Mathematical Methods for Physics and Engineering 3rd Edition by Riley, Hobson, Bence 0521679710 9780521679718

1 Preliminary algebra

1.1 Simple functions and equations

1.1.1 Polynomials and polynomial equations

1.1.2 Factorising polynomials

1.1.3 Properties of roots

1.2 Trigonometric identities

1.2.1 Single-angle identities

1.2.2 Compound-angle identities

1.2.3 Double- and half-angle identities

1.3 Coordinate geometry

1.4 Partial fractions

1.4.1 Complications and special cases

The degree of the numerator is greater than or equal to that of the denominator

Factors of the form a + x in the denominator

Repeated factors in the denominator

1.5 Binomial expansion

1.5.1 Binomial coefficients

1.5.2 Proof of the binomial expansion

1.6 Properties of binomial coefficients

1.6.1 Identities involving binomial coefficients

1.6.2 Negative and non-integral values of n

1.7 Some particular methods of proof

1.7.1 Proof by induction

1.7.2 Proof by contradiction

1.7.3 Necessary and sufficient conditions

1.8 Exercises

Polynomial equations

Trigonometric identities

Coordinate geometry

Partial fractions

Binomial expansion

Proof by induction and contradiction

Necessary and sufficient conditions

1.9 Hints and answers

2 Preliminary calculus

2.1 Differentiation

2.1.1 Differentiation from first principles

2.1.2 Differentiation of products

2.1.3 The chain rule

2.1.4 Differentiation of quotients

2.1.5 Implicit differentiation

2.1.6 Logarithmic differentiation

2.1.7 Leibnitz’ theorem

2.1.8 Special points of a function

2.1.9 Curvature of a function

2.1.10 Theorems of differentiation

Rolle’s theorem

Mean value theorem

Applications of Rolle’s theorem and the mean value theorem

2.2 Integration

2.2.1 Integration from first principles

2.2.2 Integration as the inverse of differentiation

2.2.3 Integration by inspection

2.2.4 Integration of sinusoidal functions

2.2.5 Logarithmic integration

2.2.6 Integration using partial fractions

2.2.7 Integration by substitution

2.2.8 Integration by parts

2.2.9 Reduction formulae

2.2.10 Infinite and improper integrals

2.2.11 Integration in plane polar coordinates

2.2.12 Integral inequalities

2.2.13 Applications of integration

Mean value of a function

Finding the length of a curve

Surfaces of revolution

Volumes of revolution

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

3.2.1 Addition and subtraction

3.2.2 Modulus and argument

3.2.3 Multiplication

3.2.4 Complex conjugate

3.2.5 Division

3.3 Polar representation of complex numbers

3.3.1 Multiplication and division in polar form

3.4 de Moivre’s theorem

3.4.1 Trigonometric identities

3.4.2 Finding the nth roots of unity

3.4.3 Solving polynomial equations

3.5 Complex logarithms and complex powers

3.6 Applications to differentiation and integration

3.7 Hyperbolic functions

3.7.1 Definitions

3.7.2 Hyperbolic–trigonometric analogies

3.7.3 Identities of hyperbolic functions

3.7.4 Solving hyperbolic equations

3.7.5 Inverses of hyperbolic functions

3.7.6 Calculus of hyperbolic functions

3.8 Exercises

3.9 Hints and answers

4 Series and limits

4.1 Series

4.2 Summation of series

4.2.1 Arithmetic series

4.2.2 Geometric series

4.2.3 Arithmetico-geometric series

4.2.4 The difference method

4.2.5 Series involving natural numbers

4.2.6 Transformation of series

4.3 Convergence of infinite series

4.3.1 Absolute and conditional convergence

4.3.2 Convergence of a series containing only real positive terms

Preliminary test

Comparison test

D’Alembert’s ratio test

Ratio comparison test

Quotient test

Integral test

Cauchy’s root test

Grouping terms

4.3.3 Alternating series test

4.4 Operations with series

4.5 Power series

4.5.1 Convergence of power series

4.5.2 Operations with power series

4.6 Taylor series

4.6.1 Taylor’s theorem

4.6.2 Approximation errors in Taylor series

4.6.3 Standard Maclaurin series

4.7 Evaluation of limits

Summary of methods for evaluating 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

