Mathematical Methods for Physics and Engineering-R F Riley
Textbook information
- Text book title :Classical Electrodynamics(3Ed , Wiley, 1999)
- Author : R F Riley,Hobson,Bence
- ISBN : 9780521679718
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- File size :9.25 Mb
- File format : PDF File
- Total No. of pages : 1363 pages
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Text book Content page
1 Preliminary algebra 1
- 1.1 Simple functions and equations 1
- Polynomial equations; factorisation; properties of roots
- 1.2 Trigonometric identities 10
- Single angle; compound angles; double- and half-angle identities
- 1.3 Coordinate geometry 15
- 1.4 Partial fractions 18
- Complications and special cases
- 1.5 Binomial expansion 25
- 1.6 Properties of binomial coefficients 27
- 1.7 Some particular methods of proof 30
- Proof by induction; proof by contradiction; necessary and sufficient conditions
- 1.8 Exercises 36
- 1.9 Hints and answers 39
- 2.1 Differentiation 41
- Differentiation from first principles; products; the chain rule; quotients;
- implicit differentiation; logarithmic differentiation; Leibnitz’ theorem; special
- points of a function; curvature; theorems of differentiation
- 2.2 Integration 59
- Integration from first principles; the inverse of differentiation; by inspection;
- sinusoidal functions; logarithmic integration; using partial fractions;
- substitution method; integration by parts; reduction formulae; infinite and
- improper integrals; plane polar coordinates; integral inequalities; applications
- of integration
- 2.3 Exercises 76
- 2.4 Hints and answers 81
3 Complex numbers and hyperbolic functions 83
- 3.1 The need for complex numbers 83
- 3.2 Manipulation of complex numbers 85
- Addition and subtraction; modulus and argument; multiplication; complex
- conjugate; division
- 3.3 Polar representation of complex numbers 92
- Multiplication and division in polar form
- 3.4 de Moivre’s theorem 95
- trigonometric identities; finding the nth roots of unity; solving polynomial
- equations
- 3.5 Complex logarithms and complex powers 99
- 3.6 Applications to differentiation and integration 101
- 3.7 Hyperbolic functions 102
- Definitions; hyperbolic–trigonometric analogies; identities of hyperbolic
- functions; solving hyperbolic equations; inverses of hyperbolic functions;
- calculus of hyperbolic functions
- 3.8 Exercises 109
- 3.9 Hints and answers 113
4 Series and limits 115
- 4.1 Series 115
- 4.2 Summation of series 116
- Arithmetic series; geometric series; arithmetico-geometric series; the difference
- method; series involving natural numbers; transformation of series
- 4.3 Convergence of infinite series 124
- Absolute and conditional convergence; series containing only real positive
- terms; alternating series test
- 4.4 Operations with series 131
- 4.5 Power series 131
- Convergence of power series; operations with power series
- 4.6 Taylor series 136
- Taylor’s theorem; approximation errors; standard Maclaurin series
- 4.7 Evaluation of limits 141
- 4.8 Exercises 144
- 4.9 Hints and answers
- 5.1 Definition of the partial derivative 151
- 5.2 The total differential and total derivative 153
- 5.3 Exact and inexact differentials 155
- 5.4 Useful theorems of partial differentiation 157
- 5.5 The chain rule 157
- 5.6 Change of variables 158
- 5.7 Taylor’s theorem for many-variable functions 160
- 5.8 Stationary values of many-variable functions 162
- 5.9 Stationary values under constraints 167
- 5.10 Envelopes 173
- 5.11 Thermodynamic relations 176
- 5.12 Differentiation of integrals 178
- 5.13 Exercises 179
- 5.14 Hints and answers 185
- 6.1 Double integrals 187
- 6.2 Triple integrals 190
- 6.3 Applications of multiple integrals 191
- Areas and volumes; masses, centres of mass and centroids; Pappus’ theorems;
- moments of inertia; mean values of functions
- 6.4 Change of variables in multiple integrals 199
- Change of variables in double integrals; evaluation of the integral I = ∞
- −∞ e−x2
- dx; change of variables in triple integrals; general properties of
- Jacobians
