Mathematical methods for physics and engineering / K.F. Riley, M.P. Hobson and S.J. Bence.

CONTENT


1 . Preliminary algebra

1.1 Simple functions and equations
1.2 Trigonometric identities
1.3 Coordinate geometry
1.4 Partial fractions
1.5 Binomial expansion

2. Preliminary calculus

2.1 Differentiation
2.2 Integration
2.3 Exercises
2.4 Hints and answers

3. Complex numbers and hyperbolic functions

3.1 The need for complex numbers
3.2 Manuipulation of complex numbers
3.3 Polar representation of complex numbers
3.4 de Moivre's theorem
3.5 Complex logarithms and complex powers
etc

4. Serires and Limits

4.1 Series
4.2 Summation of series
4.3 Convergence of infinite series
4.4 Operations with series
4.5 Power series
etc

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
etc

6. Multiple integrals
6.1 Double integrals
6.2 Triple integrals
6.3 Applications of multiple integrals
6.4 Change of variables in multiple integrals
6.5 Exercise
etc

7 Vector algebra

7.1 Scalars and vectors
7.2 Addition and subtraction of vectors
7.3 Multiplication by a Scalar
7.4 Basic vector and components
7.5 Magnitude of a vector
etc

8 Matrices and vector space

8.1 Vector space
8.2 Linear spaces
8.3 Matrices
8.4 Basic matrix algebra
8.5 Function of matrices
etc

9 Normal modes
9.1 Typical oscillatrory 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
10.2 Integration of vectors
10.3 Space curves
10.4 Vector functions of several arguments
10.5 Surfaces
etc

11 Line, surface and volume integrals

11.1 Line integrals
11.2 Connectivity of regions
11.3 Green's theorem in a plane
11.4 Conservative fields and potentials
11.5 Surface integrals
etc

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
etc

13 Integral transforms

13.1 Fourier transforms
13.2 Laplace transforms
13.3 Concluding remarks
13.4 Exercise
13.5 Hints and answers
etc

14 First-order ordinary differential equations

14.1 General form of soluton
14.2 First- degree first order eqyations
14.3 Higher -degree first order equations
14.4 Exercises
14.5 Hints and answers

15 Higher-order ordinary differential equations

15.1 Linear equations with constant coefficients
15.2 Linear equations with variable coefficients
15.3 General ordinary differential equations
15.4 Exercises
15.5 Hints and answers

16 Series solutions of ordinary differential equations

16.1 Second- order linear ordinarydifferential equations
16.2 Series solutions about an ordinary point
16.3 Series solutions about a regular singular point
16.4 Obtaining a second solution
16.5 Polynomial solutions
etc

17 Eigenfunction methods for differential equations

17.1 Sets of functions
17.2 Adjoint, self-adjoint and hermitian operators
17.3 Properties of hermitian operators
17.4 Sturm- Liouville equations
17.5 Superposition of eigenfunctions:Green's functions
etc

18 Special functions

18.1 Legendre functions
18.2 Associated Legendre functions
18.3 Spherical harmonics
18.4 Chebyshev functions
18.5 Bessel functions
etc

19 Quantum operators

19.1 Operator formalism
19.2 Physical examples of operators
19.3 Exercise
19.4 Hints and answers

20 Partial differentail equations : general and particular solutions

20.1 Important partial differential equations
20.2 General for of solutions
20.3 General and particular solutions
20.4 The wave equation
20.5 The diffusion equation
etc

21 Partial and differential equations : seperation 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
21.4 Integral transform methods
21.5 Inhomogeneous problems
etc

22 Calculus of variations

22.1 The Euler- Language equation
22.2 Special cases
22.3 Some extentions
22.4 Constrained varation
22.5 Physical variational principles
etc

23. Integral equations

23.1 Obtaining an integral equation from a differential equation
23.2 Types of integral equation
23.3 Operator notation and existence of solutions
23.4 Closed-form solution
23.5 Neumann series
etc

24 Complex variables

24.1 Function of complex variables
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
etc

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 Lpaplace transform
etc

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
etc

27 Numerical methods

27.1 Algebraic and transcendental equations
27.2 Convergence of iteration schemes
27.3 Simultaneous linear equations
27.4 Numerical integration
27.5 Finite differences
etc

28 Group theory

28.1 Groups
28.2 Finite groups
28.3 Non- Abelian groups
28.4 Permutation groups
28.5 Mappings between groups
etc

29 Representation theory
29.1 Dipole moments of molecules
29.2 Choosing an appropriate formalism
29.3 Equivalent representations
29.4 Reducibility of a representation
29.5 The orthogonality theorem for irreducible representations
etc

30 Probability

30.1 Venn diagram
30.2 Probability
30.3 Permutations and combinations
30.4 Random variables and distributions
30.5 Properties distribution
etc

31 Statistics

31.1 Experiments, samples and populations
31.2 Sample statistics
31.3 Estimators and sampling distribution
31.4 Some basic estimators
31.5 Maximum- likelihood methhod

Includes index : p.1305 - 1333