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Contents Preface xix 1 Useful formulae and relationships 1.1 Relationships for triangles 1 1.2 Trigonometric relationships 5 1.3 The binomial expansion (theorem) 6 1.4 The exponential e 13 1.5 Natural logarithms 15 1.6 Two-dimensional coordinate systems 17 PROBLEMS 18 2 Dimensions and dimensional analysis 2.1 Basic units and dimensions 20 2.2 Dimensional homogeneity 23 2.3 Dimensional analysis 24 2.4 Electrical and magnetic units 27 PROBLEMS 28 3 Sequences and series 3.1 Arithmetic series 29 3.2 Geometric series 30 3.3 Harmonic series 31 3.4 Tests for convergence 32 3.5 Power series 33 PROBLEMS 34 4 Differentiation 4.1 The basic idea of a derivative 35 4.2 Chain rule 37 4.3 Product rule 39 4.4 Quotient rule 41 4.5 Maxima, minima, and higher-order derivatives 42 4.6 Expressing ex as a power series in x 46 4.7 Taylor's theorem 48 PROBLEMS 51 5 Integration 5.1 Indefinite and definite integrals 53 5.2 Techniques of evaluating integrals 54 5.3 Substitution method 55 5.4 Partial fractions 58 5.5 Integration by parts 61 5.6 Integrating powers of cos x and sin x 63 5.7 The definite integral: area under the curve 65 PROBLEMS 69 6 Complex numbers 6.1 Definition of a complex number 70 6.2 Argand diagram 72 6.3 Ways of describing a complex number 73 6.4 De Moivre's theorem 74 6.5 Complex conjugate 78 6.6 Division and reduction to real-plus-imaginary form 79 6.7 Modulus-argument form as an aid to integration 80 6.8 Circuits with alternating currents and voltages 81 PROBLEMS 85 7 Ordinary differential equations 7.1 Types of ordinary differential equation 86 7.2 Separation of variables 88 7.3 Homogeneous equations 89 7.4 The integrating factor 91 7.5 Linear constant-coefficient equations 94 7.6 Simple harmonic motion 94 7.7 Damped simple harmonic motion 96 7.8 Forced vibrations 99 7.9 An LCR circuit 102 PROBLEMS 104 8 Matrices I and determinants 8.1 Definition of a matrix 106 8.2 Operations of matrix algebra 107 8.3 Types of matrix 108 8.4 Applications to lens systems 110 8.5 Application to special relativity 115 8.6 Determinants 117 8.7 Types of determinant 121 8.8 Inverse matrix 122 8.9 Linear equations 124 PROBLEMS 126 9 Vector algebra 9.1 Scalar and vector quantities 128 9.2 Products of vectors 130 9.3 Vector representations of some rotational quantities 134 9.4 Linear dependence and independence 135 9.5 A straight line in vector form 137 9.6 A plane in vector form 139 9.7 Distance of a point from a plane 141 9.8 Relationships between lines and planes 143 9.9 Differentiation of vectors 145 9.10 Motion under a central force 149 PROBLEMS 151 10 Conic sections and orbits 10.1 Kepler and Newton 152 10.2 Conic sections and the cone 153 10.3 The circle and the ellipse 154 10.4 The parabola 157 10.5 The hyperbola 158 10.6 The orbits of planets and Kepler's laws 160 10.7 The dynamics of orbits 163 10.8 Alpha-particle scattering 165 PROBLEMS 169 11 Partial differentiation 11.1 What is partial differentiation? 170 11.2 Higher partial derivatives 172 11.3 The total derivative 173 11.4 Partial differentiation and thermodynamics 176 11.5 Taylor series for a function of two variables 179 11.6 Maxima and minima in a multidimensional space 181 PROBLEMS 183 12 Probability and statistics 12.1 What is probability? 185 12.2 Combining probabilities 186 12.3 Making selections 187 12.4 The birthday problem 189 12.5 Bayes' theorem 190 12.6 Too much information? 191 12.7 Mean; variance and standard deviation; median 192 12.8 Combining different estimates 196 PROBLEMS 199 13 Coordinate systems and multiple integration 13.1 Two-dimensional coordinate systems 201 13.2 Integration in a rectangular Cartesian system 201 13.3 Integration with polar coordinates 205 13.4 Changing coordinate systems 206 13.5 Three-dimensional coordinate systems 207 13.6 Integration in three dimensions 210 13.7 Moments of inertia 214 13.8 Parallel-axis theorem 218 13.9 Perpendicular-axis theorem 219 PROBLEMS 220 14 Distributions 14.1 Kinds of distribution 221 14.2 Firing at a target 222 14.3 Normal distribution 225 14.4 Binomial distribution 229 14.5 Poisson distribution 235 PROBLEMS 237 15 Hyperbolic functions 