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 Gaussâs and Stokesâ Laws and Greenâs 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 Cauchyâ"Riemann Relations 95
10.3 Conformal Mapping 96
10.4 Cauchyâs 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 Sturmâ"Liouville Theory 123
12.3 Fourier Series 125
12.4 Fourier Transforms 127
12.5 Delta Functions 128
12.6 Greenâs 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 Newtonâs 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 Hamiltonâs Equation and Kinematics 206
16 Fluid Mechanics 210
16.1 Continuity Equation 210
16.2 Eulerâs Equation 212
16.3 Bernoulliâs 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 Maxwellâs Equations 227
18.2 Gaussâs Law 233
18.3 Electric Potential 235
18.4 Current and Resistivity 238
18.5 Dipoles and Polarization 241
18.6 Boundary Conditions and Greenâs 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 Faradayâs Law 262
18.13 Circuit Theory and Kirchoffâs Laws 266
18.14 Conservation Laws and the Stress Tensor 270
18.15 Lienardâ"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 Snellâs 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 Particleâ"Wave Duality 319
20.3 Interference of Quantum Waves 320
20.4 Schr¶dinger 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 Spinâ"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 Boseâ"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
