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Mathematical Methods for Physics and Engineering, Third Edition

Book Description

The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. New stand-alone chapters give a systematic account of the 'special functions' of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions. The remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site, www.cambridge.org/9780521679718.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface to the third edition
  6. Preface to the second edition
  7. Preface to the first edition
  8. 1 Preliminary algebra
    1. 1.1 Simple functions and equations
    2. 1.2 Trigonometric identities
    3. 1.3 Coordinate geometry
    4. 1.4 Partial fractions
    5. 1.5 Binomial expansion
    6. 1.6 Properties of binomial coefficients
    7. 1.7 Some particular methods of proof
    8. 1.8 Exercises
    9. 1.9 Hints and answers
  9. 2 Preliminary calculus
    1. 2.1 Differentiation
    2. 2.2 Integration
    3. 2.3 Exercises
    4. 2.4 Hints and answers
  10. 3 Complex numbers and hyperbolic functions
    1. 3.1 The need for complex numbers
    2. 3.2 Manipulation of complex numbers
    3. 3.3 Polar representation of complex numbers
    4. 3.4 de Moivre’s theorem
    5. 3.5 Complex logarithms and complex powers
    6. 3.6 Applications to differentiation and integration
    7. 3.7 Hyperbolic functions
    8. 3.8 Exercises
    9. 3.9 Hints and answers
  11. 4 Series and limits
    1. 4.1 Series
    2. 4.2 Summation of series
    3. 4.3 Convergence of infinite series
    4. 4.4 Operations with series
    5. 4.5 Power series
    6. 4.6 Taylor series
    7. 4.7 Evaluation of limits
    8. 4.8 Exercises
    9. 4.9 Hints and answers
  12. 5 Partial differentiation
    1. 5.1 Definition of the partial derivative
    2. 5.2 The total differential and total derivative
    3. 5.3 Exact and inexact differentials
    4. 5.4 Useful theorems of partial differentiation
    5. 5.5 The chain rule
    6. 5.6 Change of variables
    7. 5.7 Taylor’s theorem for many-variable functions
    8. 5.8 Stationary values of many-variable functions
    9. 5.9 Stationary values under constraints
    10. 5.10 Envelopes
    11. 5.11 Thermodynamic relations
    12. 5.12 Differentiation of integrals
    13. 5.13 Exercises
    14. 5.14 Hints and answers
  13. 6 Multiple integrals
    1. 6.1 Double integrals
    2. 6.2 Triple integrals
    3. 6.3 Applications of multiple integrals
    4. 6.4 Change of variables in multiple integrals
    5. 6.5 Exercises
    6. 6.6 Hints and answers
  14. 7 Vector algebra
    1. 7.1 Scalars and vectors
    2. 7.2 Addition and subtraction of vectors
    3. 7.3 Multiplication by a scalar
    4. 7.4 Basis vectors and components
    5. 7.5 Magnitude of a vector
    6. 7.6 Multiplication of vectors
    7. 7.7 Equations of lines, planes and spheres
    8. 7.8 Using vectors to find distances
    9. 7.9 Reciprocal vectors
    10. 7.10 Exercises
    11. 7.11 Hints and answers
  15. 8 Matrices and vector spaces
    1. 8.1 Vector spaces
    2. 8.2 Linear operators
    3. 8.3 Matrices
    4. 8.4 Basic matrix algebra
    5. 8.5 Functions of matrices
    6. 8.6 The transpose of a matrix
    7. 8.7 The complex and Hermitian conjugates of a matrix
    8. 8.8 The trace of a matrix
    9. 8.9 The determinant of a matrix
    10. 8.10 The inverse of a matrix
    11. 8.11 The rank of a matrix
    12. 8.12 Special types of square matrix
    13. 8.13 Eigenvectors and eigenvalues
    14. 8.14 Determination of eigenvalues and eigenvectors
    15. 8.15 Change of basis and similarity transformations
    16. 8.16 Diagonalisation of matrices
    17. 8.17 Quadratic and Hermitian forms
    18. 8.18 Simultaneous linear equations
    19. 8.19 Exercises
    20. 8.20 Hints and answers
  16. 9 Normal modes
    1. 9.1 Typical oscillatory systems
    2. 9.2 Symmetry and normal modes
    3. 9.3 Rayleigh-Ritz method
    4. 9.4 Exercises
    5. 9.5 Hints and answers
  17. 10 Vector calculus
    1. 10.1 Differentiation of vectors
    2. 10.2 Integration of vectors
    3. 10.3 Space curves
    4. 10.4 Vector functions of several arguments
    5. 10.5 Surfaces
    6. 10.6 Scalar and vector fields
    7. 10.7 Vector operators
    8. 10.8 Vector operator formulae
    9. 10.9 Cylindrical and spherical polar coordinates
    10. 10.10 General curvilinear coordinates
    11. 10.11 Exercises
