Partial Differential Equations Done Right

Construction of PDEs from physical principles

Domains and boundaries in arbitrary dimension

A review of ODEs solution methods

Outer products and tensor fields

The Taylor series for scalar fields

The Fourier series and transform for scalar fields

Flows, Hamiltonians, and the method of characteristics

Construction of the Laplace, heat, and wave equations

Separation of variables

Fourier methods for the solution to the Laplace, heat, and wave equations

Classification of elliptic, parabolic, hyperbolic PDEs

Integral transforms, the Dirac delta function, and Green's functions

Eigenvalues and eigenfunctions

The spectral theorem for Hermitian PDE operators

The Cauchy-Kovalevskaya theorem and power series methods

Finite difference and Galerkin methods