Table of contents for Mathematics for physics / Michael M. Woolfson, Malcolm S. Woolfson.

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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.