5.10.1 Envelope equations

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

6.3.1 Areas and volumes

6.3.2 Masses, centres of mass and centroids

6.3.3 Pappus’ theorems

6.3.4 Moments of inertia

6.3.5 Mean values of functions

6.4 Change of variables in multiple integrals

6.4.1 Change of variables in double integrals

6.4.2 Evaluation of the integral I…

6.4.3 Change of variables in triple integrals

6.4.4 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

7.6.1 Scalar product

7.6.2 Vector product

7.6.3 Scalar triple product

7.6.4 Vector triple product

7.7 Equations of lines, planes and spheres

7.7.1 Equation of a line

7.7.2 Equation of a plane

7.7.3 Equation of a sphere

7.8 Using vectors to find distances

7.8.1 Distance from a point to a line

7.8.2 Distance from a point to a plane

7.8.3 Distance from a line to a line

7.8.4 Distance from a line to a plane

7.9 Reciprocal vectors

7.10 Exercises

7.11 Hints and answers

8 Matrices and vector spaces

8.1 Vector spaces

8.1.1 Basis vectors

8.1.2 The inner product

8.1.3 Some useful inequalities

8.2 Linear operators

8.2.1 Properties of linear operators

8.3 Matrices

8.4 Basic matrix algebra

8.4.1 Matrix addition and multiplication by a scalar

8.4.2 Multiplication of matrices

8.4.3 The null and identity matrices

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

8.9.1 Properties of determinants

8.10 The inverse of a matrix

8.11 The rank of a matrix

8.12 Special types of square matrix

8.12.1 Diagonal matrices

8.12.2 Lower and upper triangular matrices

8.12.3 Symmetric and antisymmetric matrices

8.12.4 Orthogonal matrices

8.12.5 Hermitian and anti-Hermitian matrices

8.12.6 Unitary matrices

8.12.7 Normal matrices

8.13 Eigenvectors and eigenvalues

8.13.1 Eigenvectors and eigenvalues of a normal matrix

8.13.2 Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices

8.13.3 Eigenvectors and eigenvalues of a unitary matrix

8.13.4 Eigenvectors and eigenvalues of a general square matrix

8.13.5 Simultaneous eigenvectors

8.14 Determination of eigenvalues and eigenvectors

8.14.1 Degenerate eigenvalues

8.15 Change of basis and similarity transformations

8.16 Diagonalisation of matrices

8.17 Quadratic and Hermitian forms

8.17.1 The stationary properties of the eigenvectors

8.17.2 Quadratic surfaces

8.18 Simultaneous linear equations

8.18.1 The range and null space of a matrix

8.18.2 N simultaneous linear equations in N unknowns

8.18.3 Singular value decomposition

8.19 Exercises

8.20 Hints and 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

10.1.1 Differentiation of composite vector expressions

10.1.2 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

10.7.1 Gradient of a scalar field

10.7.2 Divergence of a vector field

10.7.3 Curl of a vector field

10.8 Vector operator formulae

10.8.1 Vector operators acting on sums and products

10.8.2 Combinations of grad, div and curl

10.9 Cylindrical and spherical polar coordinates

10.9.1 Cylindrical polar coordinates

10.9.2 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

11.1.1 Evaluating line integrals

11.1.2 Physical examples of line integrals

11.1.3 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

11.5.1 Evaluating surface integrals

11.5.2 Vector areas of surfaces

11.5.3 Physical examples of surface integrals

11.6 Volume integrals

11.6.1 Volumes of three-dimensional regions

11.7 Integral forms for grad, div and curl

11.8 Divergence theorem and related theorems

11.8.1 Green’s theorems

11.8.2 Other related integral theorems

11.8.3 Physical applications of the divergence theorem

11.9 Stokes’ theorem and related theorems

11.9.1 Related integral theorems

11.9.2 Physical applications of Stokes’ theorem

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

13.1.1 The uncertainty principle

13.1.2 Fraunhofer diffraction

13.1.3 The Dirac delta-function

13.1.4 Relation of the delta-function to Fourier transforms

13.1.5 Properties of Fourier transforms

13.1.6 Odd and even functions

13.1.7 Convolution and deconvolution

13.1.8 Correlation functions and energy spectra

13.1.9 Parseval’s theorem

13.1.10 Fourier transforms in higher dimensions

13.2 Laplace transforms

13.2.1 Laplace transforms of derivatives and integrals

