Harmonic Function / Laplace’s equation

June 5, 2018 1 minute read

harmonic 这个词的意思太多了，比如在 periodic signals 里翻译成 “谐波”。而 Harmonic Function 的翻译是 “调和函数”

本篇 quote from V7. Laplace’s Equation and Harmonic Functions, M.I.T. 18.02 Notes, Exercises and Solutions by Jeremy Orloff

Laplace operator

The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by $\nabla^2$ or $lap$, and defined by

\[\nabla^2 = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}\]

当然你可以扩展到多维：

\[\nabla^2 = \frac {\partial^{2}}{\partial x_1^{2}} + \frac {\partial^{2}}{\partial x_2^{2}} + \cdots + \frac {\partial^{2}}{\partial x_n^{2}}\]

注意 $\nabla^2$ 其实是一个参数为 $f$ 的函数，只是我们不写成 $\nabla^2(f)$ 而是直接用 $\nabla^2 f$ 表示：

\[\nabla^2 f = \frac {\partial^{2} f}{\partial x^{2}} + \frac {\partial^{2} f}{\partial y^{2}}\]

where $f(x, y)$ is a twice differentiable functions.

The notation $\nabla^2$ comes from thinking of the operator as a sort of symbolic scalar product:

\[\nabla^2 = \nabla \cdot \nabla = \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) \cdot \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}\]

Notice that the laplacian is a linear operator, that is it satisfies the two rules

$\nabla^2 (u + v) = \nabla^2 u + \nabla^2 v$
$\nabla^2 cu = c(\nabla^2 u)$

for any two twice differentiable functions $u(x, y)$ and $v(x, y)$ and any constant $c$.

Laplace equation

I.e.

\[\nabla^2 f = 0\]