数学公式

いつも忘れがちな数学公式
どこにいてもさっと目が通せるようにしておく
随時更新していく予定

3次元ラプラシアン

(1)  デカルト座標

\[ \Delta \psi(x,y,z) = \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+\frac{\partial^2 \psi}{\partial z^2} \]

(2)  円筒座標

\(x=r\cos\theta, y=r\sin\theta, z=z\) として考えると、
\begin{eqnarray*} \Delta \psi(r,\theta,z) &=& \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2}\frac{\partial^2 \psi}{\partial \theta^2}+\frac{\partial^2 \psi}{\partial z^2} \\ &=& \frac{\partial^2 \psi}{\partial r^2}+\frac{1}{r} \frac{\partial \psi}{\partial r} + \frac{1}{r^2}\frac{\partial^2 \psi}{\partial \theta^2}+\frac{\partial^2 \psi}{\partial z^2} \end{eqnarray*}

(3)  極座標

\(x=r\sin\theta\cos\phi, y=r\sin\theta\sin\phi, z=r\cos\theta \) として考えると、
\begin{eqnarray*} \Delta \psi(r,\theta,\phi) &=& \frac{1}{r^2}\frac{\partial}{\partial r} \biggl ( r^2 \frac{\partial \psi}{\partial r} \biggr) +\frac{1}{r^2\sin{\theta} }\frac{\partial }{\partial \theta} \biggl (\sin{\theta}\frac{\partial \psi}{\partial \theta} \biggr ) +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 \psi}{\partial \phi^2}\\ &=& \frac{1}{r}\frac{\partial^2}{\partial r^2} \biggl( r \psi \biggr) +\frac{1}{r^2\sin{\theta} }\frac{\partial }{\partial \theta} \biggl (\sin{\theta}\frac{\partial \psi}{\partial \theta} \biggr ) +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 \psi}{\partial \phi^2} \\ &=&\frac{\partial^2 \psi}{\partial r^2} +\frac{2}{r}\frac{\partial \psi}{\partial r} +\frac{1}{r^2}\frac{\partial^2 \psi}{\partial \theta^2} +\frac{1}{r^2}\mathrm{cot}\theta \frac{\partial \psi}{\partial \theta} +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 \psi}{\partial \phi^2} \end{eqnarray*}