    12. 10.12 Hints and answers
  18. 11 Line, surface and volume integrals
    1. 11.1 Line integrals
    2. 11.2 Connectivity of regions
    3. 11.3 Green’s theorem in a plane
    4. 11.4 Conservative fields and potentials
    5. 11.5 Surface integrals
    6. 11.6 Volume integrals
    7. 11.7 Integral forms for grad, div and curl
    8. 11.8 Divergence theorem and related theorems
    9. 11.9 Stokes’theorem and related theorems
    10. 11.10 Exercises
    11. 11.11 Hints and answers
  19. 12 Fourier series
    1. 12.1 The Dirichlet conditions
    2. 12.2 The Fourier coefficients
    3. 12.3 Symmetry considerations
    4. 12.4 Discontinuous functions
    5. 12.5 Non-periodic functions
    6. 12.6 Integration and differentiation
    7. 12.7 Complex Fourier series
    8. 12.8 Parseval’s theorem
    9. 12.9 Exercises
    10. 12.10 Hints and answers
  20. 13 Integral transforms
    1. 13.1 Fourier transforms
    2. 13.2 Laplace transforms
    3. 13.3 Concluding remarks
    4. 13.4 Exercises
    5. 13.5 Hints and answers
  21. 14 First-order ordinary differential equations
    1. 14.1 General form of solution
    2. 14.2 First-degree first-order equations
    3. 14.3 Higher-degree first-order equations
    4. 14.4 Exercises
    5. 14.5 Hints and answers
  22. 15 Higher-order ordinary differential equations
    1. 15.1 Linear equations with constant coefficients
    2. 15.2 Linear equations with variable coefficients
    3. 15.3 General ordinary differential equations
    4. 15.4 Exercises
    5. 15.5 Hints and answers
  23. 16 Series solutions of ordinary differential equations
    1. 16.1 Second-order linear ordinary differential equations
    2. 16.2 Series solutions about an ordinary point
    3. 16.3 Series solutions about a regular singular point
    4. 16.4 Obtaining a second solution
    5. 16.5 Polynomial solutions
    6. 16.6 Exercises
    7. 16.7 Hints and answers
  24. 17 Eigenfunction methods for differential equations
    1. 17.1 Sets of functions
    2. 17.2 Adjoint, self-adjoint and Hermitian operators
    3. 17.3 Properties of Hermitian operators
    4. 17.4 Sturm−Liouville equations
    5. 17.5 Superposition of eigenfunctions: Green’s functions
    6. 17.6 A useful generalisation
    7. 17.7 Exercises
    8. 17.8 Hints and answers
  25. 18 Special functions
    1. 18.1 Legendre functions
    2. 18.2 Associated Legendre functions
    3. 18.3 Spherical harmonics
    4. 18.4 Chebyshev functions
    5. 18.5 Bessel functions
    6. 18.6 Spherical Bessel functions
    7. 18.7 Laguerre functions
    8. 18.8 Associated Laguerre functions
    9. 18.9 Hermite functions
    10. 18.10 Hypergeometric functions
    11. 18.11 Confluent hypergeometric functions
    12. 18.12 The gamma function and related functions
    13. 18.13 Exercises
    14. 18.14 Hints and answers
  26. 19 Quantum operators
    1. 19.1 Operator formalism
    2. 19.2 Physical examples of operators
    3. 19.3 Exercises
    4. 19.4 Hints and answers
  27. 20 Partial differential equations: general and particular solutions
    1. 20.1 Important partial differential equations
    2. 20.2 General form of solution
    3. 20.3 General and particular solutions
    4. 20.4 The wave equation
    5. 20.5 The diffusion equation
    6. 20.6 Characteristics and the existence of solutions
    7. 20.7 Uniqueness of solutions
    8. 20.8 Exercises
    9. 20.9 Hints and answers
  28. 21 Partial differential equations: separation of variables and other methods
    1. 21.1 Separation of variables: the general method
    2. 21.2 Superposition of separated solutions
    3. 21.3 Separation of variables in polar coordinates
    4. 21.4 Integral transform methods
    5. 21.5 Inhomogeneous problems − Green’s functions
    6. 21.6 Exercises
    7. 21.7 Hints and answers
  29. 22 Calculus of variations
    1. 22.1 The Euler−Lagrange equation
    2. 22.2 Special cases
    3. 22.3 Some extensions
    4. 22.4 Constrained variation
    5. 22.5 Physical variational principles
    6. 22.6 General eigenvalue problems
    7. 22.7 Estimation of eigenvalues and eigenfunctions
    8. 22.8 Adjustment of parameters
    9. 22.9 Exercises
    10. 22.10 Hints and answers
  30. 23 Integral equations
    1. 23.1 Obtaining an integral equation from a differential equation
    2. 23.2 Types of integral equation
    3. 23.3 Operator notation and the existence of solutions
    4. 23.4 Closed-form solutions
