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*}