- 6.5 Exercises 207
- 6.6 Hints and answers 211
- 7.1 Scalars and vectors 212
- 7.2 Addition and subtraction of vectors 213
- 7.3 Multiplication by a scalar 214
- 7.4 Basis vectors and components 217
- 7.5 Magnitude of a vector 218
- 7.6 Multiplication of vectors 219
- Scalar product; vector product; scalar triple product; vector triple product
- 7.7 Equations of lines, planes and spheres 226
- 7.8 Using vectors to find distances 229
- Point to line; point to plane; line to line; line to plane
- 7.9 Reciprocal vectors 233
- 7.10 Exercises 234
- 7.11 Hints and answers 240
8 Matrices and vector spaces 241
- 8.1 Vector spaces 242
- Basis vectors; inner product; some useful inequalities
- 8.2 Linear operators 247
- 8.3 Matrices 249
- 8.4 Basic matrix algebra 250
- Matrix addition; multiplication by a scalar; matrix multiplication
- 8.5 Functions of matrices 255
- 8.6 The transpose of a matrix 255
- 8.7 The complex and Hermitian conjugates of a matrix 256
- 8.8 The trace of a matrix 258
- 8.9 The determinant of a matrix 259
- Properties of determinants
- 8.10 The inverse of a matrix 263
- 8.11 The rank of a matrix 267
- 8.12 Special types of square matrix 268
- Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian
- and anti-Hermitian; unitary; normal
- 8.13 Eigenvectors and eigenvalues 272
- Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary
- matrix; of a general square matrix
- 8.14 Determination of eigenvalues and eigenvectors 280
- Degenerate eigenvalues
- 8.15 Change of basis and similarity transformations 282
- 8.16 Diagonalisation of matrices 285
- 8.17 Quadratic and Hermitian forms 288
- Stationary properties of the eigenvectors; quadratic surfaces
- 8.18 Simultaneous linear equations 292
- Range; null space; N simultaneous linear equations in N unknowns; singular
- value decomposition
- 8.19 Exercises 307
- 8.20 Hints and answers 314
9 Normal modes 316
- 9.1 Typical oscillatory systems 317
- 9.2 Symmetry and normal modes 3229.3 Rayleigh–Ritz method 327
- 9.4 Exercises 329
- 9.5 Hints and answers 332
10 Vector calculus 334
- 10.1 Differentiation of vectors 334
- Composite vector expressions; differential of a vector
- 10.2 Integration of vectors 339
- 10.3 Space curves 340
- 10.4 Vector functions of several arguments 344
- 10.5 Surfaces 345
- 10.6 Scalar and vector fields 347
- 10.7 Vector operators 347
- Gradient of a scalar field; divergence of a vector field; curl of a vector field
- 10.8 Vector operator formulae 354
- Vector operators acting on sums and products; combinations of grad, div and
- curl
- 10.9 Cylindrical and spherical polar coordinates 357
- 10.10 General curvilinear coordinates 364
- 10.11 Exercises 369
- 10.12 Hints and answers 375
11 Line, surface and volume integrals 377
- 11.1 Line integrals 377
- Evaluating line integrals; physical examples; line integrals with respect to a
- scalar
- 11.2 Connectivity of regions 383
- 11.3 Green’s theorem in a plane 384
- 11.4 Conservative fields and potentials 387
- 11.5 Surface integrals 389
- Evaluating surface integrals; vector areas of surfaces; physical examples
- 11.6 Volume integrals 396
- Volumes of three-dimensional regions
- 11.7 Integral forms for grad, div and curl 398
- 11.8 Divergence theorem and related theorems 401
- Green’s theorems; other related integral theorems; physical applications
- 11.9 Stokes’ theorem and related theorems 406
- Related integral theorems; physical applications
- 11.10 Exercises 409
- 11.11 Hints and answers 414
12 Fourier series 415
- 12.1 The Dirichlet conditions 4 12.2 The Fourier coefficients 417
- 12.3 Symmetry considerations 419
- 12.4 Discontinuous functions 420
- 12.5 Non-periodic functions 422
- 12.6 Integration and differentiation 424
- 12.7 Complex Fourier series 424
- 12.8 Parseval’s theorem 426
- 12.9 Exercises 427
- 12.10 Hints and answers 431
13 Integral transforms 433
- 13.1 Fourier transforms 433
- The uncertainty principle; Fraunhofer diffraction; the Dirac δ-function;
- relation of the δ-function to Fourier transforms; properties of Fourier
- transforms; odd and even functions; convolution and deconvolution; correlation
