拉普拉斯算子的极坐标表示

二维极坐标： \[ \begin{equation} \Delta f = \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) f =\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r} \frac{\partial}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right) f \end{equation} \]

球坐标 \[ \begin{equation} \begin{aligned} \Delta u &=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}} \\ &=\frac{1}{r} \frac{\partial^{2}(r u)}{\partial r^{2}}+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} u}{\partial \phi^{2}} \end{aligned} \end{equation} \]

二维极坐标表示

球坐标表示