Physical Mathematics 1st Edition by Kevin Cahill 1107005213 9781107005211

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ISBN 10: 1107005213

ISBN 13: 9781107005211

Author: Kevin Cahill

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.

Physical Mathematics 1st Table of contents:

1. Linear algebra

1.1 Numbers

1.2 Arrays

1.3 Matrices

1.4 Vectors

1.5 Linear operators

1.6 Inner products

1.7 The Cauchy–Schwarz inequality

1.8 Linear independence and completeness

1.9 Dimension of a vector space

1.10 Orthonormal vectors

1.11 Outer products

1.12 Dirac notation

1.13 The adjoint of an operator

1.14 Self-adjoint or hermitian linear operators

1.15 Real, symmetric linear operators

1.16 Unitary operators

1.17 Hilbert space

1.18 Antiunitary, antilinear operators

1.19 Symmetry in quantum mechanics

1.20 Determinants

1.21 Systems of linear equations

1.22 Linear least squares

1.23 Lagrange multipliers

1.24 Eigenvectors

1.25 Eigenvectors of a square matrix

1.26 A matrix obeys its characteristic equation

1.27 Functions of matrices

1.28 Hermitian matrices

1.29 Normal matrices

1.30 Compatible normal matrices

1.31 The singular-value decomposition

1.32 The Moore–Penrose pseudoinverse

1.33 The rank of a matrix

1.34 Software

1.35 The tensor/direct product

1.36 Density operators

1.37 Correlation functions

Exercises

2. Fourier series

2.1 Complex Fourier series

2.2 The interval

2.3 Where to put the 2πs

2.4 Real Fourier series for real functions

2.5 Stretched intervals

2.6 Fourier series in several variables

2.7 How Fourier series converge

2.8 Quantum-mechanical examples

2.9 Dirac notation

2.10 Dirac’s delta function

2.11 The harmonic oscillator

2.12 Nonrelativistic strings

2.13 Periodic boundary conditions

Exercises

3. Fourier and Laplace transforms

3.1 The Fourier transform

3.2 The Fourier transform of a real function

3.3 Dirac, Parseval, and Poisson

3.4 Fourier derivatives and integrals

3.5 Fourier transforms in several dimensions

3.6 Convolutions

3.7 The Fourier transform of a convolution

3.8 Fourier transforms and Green’s functions

3.9 Laplace transforms

3.10 Derivatives and integrals of Laplace transforms

3.11 Laplace transforms and differential equations

3.12 Inversion of Laplace transforms

3.13 Application to differential equations

Exercises

4. Infinite series

4.1 Convergence

4.2 Tests of convergence

4.3 Convergent series of functions

4.4 Power series

4.5 Factorials and the gamma function

4.6 Taylor series

4.7 Fourier series as power series

4.8 The binomial series and theorem

4.9 Logarithmic series

4.10 Dirichlet series and the zeta function

4.11 Bernoulli numbers and polynomials

4.12 Asymptotic series

4.13 Some electrostatic problems

4.14 Infinite products

Exercises

5. Complex-variable theory

5.1 Analytic functions

5.2 Cauchy’s integral theorem

5.3 Cauchy’s integral formula

5.4 The Cauchy–Riemann conditions

5.5 Harmonic functions

5.6 Taylor series for analytic functions

5.7 Cauchy’s inequality

5.8 Liouville’s theorem

5.9 The fundamental theorem of algebra

5.10 Laurent series

5.11 Singularities

5.12 Analytic continuation

5.13 The calculus of residues

5.14 Ghost contours

5.15 Logarithms and cuts

5.16 Powers and roots

5.17 Conformal mapping

5.18 Cauchy’s principal value

5.19 Dispersion relations

5.20 Kramers–Kronig relations

5.21 Phase and group velocities

5.22 The method of steepest descent

5.23 The Abel–Plana formula and the Casimir effect

5.24 Applications to string theory

Exercises

6. Differential equations

6.1 Ordinary linear differential equations

6.2 Linear partial differential equations

6.3 Notation for derivatives

6.4 Gradient, divergence, and curl

6.5 Separable partial differential equations

6.6 Wave equations

