Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

Cambridge University Press, 2009. - 820 pages.An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics - differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study.Preface
Acknowledgments
Calculus of variations
What is it good for?
Functionals
The functional derivative
The Euler–Lagrange equation
Some applications
First integral
Lagrangian mechanics
One degree of freedom
Noether’s theorem
Many degrees of freedom
Continuous systems
Variable endpoints
Lagrange multipliers
Maximum or minimum?
Further exercises and problems
Function spaces
Motivation
Functions as vectors
Norms and inner products
Norms and convergence
Norms from integrals
Hilbert space
Orthogonal polynomials
Linear operators and distributions
Linear operators
Distributions and test-functions
Further exercises and problems
Linear ordinary differential equations
Existence and uniqueness of solutions
Flows for first-order equations
Linear independence
The Wronskian
Normal form
Inhomogeneous equations
Particular integral and complementary function
Variation of parameters
Singular points
Regular singular points
Further exercises and problems
Linear differential operators
Formal vs. concrete operators
The algebra of formal operators
Concrete operators
The adjoint operator
The formal adjoint
A simple eigenvalue problem
Adjoint boundary conditions
Self-adjoint boundary conditions
Completeness of eigenfunctions
Discrete spectrum
Continuous spectrum
Further exercises and problems
Green functions
Inhomogeneous linear equations
Fredholm alternative
Constructing Green functions
Sturm–Liouville equation
Initial value problems
Modified Green function
Applications of Lagrange's identity
Hermiticity of Green functions
Inhomogeneous boundary conditions
Eigenfunction expansions
ed Green function
Analytic properties of Green functions
Causality implies analyticity
Plemelj formulæ
Resolvent operator
Locality and the Gelfand–Dikii equation
Further exercises and problems
Partial differential equations
Classification of PDEs
Cauchy data
Characteristics and first-order equations
Wave equation
d’Alembert’s solution
Fourier’s solution
Causal Green function
Odd vs. even dimensions
Heat equation
Heat kernel
Causal Green function
Duhamel’s principle
Potential theory
Uniqueness and existence of solutions
Separation of variables
Eigenfunction expansions
Green functions
Boundary value problems
Kirchhoff vs. Huygens
Further exercises and problems
The mathematics of real waves
Dispersive waves
Ocean waves
Group velocity
Wakes
Hamilton’s theory of rays
Making waves
Rayleigh’s equation
Nonlinear waves
Sound in air
Shocks
Weak solutions
Solitons
Further exercises and problems
Special functions
Curvilinear coordinates
polar coordinates
cal polar coordinates
Div, grad and curl in curvilinear coordinates
Spherical harmonics
Legendre polynomials
Axisymmetric potential problems
General spherical harmonics
Bessel functions
Cylindrical Bessel functions
Orthogonality and completeness
Modified Bessel functions
Spherical Bessel functions
Singular endpoints
Weyl’s theorem
Further exercises and problems
Integral equations
Illustrations
Classification of integral equations
Integral transforms
Fourier methods
Laplace transform methods
Separable kernels
Eigenvalue problem
Inhomogeneous problem
Singular integral equations
Solution via Tchebychef polynomials
Wiener–Hopf equations I
Some functional analysis
Bounded and compact operators
Closed operators
Series solutions
Liouville–Neumann–Born series
Fredholm series
Further exercises and problems
Vectors and tensors
Covariant and contravariant vectors
Tensors
Transformation rules
Tensor character of linear maps and quadratic forms
Tensor product spaces
Symmetric and skew-symmetric tensors
Kronecker and Levi-Civita tensors
Cartesian tensors
Isotropic tensors
Stress and strain
Maxwell stress tensor
Further exercises and problems
Differential calculus on manifolds
Vector and covector fields
nguage of bundles and sections
Differentiating tensors
Lie bracket
Lie derivative
Exterior calculus
Differential forms
The exterior derivative
Physical applications
Maxwell’s equations
Hamilton’s equations
Covariant derivatives
