Cambridge University Press Foundation Mathematics for the Physical Sciences 1st edition by Riley, Hobson 0521192730 9780521192736

1 Arithmetic and geometry

1.1 Powers

1.2 Exponential and logarithmic functions

1.2.1 Logarithms

1.2.2 The exponential function and choice of logarithmic base

1.2.3 The use of logarithms

1.3 Physical dimensions

1.4 The binomial expansion

1.4.1 Binomial coefficients

1.4.2 Proof of the binomial expansion

1.4.3 Negative and non-integral values of n

1.4.4 Relationship with the exponential function

1.5 Trigonometric identities

1.5.1 Compound-angle identities

1.5.2 Double- and half-angle identities

1.6 Inequalities

2 Preliminary algebra

2.1 Polynomials and polynomial equations

2.1.1 Example: the cubic case

2.1.2 A more general case

2.1.3 Factorising polynomials

2.1.4 Properties of roots

2.2 Coordinate geometry

2.2.1 Linear graphs

2.2.2 Conic sections

2.2.3 Parametric equations

2.2.4 Plane polar coordinates

2.3 Partial fractions

2.3.1 The general method

2.3.2 Complications and special cases

2.4 Some particular methods of proof

2.4.1 Proof by induction

2.4.2 Proof by contradiction

2.4.3 Necessary and sufficient conditions

Summary

Problems

Hints and answers

3 Differential calculus

3.1 Differentiation

3.1.1 Differentiation from first principles

3.1.2 Differentiation of products

3.1.3 The chain rule

3.1.4 Differentiation of quotients

3.1.5 Implicit differentiation

3.1.6 Logarithmic differentiation

3.2 Leibnitz’s theorem

3.3 Special points of a function

3.4 Curvature of a function

3.5 Theorems of differentiation

3.5.1 Rolle’s theorem

3.5.2 Mean value theorem

3.5.3 Applications of Rolle’s theorem and the mean value theorem

3.6 Graphs

3.6.1 General considerations

3.6.2 Worked examples

Summary

Problems

Hints and answers

4 Integral calculus

4.1 Integration

4.1.1 Integration from first principles

4.1.2 Integration as the inverse of differentiation

4.2 Integration methods

4.2.1 Integration by inspection

4.2.2 Integration of sinusoidal functions

4.2.3 Logarithmic integration

4.2.4 Integration using partial fractions

4.2.5 Integration by substitution

4.3 Integration by parts

4.4 Reduction formulae

4.5 Infinite and improper integrals

4.6 Integration in plane polar coordinates

4.7 Integral inequalities

4.8 Applications of integration

4.8.1 Mean value of a function

4.8.2 Finding the length of a curve

4.8.3 Surfaces of revolution

4.8.4 Volumes of revolution

Summary

Problems

Hints and answers

5 Complex numbers and hyperbolic functions

5.1 The need for complex numbers

5.2 Manipulation of complex numbers

5.2.1 Addition and subtraction

5.2.2 Modulus and argument

5.2.3 Multiplication

5.2.4 Complex conjugate

5.2.5 Division

5.3 Polar representation of complex numbers

5.3.1 Multiplication and division in polar form

5.4 De Moivre’s theorem

5.4.1 Trigonometric identities

5.4.2 Finding the nth roots of unity

5.4.3 Solving polynomial equations

5.5 Complex logarithms and complex powers

5.6 Applications to differentiation and integration

5.7 Hyperbolic functions

5.7.1 Definitions

5.7.2 Hyperbolic–trigonometric analogies

5.7.3 Identities of hyperbolic functions

5.7.4 Solving hyperbolic equations

5.7.5 Inverses of hyperbolic functions

5.7.6 Calculus of hyperbolic functions

Summary

Problems

Hints and answers

6 Series and limits

6.1 Series

6.2 Summation of series

6.2.1 Arithmetic series

6.2.2 Geometric series

6.2.3 Arithmetico-geometric series

6.2.4 The difference method

6.2.5 Series involving natural numbers

6.2.6 Transformation of series

6.3 Convergence of infinite series

6.3.1 Absolute and conditional convergence

