拉普拉斯算符
我们令一个标量函数 $u(x, y, z)$ 的梯度的散度为它的拉普拉斯（Laplacian），合成的算符（类比复合函数）叫做拉普拉斯算符，记为 $ \boldsymbol{\nabla}^2 $．
\begin{equation}
\boldsymbol{\nabla}^2 u = \boldsymbol{\nabla}\boldsymbol{\cdot} ( \boldsymbol\nabla u) = \frac{\partial^{2}{u}}{\partial{x}^{2}} + \frac{\partial^{2}{u}}{\partial{y}^{2}} + \frac{\partial^{2}{u}}{\partial{z}^{2}}
\end{equation}
也可以记
\begin{equation}
\begin{aligned}
\boldsymbol{\nabla}^2 &= \boldsymbol{\nabla} \boldsymbol\cdot \boldsymbol{\nabla} = ( \boldsymbol{\nabla} )^2 = \left( \hat{\boldsymbol{\mathbf{x}}} \frac{\partial}{\partial{x}} + \hat{\boldsymbol{\mathbf{y}}} \frac{\partial}{\partial{y}} + \hat{\boldsymbol{\mathbf{z}}} \frac{\partial}{\partial{z}} \right) ^2\\
&= \frac{\partial^{2}{u}}{\partial{x}^{2}} + \frac{\partial^{2}{u}}{\partial{y}^{2}} + \frac{\partial^{2}{u}}{\partial{z}^{2}}
\end{aligned}
\end{equation}
这些定义也容易拓展到 $N = 1, 2, \dots$ 元函数上．