15.1 Definitions 239 15.2 Relationships linking hyperbolic functions 241 15.3 Differentiation of hyperbolic functions 242 15.4 Taylor expansions of sinh x and cosh x 242 15.5 Integration involving hyperbolic functions 244 15.6 Comments about analytical functions 245 PROBLEMS 246 16 Vector analysis 16.1 Scalar and vector fields 248 16.2 Gradient (grad) and del operators 249 16.3 Conservative fields 251 16.4 Divergence (div) 254 16.5 Laplacian operator 259 16.6 Curl of a vector field 260 16.7 Maxwell's equations and the speed of light 263 PROBLEMS 264 17 Fourier analysis 17.1 Signals 265 17.2 The nature of signals 266 17.3 Amplitude-frequency diagrams 269 17.4 Fourier transform 271 17.5 The d-function, d(x) 280 17.6 Inverse Fourier transform 282 17.7 Several cosine signals 283 17.8 Parseval's theorem 284 17.9 Fourier series 286 17.10 Determination of the Fourier coefficients a0, {an}, and {bn} 293 17.11 Fourier or waveform sythesis 296 17.12 Power in periodic signals 298 17.13 Complex form for the Fourier series 300 17.14 Amplitude and phase spectrum 301 17.15 Alternative variables for Fourier analysis 302 17.16 Applications in physics 303 17.17 Summary 305 PROBLEMS 306 18 Introduction to digital signal processing 18.1 More on sampling 309 18.2 Discrete Fourier transform (DFT) 316 18.3 Some concluding remarks 326 PROBLEMS 327 19 Numerical methods for ordinary differential equations 19.1 The need for numerical methods 329 19.2 Euler methods 329 19.3 Runge-Kutta method 332 19.4 Numerov method 335 PROBLEMS 336 20 Applications of partial differential equations 20.1 Types of partial differential equation 337 20.2 Finite differences 338 20.3 Diffusion 340 20.4 Explicit method 342 20.5 The Crank-Nicholson method 344 20.6 Poisson's and Laplace's equations 347 20.7 Numerical solution of a hot-plate problem 348 20.8 Boundary conditions for hot-plate problems 350 20.9 Wave equation 352 20.10 Finite-difference approach for a vibrating string 354 20.11 Two-dimensional vibrations 356 PROBLEMS 357 21 Quantum mechanics I: Schrödinger wave equation and observations 21.1 Transition from classical to modern physics: a brief history 359 21.2 Intuitive derivation of the Schrödinger wave equation 362 21.3 A particle in a one-dimensional box 364 21.4 Observations and operators 367 21.5 A square box and degeneracy 370 21.6 Probabilities of measurements 373 21.7 Simple harmonic oscillator 375 21.8 Three-dimensional simple harmonic oscillator 379 21.9 The free particle 381 21.10 Compatible and incompatible measurements 383 21.11 A potential barrier 385 21.12 Tunnelling 388 21.13 Other methods of solving the TISWE 389 PROBLEMS 394 22 The Maxwell-Boltzmann distribution 22.1 Deriving the Maxwell-Boltzmann distribution 395 22.2 Retention of a planetary atmosphere 398 22.3 Nuclear fusion in stars 400 PROBLEMS 404 23 The Monte Carlo method 23.1 Origin of the method 405 23.2 Random walk 406 23.3 A simple polymer model 409 23.4 Uniform distribution within a sphere and random directions 411 23.5 Generation of random numbers for non-uniform deviates 412 23.6 Equation of state of a liquid 417 23.7 Simulation of a fluid by the Monte Carlo method 420 23.8 Modelling a nuclear reactor 425 23.9 Description of a simple model reactor 428 23.10 A cautionary tale 431 PROBLEMS 431 24 Matrices II 24.1 Population studies 433 24.2 Eigenvalues and eigenvectors 435 24.3 Diagonalization of a matrix 437 24.4 A vibrating system 438 PROBLEMS 443 25 Quantum mechanics II: Angular momentum and spin 25.1 Measurement of angular momentum 444 25.2 The hydrogen atom 450 25.3 Electron spin 456 25.4 Many-electron systems 462 PROBLEMS 465 26 Sampling theory 26.1 Samples 466 26.2 Sampling proportions 469 26.3 The significance of differences 471 PROBLEMS 476 27 Straight-line relationships and the linear correlation coefficient 27.1 General considerations 478 27.2 Lines of regression 481 27.3 A numerical application 483 27.4 The linear correlation coefficient 483 27.5 A general least-squares straight line 486 27.6 Linearization of other forms of relationship 492 PROBLEMS 494 28 Interpolation 28.1 Applications of interpolation 497 28.2 Linear interpolation 498 28.3 