Connections
Cartan’s form viewpoint
Further exercises and problems
Integration on manifolds
Basic notions
Line integrals
Skew-symmetry and orientations
Integrating p-forms
Counting boxes
Relation to conventional integrals
Stokes' theorem
Applications
Pull-backs and push-forwards
Spin textures
The Hopf map
Homotopy and the Hopf map
The Hopf index
Twist and writhe
Further exercises and problems
An introduction to differential topology
Homeomorphism and diffeomorphism
Cohomology
Retractable spaces: Converse of Poincaré’s lemma
Obstructions to exactness
De Rham cohomology
Homology
Chains, cycles and boundaries
Relative homology
De Rham's theorem
Poincaré duality
Characteristic classes
Topological invariance
Chern characters and Chern classes
Hodge theory and the Morse index
The Laplacian on p-forms
Morse theory
Further exercises and problems
Groups and group representations
Basic ideas
Group axioms
Elementary properties
Group actions on sets
Representations
nd pseudo-real representations
sum and direct product
Reducibility and irreducibility
Characters and orthogonality
The group algebra
Physics applications
Quantum mechanics
Vibrational spectrum of H2O
Crystal field splittings
Further exercises and problems
Lie groups
Matrix groups
The unitary and orthogonal groups
Symplectic groups
Geometry of SU(2)
Invariant vector fields
Maurer–Cartan forms
Euler angles
Volume and metric
SO(3) … SU(2)/Z2
Peter–Weyl theorem
Lie brackets vs. commutators
Lie algebras
and quotient algebras
Adjoint representation
The Killing form
Roots and weights
Product representations
Subalgebras and branching rules
Further exercises and problems
The geometry of fibre bundles
Fibre bundles
Definitions
Physics examples
Landau levels
The Berry connection
Quantization
Working in the total space
Principal bundles and associated bundles
Connections
Monopole harmonics
Bundle connection and curvature forms
Characteristic classes as obstructions
Stora–Zumino descent equations
Complex analysis
Cauchy–Riemann equations
Conjugate pairs
Conformal mapping
Complex integration: Cauchy and Stokes
The complex integral
Cauchy’s theorem
The residue theorem
Applications
Two-dimensional vector calculus
Milne–Thomson circle theorem
Blasius and Kutta–Joukowski theorems
Applications of Cauchy's theorem
Cauchy’s integral formula
Taylor and Laurent series
Zeros and singularities
Analytic continuation
Meromorphic functions and the winding number
Principle of the argument
Rouché’s theorem
Analytic functions and topology
The point at infinity
Logarithms and branch cuts
Topology of Riemann surfaces
Conformal geometry of Riemann surfaces
Further exercises and problems
Applications of complex variables
Contour integration technology
Tricks of the trade
Branch-cut integrals
Jordan’s lemma
The Schwarz reflection principle
Kramers–Kronig relations
Hilbert transforms
Partial-fraction and product expansions
Mittag-Leffler partial-fraction expansion
Infinite product expansions
Wiener–Hopf equations II
Wiener–Hopf integral equations
Further exercises and problems
Special functions and complex variables
The Gamma function
te product for Gamma(z)
Linear differential equations
Monodromy
Hypergeometric functions
Solving ODEs via contour integrals
Bessel functions
Asymptotic expansions
Stirling’s approximation for n!
Airy functions
Elliptic functions
Further exercises and problems
Appendix A Linear algebra review
Vector space
Axioms
Linear maps
Matrices
Range–null-space theorem
The dual space
Inner-product spaces
Inner products
Euclidean vectors
Bra and ket vectors
Adjoint operator
Sums and differences of vector spaces
Direct sums
Quotient spaces
Projection-operator decompositions
Inhomogeneous linear equations
Rank and index
Fredholm alternative
Determinants
Skew-symmetric n-linear forms
The adjugate matrix
Diagonalization and canonical forms
Diagonalizing linear maps
Diagonalizing quadratic forms
Block-diagonalizing symplectic forms
Appendix B Fourier series and integrals
Fourier series
Finite Fourier series
Continuum limit
Fourier integral transforms
Inversion formula
The Riemann–Lebesgue lemma
Convolution
The convolution theorem
Apodization and Gibbs’ phenomenon
The Poisson summation formula
References
Index