Methods

Introduction Fourier series Properties of sine and cosine Definition of Fourier series The meaning of good behaviour Complex Fourier series Sine and cosine series Parseval's theorem The Wave Equation Waves on an elastic string Separation of variables Oscillation energy Solution in characteristic co-ordinates Wave reflection and transmission Green's Functions The Dirac delta function Representations Second order linear ODEs Definition of Green's function Defining properties Constructing G(x,): boundary value problems Derivation of jump conditions Example Constructing G(x,): initial value problems Example Sturm-Liouville Theory Self-adjoint form and boundary values Eigenfunction expansions Real eigenvalues Orthogonal eigenfunctions Complete eigenfunctions Example: Legendre polynomials Inhomogeneous boundary value problem Applications: Laplace's Equation Cartesians Plane polars Spherical polars The full glory of spherical polars Calculus of Variations The problem Euler-Lagrange equations Examples Principle of Least Action Generalisations Integral constraints Cartesian Tensors in R3 Tensors? Transformation laws Tensor algebra Quotient Laws Isotropic tensors Spherically symmetric integrals Symmetric and antisymmetric tensors Physical Applications