6.3.2 Convergence of a series containing only real positive terms

6.3.3 Alternating series test

6.4 Operations with series

6.5 Power series

6.5.1 Convergence of power series

6.5.2 Operations with power series

6.6 Taylor series

6.6.1 Taylor’s theorem

6.6.2 Approximation errors in Taylor series

6.6.3 Standard Maclaurin series

6.7 Evaluation of limits

Summary

Problems

Hints and answers

7 Partial differentiation

7.1 Definition of the partial derivative

7.2 The total differential and total derivative

7.3 Exact and inexact differentials

7.4 Useful theorems of partial differentiation

7.5 The chain rule

7.6 Change of variables

7.7 Taylor’s theorem for many-variable functions

7.8 Stationary values of two-variable functions

7.9 Stationary values under constraints

7.10 Envelopes

7.11 Thermodynamic relations

7.12 Differentiation of integrals

Summary

Problems

Hints and answers

8 Multiple integrals

8.1 Double integrals

8.2 Applications of multiple integrals

8.2.1 Areas and volumes

8.2.2 Masses, centres of mass and centroids

8.2.3 Pappus’s theorems

8.2.4 Moments of inertia

8.2.5 Mean values of functions

8.3 Change of variables in multiple integrals

8.3.1 Change of variables in double integrals

8.3.2 Evaluation of the integral…

8.3.3 Change of variables in triple integrals

8.3.4 General properties of Jacobians

Summary

Problems

Hints and answers

9 Vector algebra

9.1 Scalars and vectors

9.2 Addition, subtraction and multiplication of vectors

9.3 Basis vectors, components and magnitudes

9.4 Multiplication of two vectors

9.4.1 Scalar product

9.4.2 Vector product

9.5 Triple products

9.5.1 Scalar triple product

9.5.2 Vector triple product

9.6 Equations of lines, planes and spheres

9.6.1 Equation of a line

9.6.2 Equation of a plane

9.6.3 Equation of a sphere

9.7 Using vectors to find distances

9.7.1 Distance from a point to a line

9.7.2 Distance from a point to a plane

9.7.3 Distance from a line to a line

9.7.4 Distance from a line to a plane

9.8 Reciprocal vectors

Summary

Problems

Hints and answers

10 Matrices and vector spaces

10.1 Vector spaces

10.1.1 Basis vectors

10.1.2 The inner product

10.1.3 Some useful inequalities

10.2 Linear operators

10.3 Matrices

10.4 Basic matrix algebra

10.4.1 Matrix addition and multiplication by a scalar

10.4.2 Multiplication of matrices

10.4.3 The null and identity matrices

10.4.4 Functions of matrices

10.5 The transpose and conjugates of a matrix

10.5.1 The complex and Hermitian conjugates

10.6 The trace of a matrix

10.7 The determinant of a matrix

10.7.1 Properties of determinants

10.7.2 Evaluation of determinants

10.8 The inverse of a matrix

10.9 The rank of a matrix

10.10 Simultaneous linear equations

10.10.1 The number of solutions

10.10.2 N simultaneous linear equations in N unknowns

10.10.3 A geometrical interpretation

10.11 Special types of square matrix

10.11.1 Diagonal matrices

10.11.2 Lower and upper triangular matrices

10.11.3 Symmetric and antisymmetric matrices

10.11.4 Orthogonal matrices

10.11.5 Hermitian and anti-Hermitian matrices

10.11.6 Unitary matrices

10.11.7 Normal matrices

10.12 Eigenvectors and eigenvalues

10.12.1 Eigenvectors and eigenvalues of Hermitian and unitary matrices

10.12.2 Eigenvectors and eigenvalues of a general square matrix

10.12.3 Simultaneous eigenvectors

10.13 Determination of eigenvalues and eigenvectors

10.14 Change of basis and similarity transformations

10.15 Diagonalisation of matrices

10.16 Quadratic and Hermitian forms

10.16.1 The stationary properties of the eigenvectors

10.16.2 Quadratic surfaces

10.17 The summation convention

