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CONTENTS
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Preface to the third e dition page xx
P re f a ce to th e s e co n d e d i ti o n xxiii
P re f a ce to th e fi r s t e d i ti o n xxv
1 P re l i m i n a ry a l g e b ra 1
1.1 Simple functions and equations 1
Po ly n o m i a l e q u a ti o n s ; f a c to r isation; 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 Preliminary calculus 41
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
I n te g ra tio n f ro m fi r s t p r i n c i p l e s ; t h e in v e r s e o f d i ff e re n tia ti o n ; by i n s p e c
tio n ; s i n u s o i d a l f u n c ti o n s ; l o g arithmic 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 Co m p l ex n u m b e r s a n d hyp e rb o l i c f u n c t i o n s 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 149
i 5 Partial differentiation 151
5 . 1 D e fi n i t i on o f t h e p a r t i a l d e r i v a t i v e 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 Multiple integrals 187
6.1 Double integrals 187
6.2 Triple integrals 190
6.3 Applications of multiple integrals 191
A re a s a n d v o l u m e s ; m a s s e s , cen t re s o f m a s s a n d cen t ro i d s ; Pa p p u s ’ t h e o re m s ;
m o m e n t s o f i n e r ti a ; me a n v a l u e s o f f u n c ti o n s
6.4 Change of variables in multiple integrals 199
C h a n g e o f v a r i a b l e s i n d o u b l e in te g ra l s ; e v a l u a ti o n o f t h e i n teg ra l 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 Vector algebra 212
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 E qu a t i on s o f l i n e s , p l a n e s a n d s p h e r e s 226
7.8 Using vectors to find distances 229
Po in t to lin e; p o in t to p la n e; lin e to lin e; lin e to p la n e
7.9 Reciprocal vectors 233
7.10 Exercises 234
7.11 Hints and answers 240
8 M a t r i ce s a n d v e c t o r s p a ce s 241
8.1 Vector spaces 242
B a s i s ve c to r s ; i n n e r p ro d u c t ; s o m e u s e f u l i n e q u a l i ti e s
8.2 Linear operators 247
8.3 Matrices 249
8.4 Basic matrix algebra 250
Ma t r ix a d d itio n ; m u ltip lica tio n by a s ca la r ; ma t r ix mu ltip lica tio n
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 o f d eterminants
8.10 The inverse of a matrix 263
8.11 The rank of a matrix 267
8.12 Special types of square matrix 268
D i a g o n a l ; t r i a n g u l a r ; sy m m e t r i c a n d 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 322
9.3 Rayleigh–Ritz method 327
9.4 Exercises 329
9.5 Hints and answers 332
10 Vector calculus 334
10.1 Differentiation of vectors 334
Co mposite vector ex pressions; 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 scala r field; divergence o f a vector field; curl of a vector field
10.8 Vector operator formulae 354
Ve c to r o p e ra to r s a c ti n g o n s u m s a n d p ro d u c t s ; co m b i n a ti o n s o f g ra d , d i v a n d
c u r l
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
E v a l u a ti n g l i n e in te g ra l s ; p h y s i ca l 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 415
1 2 . 2 T h e Fou r i e r c o e ffi c i e n t s 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 Dira c 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, integrat
ing 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; canonical
form for second-order equations
1 5 . 3 G e n e r a l or d i n a ry d i ff e r e n t i a l e qu a t i on s 518
D e p e n d e n t v a r i a b l e a b s e n t ; i n d e p e n d e n t v a r i a b l e a b s e n t ; n o n - l i n e a r exa c t
e q u a ti o n s ; i s ob a r i c o r h o m o g e n e o u s e q u a ti o n s ; e q u a ti o n s h o m o g e n e o u s i n x
o r 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 functions
1 8 . 6 S p h e r i c a l B e s s e l fu n c t i on s 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 op erators 648
19.1 Operator formalism 648
Co mmu ta to r s
19.2 Physical examples of operators 656
Un certainty p rinciple; angula r 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; Schrodinger’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
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 747
2 1 . 5 I n h o m og e n e ou s p rob l e m s – G r e e n ’ s fu n c t i on s 751
Simila rities to Green’s functions for ordina ry differential equations; general
bounda ry -value prob lems; Dirich let 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 845
2 4 . 9 C a u ch y ’ s t h e or e m 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 v ariables 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
