假设 $r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$, 求 $\Delta r$, $\Delta r^{-1}$, $\Delta\ln r$. 这里 $\Delta$ 是 Laplace 算子, $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$.
假设 $r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$, 求 $\Delta r$, $\Delta r^{-1}$, $\Delta\ln r$. 这里 $\Delta$ 是 Laplace 算子, $\Delta=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}$.
一般的, 若 $r=\sqrt{\sum_{i=1}^{n}(x_i-x_i^0)^2}$, 且 $f(r)$ 二阶可导, 证明: $\Delta f(r)=f''(r)+\dfrac{n-1}{r}f'(r)$.