- functions and energy spectra; Parseval’s theorem; Fourier transforms in higher
- dimensions
- 13.2 Laplace transforms 453
- Laplace transforms of derivatives and integrals; other properties of Laplace
- transforms
- 13.3 Concluding remarks 459
- 13.4 Exercises 460
- 13.5 Hints and answers 466
14 First-order ordinary differential equations 468
- 14.1 General form of solution 469
- 14.2 First-degree first-order equations 470
- Separable-variable equations; exact equations; inexact equations, integrating
- factors; linear equations; homogeneous equations; isobaric equations;
- Bernoulli’s equation; miscellaneous equations
- 14.3 Higher-degree first-order equations 480
- Equations soluble for p; for x; for y; Clairaut’s equation
- 14.4 Exercises 484
- 14.5 Hints and answers 488
15 Higher-order ordinary differential equations 490
- 15.1 Linear equations with constant coefficients 492
- Finding the complementary function yc(x); finding the particular integral
- yp(x); constructing the general solution yc(x) + yp(x); linear recurrence
- relations; Laplace transform method
- 15.2 Linear equations with variable coefficients 503
- The Legendre and Euler linear equations; exact equations; partially known
- complementary function; variation of parameters; Green’s functions; canonical15.3 General ordinary differential equations 518
- Dependent variable absent; independent variable absent; non-linear exact
- equations; isobaric or homogeneous equations; equations homogeneous in x
- or y alone; equations having y = Aex as a solution
- 15.4 Exercises 523
- 15.5 Hints and answers 529
16 Series solutions of ordinary differential equations 531
- 16.1 Second-order linear ordinary differential equations 531
- Ordinary and singular points
- 16.2 Series solutions about an ordinary point 535
- 16.3 Series solutions about a regular singular point 538
- Distinct roots not differing by an integer; repeated root of the indicial
- equation; distinct roots differing by an integer
- 16.4 Obtaining a second solution 544
- The Wronskian method; the derivative method; series form of the second
- solution
- 16.5 Polynomial solutions 548
- 16.6 Exercises 550
- 16.7 Hints and answers 553
17 Eigenfunction methods for differential equations 554
- 17.1 Sets of functions 556
- Some useful inequalities
- 17.2 Adjoint, self-adjoint and Hermitian operators 559
- 17.3 Properties of Hermitian operators 561
- Reality of the eigenvalues; orthogonality of the eigenfunctions; construction
- of real eigenfunctions
- 17.4 Sturm–Liouville equations 564
- Valid boundary conditions; putting an equation into Sturm–Liouville form
- 17.5 Superposition of eigenfunctions: Green’s functions 569
- 17.6 A useful generalisation 572
- 17.7 Exercises 573
- 17.8 Hints and answers 576
18 Special functions 577
- 18.1 Legendre functions 577
- General solution for integer ; properties of Legendre polynomials
- 18.2 Associated Legendre functions 587
- 18.3 Spherical harmonics 593
- 18.4 Chebyshev functions 595
- 18.5 Bessel functions 602
- General solution for non-integer ν; general solution for integer ν; properties
- of Bessel functions18.6 Spherical Bessel functions 614
- 18.7 Laguerre functions 616
- 18.8 Associated Laguerre functions 621
- 18.9 Hermite functions 624
- 18.10 Hypergeometric functions 628
- 18.11 Confluent hypergeometric functions 633
- 18.12 The gamma function and related functions 635
- 18.13 Exercises 640
- 18.14 Hints and answers 646
19 Quantum operators 648
- 19.1 Operator formalism 648
- Commutators
- 19.2 Physical examples of operators 656
- Uncertainty principle; angular momentum; creation and annihilation operators
- 19.3 Exercises 671
- 19.4 Hints and answers 674
20 Partial differential equations: general and particular solutions 675
- 20.1 Important partial differential equations 676
- The wave equation; the diffusion equation; Laplace’s equation; Poisson’s
- equation; Schr¨odinger’s equation
- 20.2 General form of solution 680
- 20.3 General and particular solutions 681
- First-order equations; inhomogeneous equations and problems; second-order