Parabolic interpolation 501 28.4 Gauss interpolation formula 502 28.5 Cubic spline interpolation 502 28.6 Multidimensional interpolation 505 28.7 Extrapolation 507 PROBLEMS 507 29 Quadrature 29.1 Definite integrals 508 29.2 Trapezium method 508 29.3 Simpson's method (rule) 511 29.4 Romberg method 513 29.5 Gauss quadrature 515 29.6 Multidimensional quadrature 518 29.7 Monte Carlo integration 520 PROBLEMS 521 30 Linear equations 30.1 Interpretation of linearly dependent and incompatible equations 522 30.2 Gauss elimination method 525 30.3 Conditioning of a set of equations 526 30.4 Gauss-Seidel method 527 30.5 Homogeneous equations 528 30.6 Least-squares solutions 529 30.7 Refinement procedures using least squares 532 PROBLEMS 534 31 Numerical solution of equations 31.1 The nature of equations 535 31.2 Fixed-point iteration method 536 31.3 Newton-Raphson method 538 PROBLEMS 540 32 Signals and noise 32.1 Introduction 541 32.2 Signals, noise, and noisy signals 543 32.3 Mathematical and statistical description of noise 546 32.4 Auto- and cross-correlation functions 550 32.5 Detection of signals in noise 562 32.6 White noise 565 32.7 Concluding remarks 567 PROBLEMS 569 33 Digital filters 33.1 Introduction 571 33.2 Fourier transform methods 572 33.3 Constant-coefficient digital filters 576 33.4 Other filter design methods 588 33.5 Summary of main results and concluding remarks 588 PROBLEMS 589 34 Introduction to estimation theory 34.1 Introduction 591 34.2 Estimation of a constant 592 34.3 Taking into account the changes in the underlying model 596 34.4 Further methods 603 34.5 Concluding remarks 604 PROBLEMS 605 35 Linear programming and optimization 35.1 Basic ideas of linear programming 607 35.2 Simplex method 611 35.3 Non-linear optimization; gradient methods 613 35.4 Gradient method for two variables 614 35.5 A practical gradient method for any number of variables 615 35.6 Optimization with constraints-the Lagrange multiplier method 618 PROBLEMS 621 36 Laplace transforms 36.1 Defining the Laplace transform 622 36.2 Inverse Laplace transforms 624 36.3 Solving differential equations with Laplace transforms 625 36.4 Laplace transforms and transfer functions 628 PROBLEMS 634 37 Networks 37.1 Graphs and networks 635 37.2 Types of network 635 37.3 Finding cheapest paths 640 37.4 Critical path analysis 643 PROBLEMS 645 38 Simulation with particles 38.1 Types of problem 647 38.2 Binary systems 648 38.3 An electron in a magnetic field 651 38.4 N-body problems 653 38.5 Molecular dynamics 656 38.6 Modelling plasmas 657 38.7 Collisionless particle-in-cell model 664 PROBLEMS 670 39 Chaos and physical calculations 39.1 The nature of chaos 672 39.2 An example from population studies 673 39.3 Other aspects of chaos 677 PROBLEM 680 Appendices 681 Appendix 1 Table of integrals 683 Appendix 2 Inverse Fourier transform 683 Appendix 3 Fourier transform of a sampled signal 685 Appendix 4 Derivation of the discrete and inverse discrete Fourier transforms 687 Appendix 5 Program OSCILLAT 689 Appendix 6 Program EXPLICIT 689 Appendix 7 Program HEATCRNI 690 Appendix 8 Program SIMPLATE 690 Appendix 9 Program STRING 691 Appendix 10 Program DRUM 692 Appendix 11 Program SHOOT 693 Appendix 12 Program DRUNKARD 694 Appendix 13 Program POLYMER 694 Appendix 14 Program METROPOLIS 696 Appendix 15 Program REACTOR 697 Appendix 16 Program LESLIE 698 Appendix 17 Eigenvalues and eigenvectors of Hermitian matrices 699 Appendix 18 Distance of a point from a line 701 Appendix 19 Program MULGAUSS 701 Appendix 20 Program MCINT 702 Appendix 21 Program GS 703 Appendix 22 Second moments for uniform and Gaussian noise 704 Appendix 23 Convolution theorem 705 Appendix 24 Output from a filter when the input is a cosine 706 Appendix 25 Program GRADMAX 707 Appendix 26 Program NETWORK 708 Appendix 27 Program GRAVBODY 709 Appendix 28 Program ELECLENS 709 Appendix 29 Program CLUSTER 710 Appendix 30 Program FLUIDYN 711 Appendix 31 Condition for collisionless PIC 712 Appendix 32 Program PLASMA1 714 References and further reading 715 Solutions to exercises and problems 716 Index 000

Library of Congress Subject Headings for this publication:

Mathematical physics.