and Harmonic Functions
In this section, we will show how Green's theorem is closely connected with solutions to
Laplace's partial differential equation in two dimensions:
where w(x, y) is some unknown function of two variables, assumed to be twice differentiable.
Equation (1) models a variety of physical situations, as we discussed in Section P of these
notes, and shall briefly review.
The Laplace operator and harmonic functions.
The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by
V2 or lap, and defined by
The notation V2 comes from thinking of the operator as a sort of symbolic scalar product:
In terms of this operator, Laplace's equation (1) reads simply
Notice that the laplacian is a linear operator, that is it satisfies the two rules
for any two twice differentiable functions u(x, y) and v(x, y) and any constant c.
A function w(x, y) which has continuous second partial derivatives and
solves Laplace's equation (1) is called a harmonic function.
In the sequel, we will use the Greek letters q5 and
to denote harmonic functions;
functions which aren't assumed to be harmonic will be denoted by Roman letters
g, u, v,
. According to the definition,
4(x, y) is harmonic
By combining (4) with the rules (3) for using Laplace operator, we see
and cq5 are harmonic (c constant).
Examples of harmonic functions.
Here are some examples of harmonic functions.
The verifications are left to the Exercises.