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Physical Mathematics

Book Description

Unique in its clarity, examples and range, Physical Mathematics explains as simply as possible the mathematics that graduate students and professional physicists need in their courses and research. The author illustrates the mathematics with numerous physical examples drawn from contemporary research. In addition to basic subjects such as linear algebra, Fourier analysis, complex variables, differential equations and Bessel functions, this textbook covers topics such as the singular-value decomposition, Lie algebras, the tensors and forms of general relativity, the central limit theorem and Kolmogorov test of statistics, the Monte Carlo methods of experimental and theoretical physics, the renormalization group of condensed-matter physics and the functional derivatives and Feynman path integrals of quantum field theory.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 1. Linear algebra
    1. 1.1 Numbers
    2. 1.2 Arrays
    3. 1.3 Matrices
    4. 1.4 Vectors
    5. 1.5 Linear operators
    6. 1.6 Inner products
    7. 1.7 The Cauchy–Schwarz inequality
    8. 1.8 Linear independence and completeness
    9. 1.9 Dimension of a vector space
    10. 1.10 Orthonormal vectors
    11. 1.11 Outer products
    12. 1.12 Dirac notation
    13. 1.13 The adjoint of an operator
    14. 1.14 Self-adjoint or hermitian linear operators
    15. 1.15 Real, symmetric linear operators
    16. 1.16 Unitary operators
    17. 1.17 Hilbert space
    18. 1.18 Antiunitary, antilinear operators
    19. 1.19 Symmetry in quantum mechanics
    20. 1.20 Determinants
    21. 1.21 Systems of linear equations
    22. 1.22 Linear least squares
    23. 1.23 Lagrange multipliers
    24. 1.24 Eigenvectors
    25. 1.25 Eigenvectors of a square matrix
    26. 1.26 A matrix obeys its characteristic equation
    27. 1.27 Functions of matrices
    28. 1.28 Hermitian matrices
    29. 1.29 Normal matrices
    30. 1.30 Compatible normal matrices
    31. 1.31 The singular-value decomposition
    32. 1.32 The Moore–Penrose pseudoinverse
    33. 1.33 The rank of a matrix
    34. 1.34 Software
    35. 1.35 The tensor/direct product
    36. 1.36 Density operators
    37. 1.37 Correlation functions
    38. Exercises
  9. 2. Fourier series
    1. 2.1 Complex Fourier series
    2. 2.2 The interval
    3. 2.3 Where to put the 2πs
    4. 2.4 Real Fourier series for real functions
    5. 2.5 Stretched intervals
    6. 2.6 Fourier series in several variables
    7. 2.7 How Fourier series converge
    8. 2.8 Quantum-mechanical examples
    9. 2.9 Dirac notation
    10. 2.10 Dirac’s delta function
    11. 2.11 The harmonic oscillator
    12. 2.12 Nonrelativistic strings
    13. 2.13 Periodic boundary conditions
    14. Exercises
  10. 3. Fourier and Laplace transforms
    1. 3.1 The Fourier transform
    2. 3.2 The Fourier transform of a real function
    3. 3.3 Dirac, Parseval, and Poisson
    4. 3.4 Fourier derivatives and integrals
    5. 3.5 Fourier transforms in several dimensions
    6. 3.6 Convolutions
    7. 3.7 The Fourier transform of a convolution
    8. 3.8 Fourier transforms and Green’s functions
    9. 3.9 Laplace transforms
    10. 3.10 Derivatives and integrals of Laplace transforms
    11. 3.11 Laplace transforms and differential equations
    12. 3.12 Inversion of Laplace transforms
    13. 3.13 Application to differential equations
    14. Exercises
  11. 4. Infinite series
    1. 4.1 Convergence
    2. 4.2 Tests of convergence
    3. 4.3 Convergent series of functions
    4. 4.4 Power series
    5. 4.5 Factorials and the gamma function
    6. 4.6 Taylor series
    7. 4.7 Fourier series as power series
    8. 4.8 The binomial series and theorem
    9. 4.9 Logarithmic series
    10. 4.10 Dirichlet series and the zeta function
    11. 4.11 Bernoulli numbers and polynomials
    12. 4.12 Asymptotic series
    13. 4.13 Some electrostatic problems
