particle Physics 379

22.1 Nuclear Properties 379

22.2 Radioactive Decay 381

22.3 Nuclear Reactions 382

22.4 Fission and Fusion 383

22.5 Fundamental Properties of Elementary Particles 383

23 Thermodynamics and Statistical Mechanics 386

23.1 Entropy 386

23.2 Ensembles 388

23.3 Statistics 391

23.4 Partition Functions 393

23.5 Density of States 396

23.6 Temperature and Energy 397

23.7 Phonons and Photons 400

23.8 The Laws of Thermodynamics 401

23.9 The Legendre Transformation and Thermodynamic Quantities 403

23.10 Expansion of Gases 407

23.11 Heat Engines and the Carnot Cycle 409

23.12 Thermodynamic Fluctuations 410

23.13 Phase Transformations 412

23.14 The Chemical Potential and Chemical Reactions 413

23.15 The Fermi Gas 414

23.16 Bose–Einstein Condensation 416

23.17 Physical Kinetics and Transport Theory 417

24 Condensed Matter Physics 422

24.1 Crystal Structure 422

24.2 X-Ray Diffraction 423

24.3 Thermal Properties 424

24.4 Electron Theory of Metals 425

24.5 Superconductors 426

24.6 Semiconductors 427

25 Laboratory Methods 430

25.1 Interaction of Particles with Matter 430

25.2 Radiation Detection and Counting Statistics 431

25.3 Lasers 432

Index 434