6.7 First-order differential equations

6.8 Separable first-order differential equations

6.9 Hidden separability

6.10 Exact first-order differential equations

6.11 The meaning of exactness

6.12 Integrating factors

6.13 Homogeneous functions

6.14 The virial theorem

6.15 Homogeneous first-order ordinary differential equations

6.16 Linear first-order ordinary differential equations

6.17 Systems of differential equations

6.18 Singular points of second-order ordinary differential equations

6.19 Frobenius’s series solutions

6.20 Fuch’s theorem

6.21 Even and odd differential operators

6.22 Wronski’s determinant

6.23 A second solution

6.24 Why not three solutions?

6.25 Boundary conditions

6.26 A variational problem

6.27 Self-adjoint differential operators

6.28 Self-adjoint differential systems

6.29 Making operators formally self adjoint

6.30 Wronskians of self-adjoint operators

6.31 First-order self-adjoint differential operators

6.32 A constrained variational problem

6.33 Eigenfunctions and eigenvalues of self-adjoint systems

6.34 Unboundedness of eigenvalues

6.35 Completeness of eigenfunctions

6.36 The inequalities of Bessel and Schwarz

6.37 Green’s functions

6.38 Eigenfunctions and Green’s functions

6.39 Green’s functions in one dimension

6.40 Nonlinear differential equations

Exercises

7. Integral equations

7.1 Fredholm integral equations

7.2 Volterra integral equations

7.3 Implications of linearity

7.4 Numerical solutions

7.5 Integral transformations

Exercises

8. Legendre functions

8.1 The Legendre polynomials

8.2 The Rodrigues formula

8.3 The generating function

8.4 Legendre’s differential equation

8.5 Recurrence relations

8.6 Special values of Legendre’s polynomials

8.7 Schlaefli’s integral

8.8 Orthogonal polynomials

8.9 The azimuthally symmetric Laplacian

8.10 Laplacian in two dimensions

8.11 The Laplacian in spherical coordinates

8.12 The associated Legendre functions/polynomials

8.13 Spherical harmonics

Exercises

9. Bessel functions

9.1 Bessel functions of the first kind

9.2 Spherical Bessel functions of the first kind

9.3 Bessel functions of the second kind

9.4 Spherical Bessel functions of the second kind

Further reading

Exercises

10. Group theory

10.1 What is a group?

10.2 Representations of groups

10.3 Representations acting in Hilbert space

10.4 Subgroups

10.5 Cosets

10.6 Morphisms

10.7 Schur’s lemma

10.8 Characters

10.9 Tensor products

10.10 Finite groups

10.11 The regular representation

10.12 Properties of finite groups

10.13 Permutations

10.14 Compact and noncompact Lie groups

10.15 Lie algebra

10.16 The rotation group

10.17 The Lie algebra and representations of SU(2)

10.18 The defining representation of SU(2)

10.19 The Jacobi identity

10.20 The adjoint representation

10.21 Casimir operators

10.22 Tensor operators for the rotation group

10.23 Simple and semisimple Lie algebras

10.24 SU(3)

10.25 SU(3) and quarks

10.26 Cartan subalgebra

10.27 Quaternions

10.28 The symplectic group Sp (2n)

10.29 Compact simple Lie groups

10.30 Group integration

10.31 The Lorentz group

10.32 Two-dimensional representations of the Lorentz group

10.33 The Dirac representation of the Lorentz group

10.34 The Poincaré group

Further reading

Exercises

11. Tensors and local symmetries

11.1 Points and coordinates

11.2 Scalars

11.3 Contravariant vectors

11.4 Covariant vectors

11.5 Euclidean space in euclidean coordinates

11.6 Summation conventions

11.7 Minkowski space

11.8 Lorentz transformations

11.9 Special relativity

11.10 Kinematics

11.11 Electrodynamics

11.12 Tensors

11.13 Differential forms

11.14 Tensor equations

11.15 The quotient theorem

11.16 The metric tensor

11.17 A basic axiom

11.18 The contravariant metric tensor

