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Essential Mathematical Methods for the Physical Sciences

Book Description

The mathematical methods that physical scientists need for solving substantial problems in their fields of study are set out clearly and simply in this tutorial-style textbook. Students will develop problem-solving skills through hundreds of worked examples, self-test questions and homework problems. Each chapter concludes with a summary of the main procedures and results and all assumed prior knowledge is summarized in one of the appendices. Over 300 worked examples show how to use the techniques and around 100 self-test questions in the footnotes act as checkpoints to build student confidence. Nearly 400 end-of-chapter problems combine ideas from the chapter to reinforce the concepts. Hints and outline answers to the odd-numbered problems are given at the end of each chapter, with fully-worked solutions to these problems given in the accompanying Student Solutions Manual. Fully-worked solutions to all problems, password-protected for instructors, are available at www.cambridge.org/essential.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Review of background topics
  8. 1. Matrices and vector spaces
    1. 1.1 Vector spaces
    2. 1.2 Linear operators
    3. 1.3 Matrices
    4. 1.4 Basic matrix algebra
    5. 1.5 Functions of matrices
    6. 1.6 The transpose of a matrix
    7. 1.7 The complex and Hermitian conjugates of a matrix
    8. 1.8 The trace of a matrix
    9. 1.9 The determinant of a matrix
    10. 1.10 The inverse of a matrix
    11. 1.11 The rank of a matrix
    12. 1.12 Simultaneous linear equations
    13. 1.13 Special types of square matrix
    14. 1.14 Eigenvectors and eigenvalues
    15. 1.15 Determination of eigenvalues and eigenvectors
    16. 1.16 Change of basis and similarity transformations
    17. 1.17 Diagonalization of matrices
    18. 1.18 Quadratic and Hermitian forms
    19. 1.19 Normal modes
    20. 1.20 The summation convention
    21. Summary
    22. Problems
    23. Hints and answers
  9. 2. Vector calculus
    1. 2.1 Differentiation of vectors
    2. 2.2 Integration of vectors
    3. 2.3 Vector functions of several arguments
    4. 2.4 Surfaces
    5. 2.5 Scalar and vector fields
    6. 2.6 Vector operators
    7. 2.7 Vector operator formulae
    8. 2.8 Cylindrical and spherical polar coordinates
    9. 2.9 General curvilinear coordinates
    10. Summary
    11. Problems
    12. Hints and answers
  10. 3. Line, surface and volume integrals
    1. 3.1 Line integrals
    2. 3.2 Connectivity of regions
    3. 3.3 Green’s theorem in a plane
    4. 3.4 Conservative fields and potentials
    5. 3.5 Surface integrals
    6. 3.6 Volume integrals
    7. 3.7 Integral forms for grad, div and curl
    8. 3.8 Divergence theorem and related theorems
    9. 3.9 Stokes’ theorem and related theorems
    10. Summary
    11. Problems
    12. Hints and answers
  11. 4. Fourier series
    1. 4.1 The Dirichlet conditions
    2. 4.2 The Fourier coefficients
    3. 4.3 Symmetry considerations
    4. 4.4 Discontinuous functions
    5. 4.5 Non-periodic functions
    6. 4.6 Integration and differentiation
    7. 4.7 Complex Fourier series
    8. 4.8 Parseval’s theorem
    9. Summary
    10. Problems
    11. Hints and answers
  12. 5. Integral transforms
    1. 5.1 Fourier transforms
    2. 5.2 The Dirac δ-function
    3. 5.3 Properties of Fourier transforms
    4. 5.4 Laplace transforms
    5. 5.5 Concluding remarks
    6. Summary
    7. Problems
    8. Hints and answers
  13. 6. Higher-order ordinary differential equations
    1. 6.1 General considerations
    2. 6.2 Linear equations with constant coefficients
    3. 6.3 Linear recurrence relations
    4. 6.4 Laplace transform method
    5. 6.5 Linear equations with variable coefficients
    6. 6.6 General ordinary differential equations
    7. Summary
    8. Problems
    9. Hints and answers
  14. 7. Series solutions of ordinary differential equations
    1. 7.1 Second-order linear ordinary differential equations
    2. 7.2 Ordinary and singular points of an ODE
    3. 7.3 Series solutions about an ordinary point
    4. 7.4 Series solutions about a regular singular point
    5. 7.5 Obtaining a second solution
    6. 7.6 Polynomial solutions
    7. Summary
    8. Problems
    9. Hints and answers
  15. 8. Eigenfunction methods for differential equations
    1. 8.1 Sets of functions
    2. 8.2 Adjoint, self-adjoint and Hermitian operators
