球坐标下Laplace算子的公式

前几天给人找了几道微积分题目,里面有一个是计算球坐标下面的Laplace算子的表达式。这当然可以暴力用复合函数的求导来算,但是可能比较复杂。事实上可以有一种积分的办法来算出这个结果。

我们知道,在标准坐标下,Laplace算子的形式为

 \Delta u = \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}

现在计算在球坐标

\begin{cases}
x = r\sin\theta\cos\varphi \\
y = r\sin\theta\sin\varphi \\
z = r\cos\theta
\end{cases}

下Laplace算子的表达式。首先计算其Jacobi矩阵

 \frac{\partial(x, y, z)}{\partial(r, \theta, \varphi)} = \begin{pmatrix}
\sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \\
r\cos\theta\cos\varphi & r\cos\theta\sin\varphi & -r\sin\theta \\
-r\sin\theta\sin\varphi & r\sin\theta\cos\varphi & 0
\end{pmatrix}^T

对应逆矩阵为

 \frac{\partial(r, \theta, \varphi)}{\partial(x, y, z)} = \begin{pmatrix}
\sin \theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \\
\frac{1}{r}\cos\theta\cos\varphi & \frac{1}{r}\cos\theta\sin\varphi & -\frac{1}{r}\sin\theta \\
-\frac{\sin\varphi}{r\sin\theta} & \frac{\cos\varphi}{r\sin\theta} & 0
\end{pmatrix}

注意到这是一个正交矩阵。

我们利用Laplace算子的积分表达式,考虑

 \begin{aligned}\frac{ \partial{u(r, \theta, \varphi)}}{\partial(x, y, z)} \cdot \frac{\partial{v(r, \theta, \varphi)}}{\partial(x, y, z)} &= \frac{\partial{u(r, \theta, \varphi)}}{\partial(r, \theta, \varphi)}\frac{\partial(r, \theta, \varphi)}{\partial(x, y, z)}^T\frac{\partial(r, \theta, \varphi)}{\partial(x, y, z)}\frac{\partial{v(r, \theta, \varphi)}}{\partial(r, \theta, \varphi)} \\ &= u_rv_r + \frac{1}{r^2}u_\theta v_\theta + \frac{1}{r^2\sin^2\theta}u_\varphi v_\varphi \end{aligned}

对于任意的$v \in C_c^\infty$,利用分部积分

 \begin{aligned}
\int \Delta u v dxdydz &= -\int \nabla u \cdot \nabla v dxdydz \\
&= -\int \left( u_r v_r + \frac{1}{r^2} u_\theta v_\theta + \frac{1}{r^2\sin^2\theta}u_\varphi v_\varphi \right) r^2\sin\theta drd\theta d\varphi \\
&= \int \left( \left (u_r r^2\sin\theta \right )_r + ( u_\theta \sin\theta )_\theta + u_{\varphi\varphi} \right) v drd\theta d\varphi \\
\end{aligned}

再根据

 \int \Delta u v dxdydz = \int \Delta u v r^2\sin\theta drd\theta d\varphi

这样就可以得到

 \begin{aligned}
\Delta u &= \frac{1}{r^2\sin\theta}\left( \left (u_r r^2\sin\theta \right )_r + ( u_\theta \sin\theta )_\theta + u_{\varphi\varphi} \right) \\
&= \frac{1}{r} \frac{\partial^2}{\partial r^2}(ru) + \frac{1}{r^2\sin \theta} \frac{\partial}{\partial \theta}\left( \sin \theta \frac{\partial u}{\partial \theta} \right) + \frac{1}{r^2\sin^2 \theta} \frac{\partial^2 u}{\partial \varphi^2}
\end{aligned}