    14. 4.14 Infinite products
    15. Exercises
  12. 5. Complex-variable theory
    1. 5.1 Analytic functions
    2. 5.2 Cauchy’s integral theorem
    3. 5.3 Cauchy’s integral formula
    4. 5.4 The Cauchy–Riemann conditions
    5. 5.5 Harmonic functions
    6. 5.6 Taylor series for analytic functions
    7. 5.7 Cauchy’s inequality
    8. 5.8 Liouville’s theorem
    9. 5.9 The fundamental theorem of algebra
    10. 5.10 Laurent series
    11. 5.11 Singularities
    12. 5.12 Analytic continuation
    13. 5.13 The calculus of residues
    14. 5.14 Ghost contours
    15. 5.15 Logarithms and cuts
    16. 5.16 Powers and roots
    17. 5.17 Conformal mapping
    18. 5.18 Cauchy’s principal value
    19. 5.19 Dispersion relations
    20. 5.20 Kramers–Kronig relations
    21. 5.21 Phase and group velocities
    22. 5.22 The method of steepest descent
    23. 5.23 The Abel–Plana formula and the Casimir effect
    24. 5.24 Applications to string theory
    25. Exercises
  13. 6. Differential equations
    1. 6.1 Ordinary linear differential equations
    2. 6.2 Linear partial differential equations
    3. 6.3 Notation for derivatives
    4. 6.4 Gradient, divergence, and curl
    5. 6.5 Separable partial differential equations
    6. 6.6 Wave equations
    7. 6.7 First-order differential equations
    8. 6.8 Separable first-order differential equations
    9. 6.9 Hidden separability
    10. 6.10 Exact first-order differential equations
    11. 6.11 The meaning of exactness
    12. 6.12 Integrating factors
    13. 6.13 Homogeneous functions
    14. 6.14 The virial theorem
    15. 6.15 Homogeneous first-order ordinary differential equations
    16. 6.16 Linear first-order ordinary differential equations
    17. 6.17 Systems of differential equations
    18. 6.18 Singular points of second-order ordinary differential equations
    19. 6.19 Frobenius’s series solutions
    20. 6.20 Fuch’s theorem
    21. 6.21 Even and odd differential operators
    22. 6.22 Wronski’s determinant
    23. 6.23 A second solution
    24. 6.24 Why not three solutions?
    25. 6.25 Boundary conditions
    26. 6.26 A variational problem
    27. 6.27 Self-adjoint differential operators
    28. 6.28 Self-adjoint differential systems
    29. 6.29 Making operators formally self adjoint
    30. 6.30 Wronskians of self-adjoint operators
    31. 6.31 First-order self-adjoint differential operators
    32. 6.32 A constrained variational problem
    33. 6.33 Eigenfunctions and eigenvalues of self-adjoint systems
    34. 6.34 Unboundedness of eigenvalues
    35. 6.35 Completeness of eigenfunctions
    36. 6.36 The inequalities of Bessel and Schwarz
    37. 6.37 Green’s functions
    38. 6.38 Eigenfunctions and Green’s functions
    39. 6.39 Green’s functions in one dimension
    40. 6.40 Nonlinear differential equations
    41. Exercises
  14. 7. Integral equations
    1. 7.1 Fredholm integral equations
    2. 7.2 Volterra integral equations
    3. 7.3 Implications of linearity
    4. 7.4 Numerical solutions
    5. 7.5 Integral transformations
    6. Exercises
  15. 8. Legendre functions
    1. 8.1 The Legendre polynomials
    2. 8.2 The Rodrigues formula
    3. 8.3 The generating function
    4. 8.4 Legendre’s differential equation
    5. 8.5 Recurrence relations
    6. 8.6 Special values of Legendre’s polynomials
    7. 8.7 Schlaefli’s integral
    8. 8.8 Orthogonal polynomials
    9. 8.9 The azimuthally symmetric Laplacian
    10. 8.10 Laplacian in two dimensions
    11. 8.11 The Laplacian in spherical coordinates
    12. 8.12 The associated Legendre functions/polynomials
    13. 8.13 Spherical harmonics
    14. Exercises
  16. 9. Bessel functions
    1. 9.1 Bessel functions of the first kind
    2. 9.2 Spherical Bessel functions of the first kind
    3. 9.3 Bessel functions of the second kind
    4. 9.4 Spherical Bessel functions of the second kind
    5. Further reading
    6. Exercises
  17. 10. Group theory
    1. 10.1 What is a group?