13.2.2 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

14.2.1 Separable-variable equations

14.2.2 Exact equations

14.2.3 Inexact equations: integrating factors

14.2.4 Linear equations

14.2.5 Homogeneous equations

14.2.6 Isobaric equations

14.2.7 Bernoulli’s equation

14.2.8 Miscellaneous equations

14.3 Higher-degree first-order equations

14.3.1 Equations soluble for p

14.3.2 Equations soluble for x

14.3.3 Equations soluble for y

14.3.4 Clairaut’s equation

14.4 Exercises

14.5 Hints and answers

15 Higher-order ordinary differential equations

15.1 Linear equations with constant coefficients

15.1.1 Finding the complementary function…

15.1.2 Finding the particular integral…

15.1.3 Constructing the general solution…

15.1.4 Linear recurrence relations

First-order recurrence relations

Second-order recurrence relations

Higher-order recurrence relations

15.1.5 Laplace transform method

15.2 Linear equations with variable coefficients

15.2.1 The Legendre and Euler linear equations

15.2.2 Exact equations

15.2.3 Partially known complementary function

15.2.4 Variation of parameters

15.2.5 Green’s functions

15.2.6 Canonical form for second-order equations

15.3 General ordinary differential equations

15.3.1 Dependent variable absent

15.3.2 Independent variable absent

15.3.3 Non-linear exact equations

15.3.4 Isobaric or homogeneous equations

15.3.5 Equations homogeneous in x or y alone

15.3.6 Equations having…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

16.1.1 Ordinary and singular points of an ODE

16.2 Series solutions about an ordinary point

16.3 Series solutions about a regular singular point

16.3.1 Distinct roots not differing by an integer

16.3.2 Repeated root of the indicial equation

16.3.3 Distinct roots differing by an integer

16.4 Obtaining a second solution

16.4.1 The Wronskian method

16.4.2 The derivative method

16.4.3 Series form of the second solution

16.5 Polynomial solutions

16.6 Exercises

16.7 Hints and answers

17 Eigenfunction methods for differential equations

17.1 Sets of functions

17.1.1 Some useful inequalities

17.2 Adjoint, self-adjoint and Hermitian operators

17.3 Properties of Hermitian operators

17.3.1 Reality of the eigenvalues

17.3.2 Orthogonality and normalisation of the eigenfunctions

17.3.3 Completeness of the eigenfunctions

17.3.4 Construction of real eigenfunctions

17.4 Sturm–Liouville equations

17.4.1 Hermitian nature of the Sturm–Liouville operator

17.4.2 Transforming an equation into Sturm–Liouville form

17.5 Superposition of eigenfunctions: Green’s functions

17.6 A useful generalisation

17.7 Exercises

17.8 Hints and answers

18 Special functions

18.1 Legendre functions

18.1.1 Legendre functions for integer l

18.1.2 Properties of Legendre polynomials

18.2 Associated Legendre functions

18.2.1 Associated Legendre functions for integer l

18.2.2 Properties of associated Legendre functions P…

18.3 Spherical harmonics

18.4 Chebyshev functions

18.4.1 Properties of Chebyshev polynomials

18.5 Bessel functions

18.5.1 Bessel functions for non-integer v

18.5.2 Bessel functions for integer v

18.6 Spherical Bessel functions

18.7 Laguerre functions

18.7.1 Properties of Laguerre polynomials

18.8 Associated Laguerre functions

18.8.1 Properties of associated Laguerre polynomials

18.9 Hermite functions

18.9.1 Properties of Hermite polynomials

18.10 Hypergeometric functions

18.10.1 Properties of hypergeometric functions

18.11 Confluent hypergeometric functions

18.12 The gamma function and related functions

18.12.1 The gamma function

18.12.2 The beta function

18.12.3 The incomplete gamma function

18.12.4 The error function

18.13 Exercises

18.14 Hints and answers

19 Quantum operators

19.1 Operator formalism

19.1.1 Commutation and commutators

19.2 Physical examples of operators

19.2.1 Angular momentum operators

Eigenvalues of the angular momentum operators

19.2.2 Uncertainty principles

19.2.3 Annihilation and creation operators

The energy spectrum of the simple harmonic oscillator

The normalisation of the eigenstates

19.3 Exercises

19.4 Hints and answers

20 Partial differential equations: general and particular solutions

20.1 Important partial differential equations