- equations
- 20.4 The wave equation 693
- 20.5 The diffusion equation 695
- 20.6 Characteristics and the existence of solutions 699
- First-order equations; second-order equations
- 20.7 Uniqueness of solutions 705
- 20.8 Exercises 707
- 20.9 Hints and answers 711
21 Partial differential equations: separation of variables
and other methods 713
and other methods 713
- 21.1 Separation of variables: the general method 713
- 21.2 Superposition of separated solutions 717
- 21.3 Separation of variables in polar coordinates 725
- Laplace’s equation in polar coordinates; spherical harmonics; other equations
- in polar coordinates; solution by expansion; separation of variables for
- inhomogeneous equations
- 21.4 Integral transform methods 74721.5 Inhomogeneous problems – Green’s functions 751
- Similarities to Green’s functions for ordinary differential equations; general
- boundary-value problems; Dirichlet problems; Neumann problems
- 21.6 Exercises 767
- 21.7 Hints and answers 773
22 Calculus of variations 775
- 22.1 The Euler–Lagrange equation 776
- 22.2 Special cases 777
- F does not contain y explicitly; F does not contain x explicitly
- 22.3 Some extensions 781
- Several dependent variables; several independent variables; higher-order
- derivatives; variable end-points
- 22.4 Constrained variation 785
- 22.5 Physical variational principles 787
- Fermat’s principle in optics; Hamilton’s principle in mechanics
- 22.6 General eigenvalue problems 790
- 22.7 Estimation of eigenvalues and eigenfunctions 792
- 22.8 Adjustment of parameters 795
- 22.9 Exercises 797
- 22.10 Hints and answers 801
23 Integral equations 803
- 23.1 Obtaining an integral equation from a differential equation 803
- 23.2 Types of integral equation 804
- 23.3 Operator notation and the existence of solutions 805
- 23.4 Closed-form solutions 806
- Separable kernels; integral transform methods; differentiation
- 23.5 Neumann series 813
- 23.6 Fredholm theory 815
- 23.7 Schmidt–Hilbert theory 816
- 23.8 Exercises 819
- 23.9 Hints and answers 823
24 Complex variables 824
- 24.1 Functions of a complex variable 825
- 24.2 The Cauchy–Riemann relations 827
- 24.3 Power series in a complex variable 830
- 24.4 Some elementary functions 832
- 24.5 Multivalued functions and branch cuts 835
- 24.6 Singularities and zeros of complex functions 837
- 24.7 Conformal transformations 839
- 24.8 Complex integrals 8424.9 Cauchy’s theorem 849
- 24.10 Cauchy’s integral formula 851
- 24.11 Taylor and Laurent series 853
- 24.12 Residue theorem 858
- 24.13 Definite integrals using contour integration 861
- 24.14 Exercises 867
- 24.15 Hints and answers 870
25 Applications of complex variables 871
- 25.1 Complex potentials 871
- 25.2 Applications of conformal transformations 876
- 25.3 Location of zeros 879
- 25.4 Summation of series 882
- 25.5 Inverse Laplace transform 884
- 25.6 Stokes’ equation and Airy integrals 888
- 25.7 WKB methods 895
- 25.8 Approximations to integrals 905
- Level lines and saddle points; steepest descents; stationary phase
- 25.9 Exercises 920
- 25.10 Hints and answers 925
26 Tensors 927
- 26.1 Some notation 928
- 26.2 Change of basis 929
- 26.3 Cartesian tensors 930
- 26.4 First- and zero-order Cartesian tensors 932
- 26.5 Second- and higher-order Cartesian tensors 935
- 26.6 The algebra of tensors 938
- 26.7 The quotient law 939
- 26.8 The tensors δij and ijk 941
- 26.9 Isotropic tensors 944
- 26.10 Improper rotations and pseudotensors 946
- 26.11 Dual tensors 949
- 26.12 Physical applications of tensors 950
- 26.13 Integral theorems for tensors 954
- 26.14 Non-Cartesian coordinates 955
- 26.15 The metric tensor 957
- 26.16 General coordinate transformations and tensors 960
- 26.17 Relative tensors 963
- 26.18 Derivatives of basis vectors and Christoffel symbols 965
- 26.19 Covariant differentiation 968
- 26.20 Vector operators in tensor form 97126.21 Absolute derivatives along curves 975
- 26.22 Geodesics 976
- 26.23 Exercises 977
- 26.24 Hints and answers 982
27 Numerical methods 984
- 27.1 Algebraic and transcendental equations 985