    2. 10.2 Representations of groups
    3. 10.3 Representations acting in Hilbert space
    4. 10.4 Subgroups
    5. 10.5 Cosets
    6. 10.6 Morphisms
    7. 10.7 Schur’s lemma
    8. 10.8 Characters
    9. 10.9 Tensor products
    10. 10.10 Finite groups
    11. 10.11 The regular representation
    12. 10.12 Properties of finite groups
    13. 10.13 Permutations
    14. 10.14 Compact and noncompact Lie groups
    15. 10.15 Lie algebra
    16. 10.16 The rotation group
    17. 10.17 The Lie algebra and representations of SU(2)
    18. 10.18 The defining representation of SU(2)
    19. 10.19 The Jacobi identity
    20. 10.20 The adjoint representation
    21. 10.21 Casimir operators
    22. 10.22 Tensor operators for the rotation group
    23. 10.23 Simple and semisimple Lie algebras
    24. 10.24 SU(3)
    25. 10.25 SU(3) and quarks
    26. 10.26 Cartan subalgebra
    27. 10.27 Quaternions
    28. 10.28 The symplectic group Sp (2n)
    29. 10.29 Compact simple Lie groups
    30. 10.30 Group integration
    31. 10.31 The Lorentz group
    32. 10.32 Two-dimensional representations of the Lorentz group
    33. 10.33 The Dirac representation of the Lorentz group
    34. 10.34 The Poincaré group
    35. Further reading
    36. Exercises
  18. 11. Tensors and local symmetries
    1. 11.1 Points and coordinates
    2. 11.2 Scalars
    3. 11.3 Contravariant vectors
    4. 11.4 Covariant vectors
    5. 11.5 Euclidean space in euclidean coordinates
    6. 11.6 Summation conventions
    7. 11.7 Minkowski space
    8. 11.8 Lorentz transformations
    9. 11.9 Special relativity
    10. 11.10 Kinematics
    11. 11.11 Electrodynamics
    12. 11.12 Tensors
    13. 11.13 Differential forms
    14. 11.14 Tensor equations
    15. 11.15 The quotient theorem
    16. 11.16 The metric tensor
    17. 11.17 A basic axiom
    18. 11.18 The contravariant metric tensor
    19. 11.19 Raising and lowering indices
    20. 11.20 Orthogonal coordinates in euclidean n-space
    21. 11.21 Polar coordinates
    22. 11.22 Cylindrical coordinates
    23. 11.23 Spherical coordinates
    24. 11.24 The gradient of a scalar field
    25. 11.25 Levi-Civita’s tensor
    26. 11.26 The Hodge star
    27. 11.27 Derivatives and affine connections
    28. 11.28 Parallel transport
    29. 11.29 Notations for derivatives
    30. 11.30 Covariant derivatives
    31. 11.31 The covariant curl
    32. 11.32 Covariant derivatives and antisymmetry
    33. 11.33 Affine connection and metric tensor
    34. 11.34 Covariant derivative of the metric tensor
    35. 11.35 Divergence of a contravariant vector
    36. 11.36 The covariant Laplacian
    37. 11.37 The principle of stationary action
    38. 11.38 A particle in a gravitational field
    39. 11.39 The principle of equivalence
    40. 11.40 Weak, static gravitational fields
    41. 11.41 Gravitational time dilation
    42. 11.42 Curvature
    43. 11.43 Einstein’s equations
    44. 11.44 The action of general relativity
    45. 11.45 Standard form
    46. 11.46 Schwarzschild’s solution
    47. 11.47 Black holes
    48. 11.48 Cosmology
    49. 11.49 Model cosmologies
    50. 11.50 Yang–Mills theory
    51. 11.51 Gauge theory and vectors
    52. 11.52 Geometry
    53. Further reading
    54. Exercises
  19. 12. Forms
    1. 12.1 Exterior forms
    2. 12.2 Differential forms
    3. 12.3 Exterior differentiation
    4. 12.4 Integration of forms
    5. 12.5 Are closed forms exact?