11.19 Raising and lowering indices

11.20 Orthogonal coordinates in euclidean n-space

11.21 Polar coordinates

11.22 Cylindrical coordinates

11.23 Spherical coordinates

11.24 The gradient of a scalar field

11.25 Levi-Civita’s tensor

11.26 The Hodge star

11.27 Derivatives and affine connections

11.28 Parallel transport

11.29 Notations for derivatives

11.30 Covariant derivatives

11.31 The covariant curl

11.32 Covariant derivatives and antisymmetry

11.33 Affine connection and metric tensor

11.34 Covariant derivative of the metric tensor

11.35 Divergence of a contravariant vector

11.36 The covariant Laplacian

11.37 The principle of stationary action

11.38 A particle in a gravitational field

11.39 The principle of equivalence

11.40 Weak, static gravitational fields

11.41 Gravitational time dilation

11.42 Curvature

11.43 Einstein’s equations

11.44 The action of general relativity

11.45 Standard form

11.46 Schwarzschild’s solution

11.47 Black holes

11.48 Cosmology

11.49 Model cosmologies

11.50 Yang–Mills theory

11.51 Gauge theory and vectors

11.52 Geometry

Further reading

Exercises

12. Forms

12.1 Exterior forms

12.2 Differential forms

12.3 Exterior differentiation

12.4 Integration of forms

12.5 Are closed forms exact?

12.6 Complex differential forms

12.7 Frobenius’s theorem

Further reading

Exercises

13. Probability and statistics

13.1 Probability and Thomas Bayes

13.2 Mean and variance

13.3 The binomial distribution

13.4 The Poisson distribution

13.5 The Gaussian distribution

13.6 The error function erf

13.7 The Maxwell–Boltzmann distribution

13.8 Diffusion

13.9 Langevin’s theory of brownian motion

13.10 The Einstein–Nernst relation

13.11 Fluctuation and dissipation

13.12 Characteristic and moment-generating functions

13.13 Fat tails

13.14 The central limit theorem and Jarl Lindeberg

13.15 Random-number generators

13.16 Illustration of the central limit theorem

13.17 Measurements, estimators, and Friedrich Bessel

13.18 Information and Ronald Fisher

13.19 Maximum likelihood

13.20 Karl Pearson’s chi-squared statistic

13.21 Kolmogorov’s test

Further reading

Exercises

14. Monte Carlo methods

14.1 The Monte Carlo method

14.2 Numerical integration

14.3 Applications to experiments

14.4 Statistical mechanics

14.5 Solving arbitrary problems

14.6 Evolution

Further reading

Exercises

15 Functional derivatives

15.1 Functionals

15.2 Functional derivatives

15.3 Higher-order functional derivatives

15.4 Functional Taylor series

15.5 Functional differential equations

Exercises

16. Path integrals

16.1 Path integrals and classical physics

16.2 Gaussian integrals

16.3 Path integrals in imaginary time

16.4 Path integrals in real time

16.5 Path integral for a free particle

16.6 Free particle in imaginary time

16.7 Harmonic oscillator in real time

16.8 Harmonic oscillator in imaginary time

16.9 Euclidean correlation functions

16.10 Finite-temperature field theory

16.11 Real-time field theory

16.12 Perturbation theory

16.13 Application to quantum electrodynamics

16.14 Fermionic path integrals

16.15 Application to nonabelian gauge theories

16.16 The Faddeev–Popov trick

16.17 Ghosts

Further reading

Exercises

17. The renormalization group

17.1 The renormalization group in quantum field theory

17.2 The renormalization group in lattice field theory

17.3 The renormalization group in condensed-matter physics

Exercises

18 Chaos and fractals

18.1 Chaos

18.2 Attractors

18.3 Fractals

Further reading

Exercises

19. Strings

19.1 The infinities of quantum field theory

19.2 The Nambu–Goto string action

19.3 Regge trajectories

19.4 Quantized strings

19.5 D-branes

19.6 String–string scattering

19.7 Riemann surfaces and moduli

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