L e v e l l i n e s a n d s a d d l e p o i n t s ; s te e p e s t d e s ce n t s ; s ta ti o n a ry p h a s e
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 δ
ij 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 971
2 6 . 2 1 A b s o l u t e d e r i v a t i v e s a l on g c u rv e s 975
26.22 Geodesics 976
26.23 Exercises 977
26.24 Hints and answers 982
2 7 N u m e r i c a l m e t h o d s 984
27.1 Algebraic and transcendental equations 985
R e a r ra n g e m e n t o f t h e e q u a ti o n ; l i n e a r in te rp o l a ti o n ; b i n a ry ch o p p i n g ;
New to n – R a p h s o n m e t h o d
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 1092
O r t h o g o n a l i ty p ro p e r ty o f ch a ra c te r s
29.7 Counting irreps using characters 1095
S u m m a ti o n r u l e s f o r ir re p s
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 e lements in quantum mechanics; degeneracy o f
n o r m a l m o d e s ; b re a k i n g o f d e g e n e ra c i e s
29.12 Exercises 1113
29.13 Hints and answers 1117
30 Probability 1119
30.1 Venn diagrams 1119
30.2 Probability 1124
Axi o m s a n d t h e o re m s ; co n d i ti o n a l p rob a b i l i ty ; B ay e s ’ t h e o re m
30.3 Permutations and combinations 1133
30.4 Random variables and distributions 1139
Discrete random va riables; 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
3 1 . 1 E x p e r i m e n t s , s a m p l e s a n d p op u l a t i on s 1221
31.2 Sample statistics 1222
A v e ra g e s ; v a r i a n ce a n d s ta n d a rd d e v i a ti o n ; m o m e n t s ; co v a r i a n ce a n d co r re l a
tio n
31.3 Estimators and sampling distributions 1229
Co n s i s te n cy , b i a s a n d e ffi c i e n cy ; Fis h e r ’ s i n e q u a l i ty ; s ta n d a rd e r ro r s ; co n fi
dence limits
31.4 Some basic estimators 1243
Mea n ; v a r i a n ce ; s ta n d a rd d e v i a ti o n ; m o m e n t s ; co v a r i a n ce a n d co r re l a ti o n
31.5 Maximum-likelihood method 1255
M L e s ti m a to r ; t ra n s f o r m a ti o n in v a r i a n ce a n d b i a s ; e ffi c i e n cy ; e r ro r s a n d
co n fi d e n ce l i m i t s ; B ay e s i a n i n terp reta tio n ; l a rg e -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; gener
alised likelihood-ratio; Student’s t; Fisher’s F; goodness of fit
31.8 Exercises 1298
31.9 Hints and answers 1303
Index 1305
P re f a ce to th e s e co n d e d i ti o n xxiii
P re f a ce to th e fi r s t e d i ti o n xxv
1 P re l i m i n a ry a l g e b ra 1
1.1 Simple functions and equations 1
Po ly n o m i a l e q u a ti o n s ; f a c to r isation; 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 Preliminary calculus 41
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
I n te g ra tio n f ro m fi r s t p r i n c i p l e s ; t h e in v e r s e o f d i ff e re n tia ti o n ; by i n s p e c
tio n ; s i n u s o i d a l f u n c ti o n s ; l o g arithmic 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 Co m p l ex n u m b e r s a n d hyp e rb o l i c f u n c t i o n s 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 149
i 5 Partial differentiation 151
5 . 1 D e fi n i t i on o f t h e p a r t i a l d e r i v a t i v e 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 Multiple integrals 187
6.1 Double integrals 187
6.2 Triple integrals 190
6.3 Applications of multiple integrals 191
A re a s a n d v o l u m e s ; m a s s e s , cen t re s o f m a s s a n d cen t ro i d s ; Pa p p u s ’ t h e o re m s ;
m o m e n t s o f i n e r ti a ; me a n v a l u e s o f f u n c ti o n s
6.4 Change of variables in multiple integrals 199
C h a n g e o f v a r i a b l e s i n d o u b l e in te g ra l s ; e v a l u a ti o n o f t h e i n teg ra l 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 Vector algebra 212
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 E qu a t i on s o f l i n e s , p l a n e s a n d s p h e r e s 226
7.8 Using vectors to find distances 229
Po in t to lin e; p o in t to p la n e; lin e to lin e; lin e to p la n e
7.9 Reciprocal vectors 233
7.10 Exercises 234
7.11 Hints and answers 240
8 M a t r i ce s a n d v e c t o r s p a ce s 241
8.1 Vector spaces 242
B a s i s ve c to r s ; i n n e r p ro d u c t ; s o m e u s e f u l i n e q u a l i ti e s
8.2 Linear operators 247
8.3 Matrices 249
8.4 Basic matrix algebra 250
Ma t r ix a d d itio n ; m u ltip lica tio n by a s ca la r ; ma t r ix mu ltip lica tio n
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 o f d eterminants
8.10 The inverse of a matrix 263
8.11 The rank of a matrix 267
8.12 Special types of square matrix 268