    5. 23.5 Neumann series
    6. 23.6 Fredholm theory
    7. 23.7 Schmidt−Hilbert theory
    8. 23.8 Exercises
    9. 23.9 Hints and answers
  31. 24 Complex variables
    1. 24.1 Functions of a complex variable
    2. 24.2 The Cauchy−Riemann relations
    3. 24.3 Power series in a complex variable
    4. 24.4 Some elementary functions
    5. 24.5 Multivalued functions and branch cuts
    6. 24.6 Singularities and zeros of complex functions
    7. 24.7 Conformal transformations
    8. 24.8 Complex integrals
    9. 24.9 Cauchy’s theorem
    10. 24.10 Cauchy’s integral formula
    11. 24.11 Taylor and Laurent series
    12. 24.12 Residue theorem
    13. 24.13 Definite integrals using contour integration
    14. 24.14 Exercises
    15. 24.15 Hints and answers
  32. 25 Applications of complex variables
    1. 25.1 Complex potentials
    2. 25.2 Applications of conformal transformations
    3. 25.3 Location of zeros
    4. 25.4 Summation of series
    5. 25.5 Inverse Laplace transform
    6. 25.6 Stokes’ equation and Airy integrals
    7. 25.7 WKB methods
    8. 25.8 Approximations to integrals
    9. 25.9 Exercises
    10. 25.10 Hints and answers
  33. 26 Tensors
    1. 26.1 Some notation
    2. 26.2 Change of basis
    3. 26.3 Cartesian tensors
    4. 26.4 First- and zero-order Cartesian tensors
    5. 26.5 Second- and higher-order Cartesian tensors
    6. 26.6 The algebra of tensors
    7. 26.7 The quotient law
    8. 26.8 The tensors δij and ∈ijk
    9. 26.9 Isotropic tensors
    10. 26.10 Improper rotations and pseudotensors
    11. 26.11 Dual tensors
    12. 26.12 Physical applications of tensors
    13. 26.13 Integral theorems for tensors
    14. 26.14 Non-Cartesian coordinates
    15. 26.15 The metric tensor
    16. 26.16 General coordinate transformations and tensors
    17. 26.17 Relative tensors
    18. 26.18 Derivatives of basis vectors and Christoffel symbols
    19. 26.19 Covariant differentiation
    20. 26.20 Vector operators in tensor form
    21. 26.21 Absolute derivatives along curves
    22. 26.22 Geodesics
    23. 26.23 Exercises
    24. 26.24 Hints and answers
  34. 27 Numerical methods
    1. 27.1 Algebraic and transcendental equations
    2. 27.2 Convergence of iteration schemes
    3. 27.3 Simultaneous linear equations
    4. 27.4 Numerical integration
    5. 27.5 Finite differences
    6. 27.6 Differential equations
    7. 27.7 Higher-order equations
    8. 27.8 Partial differential equations
    9. 27.9 Exercises
    10. 27.10 Hints and answers
  35. 28 Group theory
    1. 28.1 Groups
    2. 28.2 Finite groups
    3. 28.3 Non-Abelian groups
    4. 28.4 Permutation groups
    5. 28.5 Mappings between groups
    6. 28.6 Subgroups
    7. 28.7 Subdividing a group
    8. 28.8 Exercises
    9. 28.9 Hints and answers
  36. 29 Representation theory
    1. 29.1 Dipole moments of molecules
    2. 29.2 Choosing an appropriate formalism
    3. 29.3 Equivalent representations
    4. 29.4 Reducibility of a representation
    5. 29.5 The orthogonality theorem for irreducible representations
    6. 29.6 Characters
    7. 29.7 Counting irreps using characters
    8. 29.8 Construction of a character table
    9. 29.9 Group nomenclature
    10. 29.10 Product representations
    11. 29.11 Physical applications of group theory
    12. 29.12 Exercises
    13. 29.13 Hints and answers
  37. 30 Probability
    1. 30.1 Venn diagrams
    2. 30.2 Probability
    3. 30.3 Permutations and combinations
    4. 30.4 Random variables and distributions
    5. 30.5 Properties of distributions
    6. 30.6 Functions of random variables
    7. 30.7 Generating functions
    8. 30.8 Important discrete distributions
    9. 30.9 Important continuous distributions
    10. 30.10 The central limit theorem
    11. 30.11 Joint distributions
    12. 30.12 Properties of joint distributions
    13. 30.13 Generating functions for joint distributions
    14. 30.14 Transformation of variables in joint distributions
    15. 30.15 Important joint distributions
    16. 30.16 Exercises
    17. 30.17 Hints and answers
  38. 31 Statistics
    1. 31.1 Experiments, samples and populations
    2. 31.2 Sample statistics
    3. 31.3 Estimators and sampling distributions
    4. 31.4 Some basic estimators
    5. 31.5 Maximum-likelihood method
    6. 31.6 The method of least squares
    7. 31.7 Hypothesis testing
    8. 31.8 Exercises
    9. 31.9 Hints and answers
  39. Index