拉普拉斯算子
拉普拉斯算子(Laplacian)是一个微分算子(算符),是一个标量算符,由标量函数在欧几里德空间上的梯度的散度给出.
通常用\(\nabla^2\) 或 \(\Delta\) 表示。在欧几里得空间中,拉普拉斯算子通常表示为对每个变量的二阶偏导数之和,其公式为:
\[
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
\]
举例
标量场\(f(x,y,z) = xy^2 + z^3\)的拉普拉斯函数为:
\[
\begin{align*}
\nabla^2 f(x,y,z) &= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \\
&= \frac{\partial^2}{\partial x^2} (xy^2 + z^3) + \frac{\partial^2}{\partial y^2} (xy^2 + z^3) + \frac{\partial^2}{\partial z^2} (xy^2 + z^3) \\
&= \frac{\partial}{\partial x}y^2 + \frac{\partial}{\partial y} 2yx + \frac{\partial}{\partial z} 3z^2 \\
&= 0 + 2x + 6z \\
&= 2x + 6z
\end{align*}
\]