Since the publication of the first edition of this book, both through teaching the material it covers and as a result of receiving helpful comments from colleagues, we have become aware of the desirability of changes in a number of areas. The most important of these is that the mathematical preparation of current senior college and university entrants is now less thorough than it used to be. To match this, we decided to include a preliminary chapter covering areas such as polynomial equations, trigonometric identities, coordinate geometry, partial fractions, binomial expansions, necessary and sufficient condition and proof by induction and contradiction.

Preface to the second edition

Preface to the first edition

1  Preliminary algebra

1.1  Simple functions and equations

Polynomial equations; factorisation; properties of roots

1.2  Trigonometric identities

Single angle; compound-angles; double- and half-angle identities

1.3  Coordinate geometry

1.4  Partial fractions

Complications and special cases

1.5  Binomial expansion

1.6  Properties of binomial coefficients

1.7  Some particular methods of proof

Proof by induction; proof by contradiction; necessary and sufficient conditions

1.8  Exercises

1.9  Hints and answers

2  Preliminary calculus

2. 1  Differentiation

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

Integration from first principles; the inverse of differentiation; by inspection; sinusoidal Jhnctions; 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

2.4  Hints and answers

3  Complex numbers and hyperbolic functions

3.1  The need for complex numbers

3.2  Manipulation of complex numbers

Addition and subtraction; modulus and argument; multiplication; complex conjugate; division

3.3  Polar representation of complex numbers Multiplication and division in polar form

3.4  de Moivre's theorem

trigonometric identities;finding the nth roots of unity: solving polynomial equations

3.5  Complex logarithms and complex powers

3.6  Applications to differentiation and integration

3.7  Hyperbolic functions

Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions: solving hyperbolic equations; inverses of hyperbolic functions;calculus of hyperbolic functions

3.8  Exercises

3.9  Hints and answers

4  Series and limits

4.1  Series

4.2  Summation of series

Arithmetic series; geometric series; arithmetico-geometric series; the difference method; series involving natural numbers; transformation of series

4.3  Convergence of infinite series

Absolute and conditional convergence; series containing only real positive terms; alternating series test

4.4  Operations with series

4.5  Power series

Convergence of power series; operations with power series

4.6  Taylor series

Taylor's theorem; approximation errors; standard Maclaurin series

4.7  Evaluation of limits

4.8  Exercises

4.9  Hints and answers

5  Partial differentiation

5.1  Definition of the partial derivative

5.2  The total differential and total derivative

5.3  Exact and inexact differentials

5.4  Useful theorems of partial differentiation

5.5  The chain rule

5.6  Change of variables

5.7  Taylor's theorem for many-variable functions

5.8  Stationary values of many-variable functions

5.9  Stationary values under constraints

5.10 Envelopes

5.11 Thermodynamic relations

5.12 Differentiation of integrals

5.13 Exercises

5.14 Hints and answers

6  Multiple integrals

6.1  Double integrals

6.2 Triple integrals

6.3  Applications of multiple integrals

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

Change of variables in double integrals; evaluation of the integral I =change of variables in triple integrals; general properties of Jacobians

6.5  Exercises

6.6  Hints and answers

7  Vector algebra

7.1  Scalars and vectors

7.2  Addition and subtraction of vectors

7.3  Multiplication by a scalar

7.4  Basis vectors and components

7.5  Magnitude of a vector

7.6  Multiplication of vectors

Scalar product; vector product; scalar triple product; vector triple product

7.7  Equations of lines, planes and spheres

7.8  Using vectors to find distances

Point to line; point to plane; line to line; line to plane

7.9  Reciprocal vectors

7.10 Exercises

7.11 Hints and answers

8  Matrices and vector spaces

8.1  Vector spaces

Basis vectors; inner product; some useful inequalities

8.2  Linear operators

8.3  Matrices

8.4  Basic matrix algebra

Matrix addition; multiplication by a scalar; matrix multiplication

8.5  Functions of matrices

8,6  The transpose of a matrix

8.7  The complex and Hermitian conjugates of a matrix

8.8  The trace of a matrix

8.9  The determinant of a matrix

Properties of determinants

8.10 The inverse of a matrix

8.11 The rank of a matrix

8.12 Special types of square matrix

Diagonal; triangular; symmetric and antisymmetric ; orthogonal; Hermitian and anti-Hermitian; unitary; normal