D i a g o n a l ; t r i a n g u l a r ; sy m m e t r i c a n d 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 322
9.3 Rayleigh–Ritz method 327
9.4 Exercises 329
9.5 Hints and answers 332
10 Vector calculus 334
10.1 Differentiation of vectors 334
Co mposite vector ex pressions; 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 scala r field; divergence o f a vector field; curl of a vector field
10.8 Vector operator formulae 354
Ve c to r o p e ra to r s a c ti n g o n s u m s a n d p ro d u c t s ; co m b i n a ti o n s o f g ra d , d i v a n d
c u r l
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
E v a l u a ti n g l i n e in te g ra l s ; p h y s i ca l 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 415
1 2 . 2 T h e Fou r i e r c o e ffi c i e n t s 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 Dira c 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, integrat
ing 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; canonical
form for second-order equations
1 5 . 3 G e n e r a l or d i n a ry d i ff e r e n t i a l e qu a t i on s 518
D e p e n d e n t v a r i a b l e a b s e n t ; i n d e p e n d e n t v a r i a b l e a b s e n t ; n o n - l i n e a r exa c t
e q u a ti o n s ; i s ob a r i c o r h o m o g e n e o u s e q u a ti o n s ; e q u a ti o n s h o m o g e n e o u s i n x
o r 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 functions
1 8 . 6 S p h e r i c a l B e s s e l fu n c t i on s 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 op erators 648
19.1 Operator formalism 648
Co mmu ta to r s
19.2 Physical examples of operators 656
Un certainty p rinciple; angula r 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; Schrodinger’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
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 747
2 1 . 5 I n h o m og e n e ou s p rob l e m s – G r e e n ’ s fu n c t i on s 751
Simila rities to Green’s functions for ordina ry differential equations; general
bounda ry -value prob lems; Dirich let 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 845
2 4 . 9 C a u ch y ’ s t h e or e m 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 v ariables 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
L e v e l l i n e s a n d s a d d l e p o i n t s ; s te e p e s t d e s ce n t s ; s ta ti o n a ry p h a s e
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 δ
ij 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 971
2 6 . 2 1 A b s o l u t e d e r i v a t i v e s a l on g c u rv e s 975
26.22 Geodesics 976
26.23 Exercises 977
26.24 Hints and answers 982
2 7 N u m e r i c a l m e t h o d s 984
27.1 Algebraic and transcendental equations 985
R e a r ra n g e m e n t o f t h e e q u a ti o n ; l i n e a r in te rp o l a ti o n ; b i n a ry ch o p p i n g ;
New to n – R a p h s o n m e t h o d
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 1092
O r t h o g o n a l i ty p ro p e r ty o f ch a ra c te r s
29.7 Counting irreps using characters 1095
S u m m a ti o n r u l e s f o r ir re p s
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 e lements in quantum mechanics; degeneracy o f
n o r m a l m o d e s ; b re a k i n g o f d e g e n e ra c i e s
29.12 Exercises 1113
29.13 Hints and answers 1117
30 Probability 1119
30.1 Venn diagrams 1119
30.2 Probability 1124
Axi o m s a n d t h e o re m s ; co n d i ti o n a l p rob a b i l i ty ; B ay e s ’ t h e o re m
30.3 Permutations and combinations 1133
30.4 Random variables and distributions 1139
Discrete random va riables; 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
3 1 . 1 E x p e r i m e n t s , s a m p l e s a n d p op u l a t i on s 1221
31.2 Sample statistics 1222
A v e ra g e s ; v a r i a n ce a n d s ta n d a rd d e v i a ti o n ; m o m e n t s ; co v a r i a n ce a n d co r re l a
tio n
31.3 Estimators and sampling distributions 1229
Co n s i s te n cy , b i a s a n d e ffi c i e n cy ; Fis h e r ’ s i n e q u a l i ty ; s ta n d a rd e r ro r s ; co n fi
dence limits
31.4 Some basic estimators 1243
Mea n ; v a r i a n ce ; s ta n d a rd d e v i a ti o n ; m o m e n t s ; co v a r i a n ce a n d co r re l a ti o n
31.5 Maximum-likelihood method 1255
M L e s ti m a to r ; t ra n s f o r m a ti o n in v a r i a n ce a n d b i a s ; e ffi c i e n cy ; e r ro r s a n d
co n fi d e n ce l i m i t s ; B ay e s i a n i n terp reta tio n ; l a rg e -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; gener
alised likelihood-ratio; Student’s t; Fisher’s F; goodness of fit
31.8 Exercises 1298
31.9 Hints and answers 1303
Index 1305
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