Advanced Engineering Mathematics, Lecture 7.1: Harmonic functions and Laplace's equation

Advanced Engineering Mathematics, Lecture 7.1: Harmonic functions and Laplace's equation. In this lecture we see what the heat and wave equations look like in higher dimensions, and this involves the Laplacian of u, denoted Δu, which is the sum of the second spacial derivatives. Because heat dissipates, steady-state solutions occur for the heat equation when u_t=0, which means that Δu=0. This PDE is called "Laplace's equation", and functions that satisfy it are called "harmonic. The graphs of harmonic functions can be thought of as being "as flat as possible", and that for any bounded region, the minimum and maximal values are achieved on the boundary. For another physical example of a harmonic function, take a metal coat hanger wire, bend it into an irregular circular shape, and dip it into a bucket of soap. The resulting surface that is the soap film is approximately the graph of a harmonic function. We conclude this lecture by solving three boundary value problems for Laplace's equation in a square region, and show how superposition can be used when multiple sides have inhomogeneous boundary conditions. Course webpage (with lecture notes, homework, worksheets, etc.): http://www.math.clemson.edu/~macaule/... Prerequisite: http://www.math.clemson.edu/~macaule/...