Summary

Problems

Hints and answers

11 Vector calculus

11.1 Differentiation of vectors

11.1.1 Differentiation of composite vector expressions

11.1.2 Differential of a vector

11.2 Integration of vectors

11.3 Vector functions of several arguments

11.4 Surfaces

11.5 Scalar and vector fields

11.6 Vector operators

11.6.1 Gradient of a scalar field

11.6.2 Divergence of a vector field

11.6.3 Curl of a vector field

11.7 Vector operator formulae

11.7.1 Vector operators acting on sums and products

11.7.2 Combinations of grad, div and curl

11.8 Cylindrical and spherical polar coordinates

11.8.1 Cylindrical polar coordinates

11.8.2 Spherical polar coordinates

11.9 General curvilinear coordinates

Summary

Problems

Hints and answers

12 Line, surface and volume integrals

12.1 Line integrals

12.1.1 Evaluating line integrals

12.1.2 Physical examples of line integrals

12.1.3 Line integrals with respect to a scalar

12.2 Connectivity of regions

12.3 Green’s theorem in a plane

12.4 Conservative fields and potentials

12.5 Surface integrals

12.5.1 Evaluating surface integrals

12.5.2 Vector areas of surfaces

12.5.3 Physical examples of surface integrals

12.6 Volume integrals

12.7 Integral forms for grad, div and curl

12.8 Divergence theorem and related theorems

12.8.1 Green’s theorems

12.8.2 Other related integral theorems

12.8.3 Physical applications of the divergence theorem

12.9 Stokes’ theorem and related theorems

12.9.1 Related integral theorems

12.9.2 Physical applications of Stokes’ theorem

Summary

Problems

Hints and answers

13 Laplace transforms

13.1 Laplace transforms

13.2 The Dirac delta-function and Heaviside step function

13.3 Laplace transforms of derivatives and integrals

13.4 Other properties of Laplace transforms

Summary

Problems

Hints and answers

14 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 Bernoulli’s equation

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

14.4 Higher order linear ODEs

14.5 Linear equations with constant coefficients

14.5.1 Finding the complementary function yc(x)

14.5.2 Finding the particular integral yp(x)

14.5.3 Constructing the general solution yc(x)+yp(x)

14.5.4 Laplace transform method

14.6 Linear recurrence relations

14.6.1 First-order recurrence relations

14.6.2 Second-order recurrence relations

14.6.3 Higher order recurrence relations

Summary

Problems

Hints and answers

15 Elementary probability

15.1 Venn diagrams

15.2 Probability

15.2.1 Axioms and theorems

15.2.2 Conditional probability

15.2.3 Bayes’ theorem

15.3 Permutations and combinations

15.3.1 Permutations

15.3.2 Combinations

15.4 Random variables and distributions

15.4.1 Discrete random variables

15.4.2 Continuous random variables

15.4.3 Sets of random variables

15.5 Properties of distributions

15.5.1 Mean

15.5.2 Mode and median

15.5.3 Variance and standard deviation

15.5.4 Moments

15.6 Functions of random variables

15.6.1 Continuous random variables

15.6.2 Expectation values and variances

15.7 Important discrete distributions

15.7.1 The binomial distribution

15.7.2 The multinomial distribution

15.7.3 The geometric and negative binomial distributions

15.7.4 The hypergeometric distribution

15.7.5 The Poisson distribution

15.8 Important continuous distributions

15.8.1 The uniform distribution

15.8.2 The Cauchy and Breit–Wigner distributions

15.8.3 The Gaussian distribution

15.8.4 The exponential distribution

15.8.5 The chi-squared distribution

15.9 Joint distributions

15.9.1 Bivariate distributions

15.9.2 Properties of joint distributions

15.9.3 Means

15.9.4 Variances

15.9.5 Covariance and correlation