8.13 Eigenvectors and eigenvalues

Ora normal matrix; of Hermitian and anti~Herrnitian matrices; ora unitary matrix; ora general square matrix

8.14 Determination of eigenvalues and eigenvectors

Degenerate eigenvalues

8.15 Change of basis and similarity transformations

8.16 Diagonalisation of matrices

8.17 Quadratic and Hermitian forms

Stationary properties of the eigenvectors ; quadratic surfaces

8.18 Simultaneous linear equations

Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition

8.19 Exercises

8.20 Hintsand answers

9  Normal modes

9.1  Typical oscillatory systems

9.2  Symmetry and normal modes

9.3  Rayleigh-Ritz method

9.4  Exercises

9.5  Hints and answers

10  Vector calculus

10.1 Differentiation of vectors

Composite vector expressions; differential of a vector

10.2 Integration of vectors

10.3 Space curves

10.4 Vector functions of several arguments

10.5 Surfaces

10.6 Scalar and vector fields

10.7 Vector operators

Gradient of a scalar field: divergence of a vector field: curl of a vector field

10.8 Vector operator formulae

Vector operators acting on sums and products; combinations of grad, div and curl

10.9 Cylindrical and spherical polar coordinates

10.10 General curvilinear coordinates

10.11 Exercises

10.12 Hints and answers

11  Line, surface and volume integrals

11.1 Line integrals

Evaluating line integrals; physical examples; line integrals with respect to a scalar

11.2 Connectivity of regions

11.3 Green's theorem in a plane

11.4 Conservative fields and potentials

11.5 Surface integrals

Evaluating surface integrals; vector areas of surfaces; physical examples

11.6 Volume integrals

Volumes of three-dimensional regions

11.7 Integral forms for grad, div and curl

11.8 Divergence theorem and related theorems

Green's theorems; other related integral theorems; physical applications

11.9 Stokes' theorem and related theorems

Related integral theorems: physical applications

11.10 Exercises

11.11 Hints and answers

12  Fourier series

12.1 The Dirichlet conditions

12.2 The Fourier coefficients

12.3 Symmetry considerations

12.4 Discontinuous functions

12.5 Non-periodic functions

12.6 Integration and differentiation

12.7 Complex Fourier series

12.8 Parseval's theorem

12.9 Exercises

12.10 Hints and answers

13  Integral transforms

13.1 Fourier transforms

The uncertainty principle; Fraunhofer diffraction: the Dirac &-function: relation of the 6-function to Fourier transforms; properties of Fourier transJorms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; Parseval's theorem; Fourier transforms in higher dimensions

13.2 Laplace transforms

Laplace transforms of derivatives and integrals; other properties of Laplace transforms

13.3 Concluding remarks

13.4 Exercises

13.5 Hints and answers

14  First-order ordinary differential equations

14.1 General form of solution

14.2 First-degree first-order equations

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

Equations soluble for p; for x; for y; Clairaut's equation

14.4 Exercises

14.5 Hints and answers

15  Higher-order ordinary differential equations

15.1 Linear equations with constant coefficients

Finding the complementary function yc(x): finding the particular integral yp(x); constructing the general solution ye(x)+ yp(x): linear recurrence relations: Laplace transform method

15.2 Linear equations with variable coefficients

The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations

15.3 General ordinary differential equations

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

15.5 Hints and answers

16  Series solutions of ordinary differential equations

16.1 Second-order linear ordinary differential equations

Ordinary and singular points

16.2 Series solutions about an ordinary point

16.3 Series solutions about a regular singular point

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

The Wronskian method; the derivative method; series form of the second solution

16.5 Polynomial solutions

16.6 Legendre's equation

General solution for integer 1 ; properties of Legendre polynomials

16.7 Bessers equation

General solution for non-integer v; general solution for integer v; properties of Bessel functions