20.1.1 The wave equation

20.1.2 The diffusion equation

20.1.3 Laplace’s equation

20.1.4 Poisson’s equation

20.1.5 Schrodinger’s equation

20.2 General form of solution

20.3 General and particular solutions

20.3.1 First-order equations

20.3.2 Inhomogeneous equations and problems

20.3.3 Second-order equations

20.4 The wave equation

20.5 The diffusion equation

20.6 Characteristics and the existence of solutions

20.6.1 First-order equations

20.6.2 Second-order equations

20.7 Uniqueness of solutions

20.8 Exercises

20.9 Hints and answers

21 Partial differential equations: separation of variables and other methods

21.1 Separation of variables: the general method

21.2 Superposition of separated solutions

21.3 Separation of variables in polar coordinates

21.3.1 Laplace’s equation in polar coordinates

Laplace’s equation in plane polars

Laplace’s equation in cylindrical polars

Laplace’s equation in spherical polars

21.3.2 Other equations in polar coordinates

Helmholtz’s equation in plane polars

Helmholtz’s equation in cylindrical polars

Helmholtz’s equation in spherical polars

21.3.3 Solution by expansion

21.3.4 Separation of variables for inhomogeneous equations

21.4 Integral transform methods

21.5 Inhomogeneous problems – Green’s functions

21.5.1 Similarities to Green’s functions for ODEs

21.5.2 General boundary-value problems

21.5.3 Dirichlet problems

21.5.4 Neumann problems

21.6 Exercises

21.7 Hints and answers

22 Calculus of variations

22.1 The Euler–Lagrange equation

22.2 Special cases

22.2.1 F does not contain y explicitly

22.2.2 F does not contain x explicitly

22.3 Some extensions

22.3.1 Several dependent variables

22.3.2 Several independent variables

22.3.3 Higher-order derivatives

22.3.4 Variable end-points

22.4 Constrained variation

22.5 Physical variational principles

22.5.1 Fermat’s principle in optics

22.5.2 Hamilton’s principle in mechanics

22.6 General eigenvalue problems

22.7 Estimation of eigenvalues 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

23.4.1 Separable kernels

23.4.2 Integral transform methods

23.4.3 Differentiation

23.5 Neumann series

23.6 Fredholm theory

23.7 Schmidt–Hilbert theory

23.8 Exercises

23.9 Hints and answers

24 Complex variables

24.1 Functions of a complex variable

24.2 The Cauchy–Riemann relations

24.3 Power series in a complex variable

24.4 Some elementary functions

24.5 Multivalued functions and branch cuts

24.6 Singularities and zeros of complex functions

24.7 Conformal transformations

24.8 Complex integrals

24.9 Cauchy’s theorem

24.10 Cauchy’s integral formula

24.11 Taylor and Laurent series

24.12 Residue theorem

24.13 Definite integrals using contour integration

24.13.1 Integrals of sinusoidal functions

24.13.2 Some infinite integrals

24.13.3 Integrals of multivalued functions

24.14 Exercises

24.15 Hints and answers

25 Applications of complex variables

25.1 Complex potentials

25.2 Applications of conformal transformations

25.3 Location of zeros

25.4 Summation of series

25.5 Inverse Laplace transform

25.6 Stokes’ equation and Airy integrals

25.6.1 The solutions of Stokes’ equation

25.6.2 Series solution of Stokes’ equation

25.6.3 Contour integral solutions

25.7 WKB methods

25.7.1 Phase memory

25.7.2 Constructing the WKB solutions

25.7.3 Accuracy of the WKB solutions

25.7.4 The Stokes phenomenon

25.8 Approximations to integrals

25.8.1 Level lines and saddle points

25.8.2 Steepest descents method

25.8.3 Stationary phase method

25.9 Exercises

25.10 Hints and answers

26 Tensors

26.1 Some notation

26.2 Change of basis

26.3 Cartesian tensors

26.4 First- and zero-order Cartesian tensors

26.5 Second- and higher-order Cartesian tensors

26.6 The algebra of tensors

26.7 The quotient law

26.8 The tensors delta and epsilon

26.9 Isotropic tensors

26.10 Improper rotations and pseudotensors

26.11 Dual tensors

26.12 Physical applications of tensors

26.13 Integral theorems for tensors

26.14 Non-Cartesian coordinates

26.15 The metric tensor

26.16 General coordinate transformations and tensors

26.17 Relative tensors

26.18 Derivatives of basis vectors and Christoffel symbols

26.19 Covariant differentiation

26.20 Vector operators in tensor form