Mathematical Methods for Physics

Techniques Built on the Problems That Need Them in 25 chapters — from “The derivative as a physical instrument” to “Root finding, interpolation, numerical linear algebra and quadrature”.

Chapters

  1. The derivative as a physical instrument
    • The derivative of measured data
    • Standard derivatives recovered
    • Implicit and parametric differentiation
    • Stationary points and stiffness
    • The differential as an error estimator
    • Summary
    • Problems
  2. Series, limits and controlled approximation
    • The satellite and the differential
    • Sequences, partial sums, geometric series
    • Convergence tests
    • The radius of convergence
    • Taylor's theorem with remainder
    • Order notation and standard expansions
    • Regimes from Taylor's theorem
    • Summary
    • Problems
  3. Accumulation: integration and the techniques that make it possible
    • The integral as an accumulated sum
    • The fundamental theorem of calculus
    • Improper integrals
    • Substitution, reduction formulas, partial fractions
    • Integral test and Gaussian integral
    • Summary
    • Problems
  4. Partial differentiation, chain rules and Jacobians
    • The partial derivative
    • Total differential and error propagation
    • Exact differentials
    • The chain rule along a path
    • Chain rule for a change of variables
    • The Jacobian
    • Taylor expansion in two variables
    • Summary
    • Problems
  5. Multiple integrals and change of variables
    • The double integral
    • Iterated integrals and their limits
    • Order of integration
    • Change of variables and plane polar coordinates
    • Squaring the Gaussian integral
    • Triple integrals in cylindrical and spherical coordinates
    • Moment of inertia
    • Summary
    • Problems
  6. Complex numbers, the complex exponential and phasors
    • Complex numbers
    • Multiplication as rotation
    • Euler's formula
    • De Moivre's theorem and the nearest singularity
    • Phasors and the voltage across an inductor
    • Complex impedance
    • Summary
    • Problems
  7. First-order differential equations and physical modelling
    • Ordinary differential equations
    • Separable equations
    • The time constant
    • The integrating factor
    • Direction fields and fitted models
    • Summary
    • Problems
  8. Second-order linear equations: damping, driving and resonance
    • The damped driven oscillator
    • The auxiliary equation
    • The three damping regimes
    • The small-oscillation frequency
    • Complementary function and particular integral
    • The frequency response
    • Resonance and its width
    • Summary
    • Problems
  9. Higher-order linear equations, the Wronskian and variation of parameters
    • Higher-order linear equations
    • The Wronskian
    • Abel's identity and the repeated root
    • Reduction of order
    • Variation of parameters
    • Summary
    • Problems
  10. Matrices, determinants and systems of linear equations
    • A circuit as a matrix
    • The matrix as a linear map
    • Matrix multiplication and order
    • The determinant, twice
    • The cross product as a determinant
    • Gaussian elimination and rank
    • The inverse matrix
    • Symmetric, orthogonal and Hermitian matrices
    • Summary
    • Problems
  11. Eigenvalue problems, diagonalisation and normal modes
    • Two coupled oscillators
    • The eigenvalue problem
    • The characteristic equation and normal modes
    • Symmetry and the spectral theorem
    • Diagonalisation and normal coordinates
    • Beating of two modes and stability
    • Summary
    • Problems
  12. Curvilinear and orthogonal coordinates
    • Curvilinear coordinates
    • Scale factors
    • Position-dependent unit vectors
    • Velocity and acceleration in plane polars
    • Line, area and volume elements
    • Summary
    • Problems
  13. Gradient, divergence and curl
    • Scalar and vector fields
    • Gradient vector and directional derivative
    • Divergence of a vector field
    • Curl as circulation per unit area
    • The Laplacian and the two identities
    • Curl test for a scalar potential
    • Operators in curvilinear coordinates
    • Summary
    • Problems
  14. Line, surface and volume integrals
    • Parameterised curve and arc length
    • Line integral of a vector field
    • Path independence and the potential
    • Directed area element
    • Surface integral and flux
    • Volume integral of a field and symmetry checks
    • Summary
    • Problems
  15. The integral theorems: Gauss, Stokes and Green
    • Point-charge flux and divergence of a vector field
    • Cancellation of interior boundaries
    • Divergence theorem and Gauss's law
    • Stokes's theorem
    • Green's theorem and the planimeter
    • Local form of an integral law
    • Multiply connected region
    • Direct calculation versus theorem
    • Summary
    • Problems
  16. Vector spaces, bases, linear maps and dual vectors
    • What makes a vector space
    • Basis and dimension without arrows
    • The inner product of functions
    • Orthonormal expansion by projection
    • Gram-Schmidt, matrices of maps, change of basis
    • Dual vectors and the gradient
    • Summary
    • Problems
  17. Fourier series and periodic decomposition
    • Fundamental and harmonics of a note
    • The orthogonality relations
    • Fourier coefficients in real and complex form
    • Symmetry shortcuts and the Gibbs phenomenon
    • Parseval's theorem and the power spectrum
    • Term-by-term differentiation and periodic drives
    • Summary
    • Problems
  18. Fourier transforms, spectra and convolution
    • The Fourier transform from a series
    • Standard transform pairs and three theorems
    • The reciprocal-width relation
    • Convolution as instrumental smearing
    • The convolution theorem and its limits
    • Sampling and the discrete Fourier transform
    • Summary
    • Problems
  19. Delta functions, distributions and generalised functions
    • Delta sequences and the delta function
    • The transform of a constant
    • Properties of the delta function
    • The three-dimensional delta function
    • The point source and the impulse response
    • Summary
    • Problems
  20. The classical partial differential equations of physics and their boundary conditions
    • Two bars and a partial differential equation
    • The diffusion equation from conservation
    • The wave equation from a force balance
    • Laplace's and Poisson's equations
    • Initial and boundary data
    • Well-posedness and uniqueness
    • Summary
    • Problems
  21. Separation of variables in Cartesian, cylindrical and spherical geometry
    • One method for three partial differential equations
    • The product guess and the separation constant
    • Eigenvalues fixed by the boundary
    • Cartesian separation from end to end
    • The drum and Bessel's equation
    • The sphere and Legendre's equation
    • Non-separable problems
    • Summary
    • Problems
  22. Probability, random variables and distributions
    • Counts as random variables
    • Probability density and its units
    • The binomial, Poisson and Gaussian distributions
    • The variance of a sum
    • The central limit theorem, and the root-N rule
    • Summary
    • Problems
  23. Estimation, fitting and hypothesis testing
    • Sample and population
    • The likelihood
    • Least squares, and when not to use it
    • The weighted straight-line fit
    • Chi-squared, confidence intervals and hypothesis tests
    • Summary
    • Problems
  24. Floating-point arithmetic, conditioning and numerical stability
    • Floating-point numbers and machine epsilon
    • Catastrophic cancellation
    • Condition numbers of functions and matrices
    • Numerical stability and accumulated rounding
    • Ill-conditioned normal equations
    • Summary
    • Problems
  25. Root finding, interpolation, numerical linear algebra and quadrature
    • Bisection and Newton's method
    • Polynomial interpolation and the Runge phenomenon
    • Pivoting and the cost of elimination
    • Numerical differentiation and the best step
    • Trapezium, Simpson and Gauss quadrature rules
    • Choosing an algorithm
    • Summary
    • Problems