Applications: Harmonic Functions
Definition
Harmonic functions are functions that satisfy the equation
$ \frac{\partial^{2} f}{\partial x_{1}^{2}}+\frac{\partial^{2} f}{\partial x_{2}^{2}}+\cdots+\frac{\partial^{2} f}{\partial x_{n}^{2}}=0 $, or $ \large\Delta f = \nabla^{2} f = 0 $.
A concurrent definition for a harmonic function is the idea that the value at a point in a function is always equal to the average of the values along a circle surrounding that point. This leads to an interesting conclusion about the Laplace Operator itself, in that when $ \Delta f = 0 $, the above statement is true.