16.8 General remarks

16.9 Exercises

16.10 Hints and answers

17  Eigenfunction methods for differential equations

17.1 Sets of functions

Some useful inequalities

17.2 Adjoint and Hermitian operators

17.3 The properties of Hermitian operators

Reality of the eigenvalues; orthogonality of the eigenfunctions; construction of real eigenfunctions

17.4 Sturm-Liouville equations

Valid boundary conditions; putting an equation into Sturm-Liouville form

17.5 Examples of Sturm-Liouville equations

Legendre's equation; the associated Legendre equation; Bessel's equation; the simple harmonic equation; Hermite's equation; Laguerre's equation; Chebyshev's equation

17.6 Superposition of eigenfunctions: Green's functions

17.7 A useful generalisation

17.8 Exercises

17.9 Hints and answers

18  Partial differential equations: general and particular solutions

18.1 Important partial differential equations

The wave equation; the diffusion equation; Laplace's equation; Poisson's equation; SchrOdinger's equation

18.2 General form of solution

18.3 General and particular solutions

First-order equations; inhomogeneous equations and problems; second-order equations

18.4 The wave equation

18.5 The diffusion equation

18.6 Characteristics and the existence of solutions

First-order equations; second-order equations

18.7 Uniqueness of solutions

18.8 Exercises

18.9 Hints and answers

19  Partial differential equations: separation of variables and other methods

19.1 Separation of variables: the general method

19.2 Superposition of separated solutions

19.3 Separation of variables in polar coordinates

Laplace's equation in polar coordinates: spherical harmonics: other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations

19.4 Integral transform methods

19.5 Inhomogeneous problems-Green's functions

Similarities to Green's functions for ordinary differential equations: general boundary-value problems: Dirichlet problems; Neumann problems

19.6 Exercises

19.7 Hints and answers

20  Complex variables

20.1 Functions of a complex variable

20.2 The Cauchy-Riemann relations

20.3 Power series in a complex variable

20.4 Some elementary functions

20.5 Multivalued functions and branch cuts

20.6 Singularities and zeroes of complex functions

20.7 Complex potentials

20.8 Conformal transformations

20.9 Applications ofconformal transformations

20.10 Complex integrals

20.11 Cauchy's theorem

20.12 Cauchy's integral formula

20.13 Taylor and Laurent series

20.14 Residue theorem

20.15 Location of zeroes

20.16 Integrals of sinusoidal functions

20.17 Some infinite integrals

20.18 Integrals of multivalued functions

20.19 Summation of series

20.20 Inverse Laplace transform

20.21 Exercises

20.22 Hints and answers

21  Tensors

21.1 Some notation

21.2 Change of basis

21.3 Cartesian tensors

21.4 First- and zero-order Cartesian tensors

21.5 Second- and higher-order Cartesian tensors

21.6 The algebra of tensors

21.7 The quotient law

21.8 The tensors and

21.9 Isotropic tensors

21.10 Improper rotations and pseudotensors

21.11 Dual tensors

21.t2 Physical applications of tensors

21.13 Integral theorems for tensors

21.14 Non-Cartesian coordinates

21.15 The metric tensor

21.16 General coordinate transformations and tensors

21.17 Relative tensors

21.18 Derivatives of basis vectors and Christoffel symbols

21.19 Covariant differentiation

21.20 Vector operators in tensor form

21.21 Absolute derivatives along curves

21.22 Geodesics

21.23 Exercises

21.24 Hints and answers

22  Calculus of variations

22.1 The Euler-Lagrange equation

22.2 Special cases

F does not contain y explicitly; F does not contain x explicitly

22.3 Some extensions

Several dependent variables; several independent variables; higher-order derivatives: variable end-points