- Rearrangement of the equation; linear interpolation; binary chopping;
- Newton–Raphson method
- 27.2 Convergence of iteration schemes 992
- 27.3 Simultaneous linear equations 994
- Gaussian elimination; Gauss–Seidel iteration; tridiagonal matrices
- 27.4 Numerical integration 1000
- Trapezium rule; Simpson’s rule; Gaussian integration; Monte Carlo methods
- 27.5 Finite differences 1019
- 27.6 Differential equations 1020
- Difference equations; Taylor series solutions; prediction and correction;
- Runge–Kutta methods; isoclines
- 27.7 Higher-order equations 1028
- 27.8 Partial differential equations 1030
- 27.9 Exercises 1033
- 27.10 Hints and answers 1039
28 Group theory 1041
- 28.1 Groups 1041
- Definition of a group; examples of groups
- 28.2 Finite groups 1049
- 28.3 Non-Abelian groups 1052
- 28.4 Permutation groups 1056
- 28.5 Mappings between groups 1059
- 28.6 Subgroups 1061
- 28.7 Subdividing a group 1063
- Equivalence relations and classes; congruence and cosets; conjugates and
- classes
- 28.8 Exercises 1070
- 28.9 Hints and answers 1074
29 Representation theory 1076
29.1 Dipole moments of molecules 1077
29.2 Choosing an appropriate formalism 1078
29.3 Equivalent representations 1084
29.4 Reducibility of a representation 1086
29.5 The orthogonality theorem for irreducible representations 1090
29.6 Characters 109230 Probability 1119
Orthogonality property of characters
29.7 Counting irreps using characters 1095
Summation rules for irreps
29.8 Construction of a character table 1100
29.9 Group nomenclature 1102
29.10 Product representations 1103
29.11 Physical applications of group theory 1105
Bonding in molecules; matrix elements in quantum mechanics; degeneracy of
normal modes; breaking of degeneracies
29.12 Exercises 1113
29.13 Hints and answers 1117
- 30.1 Venn diagrams 1119
- 30.2 Probability 1124
- Axioms and theorems; conditional probability; Bayes’ theorem
- 30.3 Permutations and combinations 1133
- 30.4 Random variables and distributions 1139
- Discrete random variables; continuous random variables
- 30.5 Properties of distributions 1143
- Mean; mode and median; variance and standard deviation; moments; central
- moments
- 30.6 Functions of random variables 1150
- 30.7 Generating functions 1157
- Probability generating functions; moment generating functions; characteristic
- functions; cumulant generating functions
- 30.8 Important discrete distributions 1168
- Binomial; geometric; negative binomial; hypergeometric; Poisson
- 30.9 Important continuous distributions 1179
- Gaussian; log-normal; exponential; gamma; chi-squared; Cauchy; Breit–
- Wigner; uniform
- 30.10 The central limit theorem 1195
- 30.11 Joint distributions 1196
- Discrete bivariate; continuous bivariate; marginal and conditional distributions
- 30.12 Properties of joint distributions 1199
- Means; variances; covariance and correlation
- 30.13 Generating functions for joint distributions 1205
- 30.14 Transformation of variables in joint distributions 1206
- 30.15 Important joint distributions 1207
- Multinominal; multivariate Gaussian
- 30.16 Exercises 1211 30.17 Hints and answers 1219
31 Statistics 1221
- 31.1 Experiments, samples and populations 1221
- 31.2 Sample statistics 1222
- Averages; variance and standard deviation; moments; covariance and correlation
- 31.3 Estimators and sampling distributions 1229
- Consistency, bias and efficiency; Fisher’s inequality; standard errors; confidence
- limits
- 31.4 Some basic estimators 1243
- Mean; variance; standard deviation; moments; covariance and correlation
- 31.5 Maximum-likelihood method 1255
- ML estimator; transformation invariance and bias; efficiency; errors and
- confidence limits; Bayesian interpretation; large-N behaviour; extended
- ML method
- 31.6 The method of least squares 1271
- Linear least squares; non-linear least squares
- 31.7 Hypothesis testing 1277
- Simple and composite hypotheses; statistical tests; Neyman–Pearson; generalised
- likelihood-ratio; Student’s t; Fisher’s F; goodness of fit
- 31.8 Exercises 1298
- 31.9 Hints and answers 1303