Differential Equations, Lecture 7.5: Harmonic functions

Lecture 7.5: Harmonic functions.

In this lecture we see what the heat and wave equations look like in higher dimensions, and this involves the Laplacian of u, which is the sum of the second derivatives. Steady-state solutions occur for the heat equation when u_t=0, which means that the Laplacian of u is zero. Functions that satisfy this are called "harmonic", and their graphs can be thought of as being "as flat as possible". For example, taking a metal coat hanger wire, bending it into an odd circular shape, and dipping it into a bucket of soap. The resulting surface that is the soap film is approximately the graph of a harmonic function.