22.4 Constrained variation

22.5 Physical variational principles

Fermat's principle in optics; Hamilton's principle in mechanics

22.6 General eigenvalue problems

22.7 Estimation ofeigenvalues and eigenfunctions

22.8 Adjustment of parameters

22.9 Exercises

22.10 Hints and answers

23  Integral equations

23.1 Obtaining an integral equation from a differential equation

23.2 Types of integral equation

23.3 Operator notation and the existence of solutions

23.4 Closed-form solutions

Separable kernels; integral transform methods; differentiation

23.5 Neumann series

23.6 Fredholm theory

23.7 Schmidt-Hilbert theory

23.8 Exercises

23.9 Hints and answers

24  Group theory

24.1 Groups

Definition of a group; examples of groups

24.2 Finite groups

24.3 Non-Abelian groups

24.4 Permutation groups

24.5 Mappings between groups

24.6 Subgroups

24.7 Subdividing a group

Equivalence relations and classes; congruence and cosets; conjugates and classes

24.8 Exercises

24.9 Hints and answers

25  Representation theory

25.1 Dipole moments of molecules

25.2 Choosing an appropriate formalism

25.3 Equivalent representations

25.4 Reducibility of a representation

25.5 The orthogonality theorem for irreducible representations

25.6 Characters

Orthogonality property of characters

25.7 Counting irreps using characters

Summation rules for irreps

25.8 Construction of a character table

25.9 Group nomenclature

25.10 Product representations

25.11 Physical applications of group theory

Bonding in molecules: matrix elements in quantum mechanics: degeneracy of normal modes: breaking of degeneracies

25.12 Exercises

25.13 Hints and answers

26  Probability

26.1 Venn diagrams

26.2 Probability

Axioms and theorems; conditional probability; Bayes' theorem

26.3 Permutations and combinations

26.4 Random variables and distributions

Discrete random variables; continuous random variables

26.5 Properties of distributions

Mean: mode and median: variance and standard deviation: moments:

central moments

26.6 Functions of random variables

2617 Generating functions

Probability generating functions; moment generating functions; characteristic functions; cumulant generating functions

26.8 Important discrete distributions

Binomial; geometric; negative binomial; hypergeometric ; Poisson

26.9 Important continuous distributions

Gaussian : log-normah exponential; gamma; chi-squared; Cauchy ; BreitWigner : uniform

26.10 The central limit theorem

26.11 Joint distributions

Discrete bivariate ; continuous bivariate ; marginal and conditional distributions

26.12 Properties of joint distributions

Means; variances; covariance and correlation

26.13 Generating functions for joint distributions

26.14 Transformation of variables in joint distributions

26.15 Important joint distributions

Multinominah multivariate Gaussian

26.16 Exercises

26.17 Hints and answers

27  Statistics

27.1 Experiments, samples and populations

27.2 Sample statistics

Averages; variance and standard deviation; moments; covariance and correlation

27.3 Estimators and sampling distributions

Consistency, bias and efficiency; Fisher's inequality: standard errors; confidence limits

27.4 Some basic estimators

Mean; variance: standard deviation; moments; covariance and correlation

27.5 Maximum-likelihood method

ML estimator; trans]ormation invariance and bias; efficiency; errors and confidence limits; Bayesian interpretation; large-N behaviour; extended ML method

27.6 The method of least squares

Linear least squares; non-linear least squares

27.7 Hypothesis testing

Simple and composite hypotheses; statistical tests; Neyman-Pearson; generalised likelihood-ratio: Student's t: Fisher's F: goodness of fit

27.8 Exercises

27.9 Hints and answers

28  Numerical methods

28.1 Algebraic and transcendental equations

Rearrangement of the equation; linear interpolation; binary chopping; Newton-Raphson method

28.2 Convergence of iteration schemes

28.3 Simultaneous linear equations

Gaussian elimination; Gauss-Seidel iteration; tridiagonal matrices

28.4 Numerical integration

Trapezium rule; Simpson's rule; Gaussian integration; Monte Carlo methods

28.5 Finite differences

28.6 Differential equations

Difference equations; Taylor series solutions; prediction and correction; Runge-Kutta methods; isoclines

28.7 Higher-order equations

28.8 Partial differential equations

28.9 Exercises

28.10 Hints and answers

Appendix Gamma, beta and error functions

A1.1 The gamma function

Al.2 The beta function

A1.3 The error function

Index