    6. 12.6 Complex differential forms
    7. 12.7 Frobenius’s theorem
    8. Further reading
    9. Exercises
  20. 13. Probability and statistics
    1. 13.1 Probability and Thomas Bayes
    2. 13.2 Mean and variance
    3. 13.3 The binomial distribution
    4. 13.4 The Poisson distribution
    5. 13.5 The Gaussian distribution
    6. 13.6 The error function erf
    7. 13.7 The Maxwell–Boltzmann distribution
    8. 13.8 Diffusion
    9. 13.9 Langevin’s theory of brownian motion
    10. 13.10 The Einstein–Nernst relation
    11. 13.11 Fluctuation and dissipation
    12. 13.12 Characteristic and moment-generating functions
    13. 13.13 Fat tails
    14. 13.14 The central limit theorem and Jarl Lindeberg
    15. 13.15 Random-number generators
    16. 13.16 Illustration of the central limit theorem
    17. 13.17 Measurements, estimators, and Friedrich Bessel
    18. 13.18 Information and Ronald Fisher
    19. 13.19 Maximum likelihood
    20. 13.20 Karl Pearson’s chi-squared statistic
    21. 13.21 Kolmogorov’s test
    22. Further reading
    23. Exercises
  21. 14. Monte Carlo methods
    1. 14.1 The Monte Carlo method
    2. 14.2 Numerical integration
    3. 14.3 Applications to experiments
    4. 14.4 Statistical mechanics
    5. 14.5 Solving arbitrary problems
    6. 14.6 Evolution
    7. Further reading
    8. Exercises
  22. 15 Functional derivatives
    1. 15.1 Functionals
    2. 15.2 Functional derivatives
    3. 15.3 Higher-order functional derivatives
    4. 15.4 Functional Taylor series
    5. 15.5 Functional differential equations
    6. Exercises
  23. 16. Path integrals
    1. 16.1 Path integrals and classical physics
    2. 16.2 Gaussian integrals
    3. 16.3 Path integrals in imaginary time
    4. 16.4 Path integrals in real time
    5. 16.5 Path integral for a free particle
    6. 16.6 Free particle in imaginary time
    7. 16.7 Harmonic oscillator in real time
    8. 16.8 Harmonic oscillator in imaginary time
    9. 16.9 Euclidean correlation functions
    10. 16.10 Finite-temperature field theory
    11. 16.11 Real-time field theory
    12. 16.12 Perturbation theory
    13. 16.13 Application to quantum electrodynamics
    14. 16.14 Fermionic path integrals
    15. 16.15 Application to nonabelian gauge theories
    16. 16.16 The Faddeev–Popov trick
    17. 16.17 Ghosts
    18. Further reading
    19. Exercises
  24. 17. The renormalization group
    1. 17.1 The renormalization group in quantum field theory
    2. 17.2 The renormalization group in lattice field theory
    3. 17.3 The renormalization group in condensed-matter physics
    4. Exercises
  25. 18 Chaos and fractals
    1. 18.1 Chaos
    2. 18.2 Attractors
    3. 18.3 Fractals
    4. Further reading
    5. Exercises
  26. 19. Strings
    1. 19.1 The infinities of quantum field theory
    2. 19.2 The Nambu–Goto string action
    3. 19.3 Regge trajectories
    4. 19.4 Quantized strings
    5. 19.5 D-branes
    6. 19.6 String–string scattering
    7. 19.7 Riemann surfaces and moduli
    8. Further reading
    9. Exercises
  27. References
  28. Index