Conditions of Validity

Vector Calculus, Linear Operators and the Boundary-Value Problems of Physics in 18 chapters — from “Differentiability, the implicit function theorem and the licence to change variables” to “Integral equations, kernels and the Fredholm alternative for the last time”.

Chapters

  1. Differentiability, the implicit function theorem and the licence to change variables
    • Partial derivatives and their unstated hypotheses
    • Differentiability and continuous partials
    • Clairaut's theorem and the exactness test
    • Chain rule and inverse function theorem
    • Implicit function theorem and thermodynamic identities
    • Legendre transform and convexity
    • Multivariable Taylor expansion and the Hessian
    • Summary
    • Problems
  2. Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist
    • Iterated integrals that disagree
    • Zero content and existence of the integral
    • Improper multiple integrals and the r−p test
    • Fubini's theorem and absolute integrability
    • The change-of-variables theorem
    • Excision at a singular point
    • Multidimensional Gaussian integral and n-ball volume
    • Summary
    • Problems
  3. A coordinate system is a map: the metric, the scale factors and where a chart fails
    • A new coordinate system and the change-of-variables theorem
    • Coordinate basis and metric coefficients
    • Reciprocal basis and skew coordinates
    • Chart failure at singular metrics and folds
    • Summary
    • Problems
  4. Gradient, divergence and curl without coordinates, and with any coordinates
    • Gradient and divergence on trial
    • The coordinate-free gradient
    • Physical components and divergence as a limit
    • Curl as a limit and the general Laplacian
    • The Levi-Civita symbol and second-derivative identities
    • Summary
    • Problems
  5. Gauss, Stokes and Green proved, and the first inversion of a differential operator
    • A disputed flux
    • Proof of the divergence theorem
    • Excision of a point source
    • Stokes's theorem and converse potential theorems
    • Green's identities and the energy uniqueness theorem
    • The Helmholtz decomposition
    • Summary
    • Problems
  6. The four subspaces and the Fredholm alternative
    • Matrices as maps and a fragile answer
    • The rank-nullity theorem and four fundamental subspaces
    • The Fredholm alternative
    • The condition number
    • Summary
    • Problems
  7. Vector spaces, linear maps and the dual: the Riesz theorem
    • Linear functionals and the dual space
    • Change of basis for a functional
    • The Riesz representation theorem
    • Basis-independent properties and function spaces
    • Problems
  8. Inner products, orthogonality and the adjoint
    • Inner product spaces and Cauchy-Schwarz
    • Gram-Schmidt orthogonalisation and the projection theorem
    • Least squares and the four subspaces
    • The adjoint of an operator
    • Self-adjoint operators and their domains
    • Problems
  9. The spectral theorem, degeneracy and the matrices that will not diagonalise
    • The reality and orthogonality theorem
    • Normal operators and the spectral theorem
    • Degeneracy and the choice of eigenbasis
    • The spectral resolution and the matrix exponential
    • A defective matrix and critical damping
    • Singular value decomposition and principal axes
    • Problems
  10. Two quadratic forms at once: the generalised eigenvalue problem
    • The generalised eigenvalue problem for unequal masses
    • The mass inner product
    • Simultaneous diagonalisation and generalised mass
    • The semi-definite mass matrix
    • Summary
    • Problems
  11. Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise
    • Hammer blows and linear ODEs with constant coefficients
    • The existence and uniqueness theorem
    • Why the Wronskian independence test works
    • The initial-value Green function
    • Convolution solution and impulse response
    • The fundamental matrix as an exponential
    • Summary
    • Problems
  12. Series solutions, singular points and where the special functions come from
    • Chart failure at the poles
    • Classification of singular points
    • The indicial equation and exponent-difference cases
    • Quantisation by boundedness
    • Legendre and Bessel functions
    • The irregular singular point
    • Summary
    • Problems
  13. Sturm-Liouville theory: the spectral theorem for a differential operator
    • Gram-Schmidt orthogonalisation meets Legendre's equation
    • Sturm-Liouville form and its weight
    • Self-adjoint boundary conditions and the Sturm-Liouville theorem
    • The weight as a mass density
    • Oscillation theorem and singular Sturm-Liouville problems
    • Completeness of eigenfunctions
    • Summary
    • Problems
  14. Eigenfunction expansions, convergence and the Rayleigh quotient
    • A tilted well and the Sturm-Liouville theorem
    • Eigenfunction expansion and its coefficient decay rate
    • Inhomogeneous eigenfunction expansion and resonance
    • Rayleigh quotient and the Rayleigh-Ritz method
    • Summary
    • Problems
  15. The three equations, their characteristics and what makes a problem well posed
    • Diffusion, wave and Laplace equations revisited
    • Characteristic directions and PDE type
    • Well-posedness and the backward heat problem
    • Maximum principle and mean-value property
    • Maximum-principle uniqueness beside the energy proof
    • Summary
    • Problems
  16. Separation of variables as a chain of Sturm-Liouville problems
    • Separation of variables and three drums
    • Separated Sturm-Liouville problems in orthogonal charts
    • Cylindrical and spherical boundary-value problems
    • The level-set criterion and failure of separation
    • Summary
    • Problems
  17. Green functions: inverting the operator
    • The impulse response and a charge between plates
    • Boundary-value Green function and reciprocity
    • The bilinear eigenfunction expansion and representation formula
    • The method of images and the plate force
    • Helmholtz Green function and the heat kernel
    • The modified Green function
    • Summary
    • Problems
  18. Integral equations, kernels and the Fredholm alternative for the last time
    • Scattering and the Helmholtz Green function
    • Fredholm and Volterra equations from boundary-value problems
    • The Neumann series and the Born series
    • Separable kernels and finite linear systems
    • Symmetric kernels and the Hilbert-Schmidt expansion
    • Fredholm alternative for kernels and first-kind equations
    • Summary
    • Problems