    3. 8.3 Properties of Hermitian operators
    4. 8.4 Sturm–Liouville equations
    5. 8.5 Superposition of eigenfunctions: Green’s functions
    6. Summary
    7. Problems
    8. Hints and answers
  16. 9. Special functions
    1. 9.1 Legendre functions
    2. 9.2 Associated Legendre functions
    3. 9.3 Spherical harmonics
    4. 9.4 Chebyshev functions
    5. 9.5 Bessel functions
    6. 9.6 Spherical Bessel functions
    7. 9.7 Laguerre functions
    8. 9.8 Associated Laguerre functions
    9. 9.9 Hermite functions
    10. 9.10 The gamma function and related functions
    11. Summary
    12. Problems
    13. Hints and answers
  17. 10. Partial differential equations
    1. 10.1 Important partial differential equations
    2. 10.2 General form of solution
    3. 10.3 General and particular solutions
    4. 10.4 The wave equation
    5. 10.5 The diffusion equation
    6. 10.6 Boundary conditions and the uniqueness of solutions
    7. Summary
    8. Problems
    9. Hints and answers
  18. 11. Solution methods for PDEs
    1. 11.1 Separation of variables: the general method
    2. 11.2 Superposition of separated solutions
    3. 11.3 Separation of variables in polar coordinates
    4. 11.4 Integral transform methods
    5. 11.5 Inhomogeneous problems – Green’s functions
    6. Summary
    7. Problems
    8. Hints and answers
  19. 12. Calculus of variations
    1. 12.1 The Euler–Lagrange equation
    2. 12.2 Special cases
    3. 12.3 Some extensions
    4. 12.4 Constrained variation
    5. 12.5 Physical variational principles
    6. 12.6 General eigenvalue problems
    7. 12.7 Estimation of eigenvalues and eigenfunctions
    8. 12.8 Adjustment of parameters
    9. Summary
    10. Problems
    11. Hints and answers
  20. 13. Integral equations
    1. 13.1 Obtaining an integral equation from a differential equation
    2. 13.2 Types of integral equation
    3. 13.3 Operator notation and the existence of solutions
    4. 13.4 Closed-form solutions
    5. 13.5 Neumann series
    6. 13.6 Fredholm theory
    7. 13.7 Schmidt–Hilbert theory
    8. Summary
    9. Problems
    10. Hints and answers
  21. 14. Complex variables
    1. 14.1 Functions of a complex variable
    2. 14.2 The Cauchy–Riemann relations
    3. 14.3 Power series in a complex variable
    4. 14.4 Some elementary functions
    5. 14.5 Multivalued functions and branch cuts
    6. 14.6 Singularities and zeros of complex functions
    7. 14.7 Conformal transformations
    8. 14.8 Complex integrals
    9. 14.9 Cauchy’s theorem
    10. 14.10 Cauchy’s integral formula
    11. 14.11 Taylor and Laurent series
    12. 14.12 Residue theorem
    13. Summary
    14. Problems
    15. Hints and answers
  22. 15. Applications of complex variables
    1. 15.1 Complex potentials
    2. 15.2 Applications of conformal transformations
    3. 15.3 Definite integrals using contour integration
    4. 15.4 Summation of series
    5. 15.5 Inverse Laplace transform
    6. 15.6 Some more advanced applications
    7. Summary
    8. Problems
    9. Hints and answers
  23. 16. Probability
    1. 16.1 Venn diagrams
    2. 16.2 Probability
    3. 16.3 Permutations and combinations
    4. 16.4 Random variables and distributions
    5. 16.5 Properties of distributions
    6. 16.6 Functions of random variables
    7. 16.7 Generating functions
    8. 16.8 Important discrete distributions
    9. 16.9 Important continuous distributions
    10. 16.10 The central limit theorem
    11. 16.11 Joint distributions
    12. 16.12 Properties of joint distributions
    13. Summary
    14. Problems
    15. Hints and answers
  24. 17. Statistics
    1. 17.1 Experiments, samples and populations
    2. 17.2 Sample statistics
    3. 17.3 Estimators and sampling distributions
    4. 17.4 Some basic estimators
    5. 17.5 Data modeling
    6. 17.6 Hypothesis testing
    7. Summary
    8. Problems
    9. Hints and answers
  25. A: Review of background topics
    1. A.1 Arithmetic and geometry
    2. A.2 Preliminary algebra
    3. A.3 Differential calculus
    4. A.4 Integral calculus
    5. A.5 Complex numbers and hyperbolic functions
    6. A.6 Series and limits
    7. A.7 Partial differentiation
    8. A.8 Multiple integrals
    9. A.9 Vector algebra
    10. A.10 First-order ordinary differential equations
  26. B: Inner products
  27. C: Inequalities in linear vector spaces
  28. D: Summation convention
  29. E: The Kronecker delta and Levi–Civita symbols
  30. F: Gram–Schmidt orthogonalization
  31. G: Linear least squares
  32. H: Footnote answers
  33. Index