最近は地上波にも登場するようになった「予備校のノリで学ぶ「大学の数学・物理」」通称ヨビノリというYoutubeのチャンネルがあります．1000本近い動画があって，大学の数学，物理をわかりやすく解説されています．今のところ配信されている動画はすべて見ていて，そろそろサポーターになったほうが良いかなと思うくらいお世話になっています．
で，今回ラプラシアンを極座標にしたものを導出する動画が配信されていました．人生で一度はやろう，といわれていますが，これまで30年間やったことがありませんでしたので，やってみました．
動画で解説しているように手計算で，省略されたところの計算を確認しつつ，最初のほうで大きなミスをして最後に気づいてやり直したりしつつ，（途中，動画の中の計算ミス（符号，記号）も見つけて），A4用紙10枚くらいを使ってやっとこさ計算し終えました．
結構頑張ったので，手計算したものをTeXにして貼っておきます．間違っていたら教えて下さい．
ラプラシアンの極座標表示
直交座標である点の座標を次のように表すとき
\[
\begin{align}
x &= r\sin\theta\cos\phi \\
y &= r\sin\theta\sin\phi \\
z &= r\cos\theta |
\end{align}
\]
ラプラシアンは次のように表される
\[
\Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
\]
極座標では次のように表される
\[
\Delta = \nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial\theta}(\sin\theta \frac{\partial}{\partial\theta}) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2}{\partial\phi^2}
\]
なお，偏微分は次を前提とする
\[
\frac{\partial}{\partial x} f(r,\theta,\phi) = \frac{\partial r}{\partial x}\frac{\partial f}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial f}{\partial \theta} + \frac{\partial \phi}{\partial x}\frac{\partial f}{\partial \phi}
\]
\(f(r,\theta,\phi)\)は省略され
\[
\frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi}
\]
と表される
以下で極座標形式におけるラプラシアンを導出する
まず，\(\nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}\)を求める
\[
\begin{align} r^2 &= x^2 + y^2 + z^2 \\ \cos\theta &= \frac{z}{r} \\ \tan y &= \frac{y}{x} \end{align}
\]
を利用する
まず次を求める
\[
\frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi}
\]
①\(r^2 = x^2 + y^2 + z^2\)より
\[
\begin{align} \frac{\partial}{\partial x}r^2 &= \frac{\partial}{\partial x}(x^2 + y^2 +z^2)\\ 2r \frac{\partial r}{\partial x} &= 2x \\ \therefore \frac{\partial r}{\partial x} &= \frac{x}{r} \\ &= \frac{r\sin\theta\cos\phi}{r} \\ &= \sin\theta\cos\phi \end{align}
\]
②\(\cos\theta = \frac{z}{r}\)より
\[
\begin{align} \frac{\partial}{\partial x} \cos\theta &= \frac{\partial}{\partial x}\frac{z}{r} \\ -\sin\theta \frac{\partial \theta}{\partial x} &= -\frac{z}{r^2}\frac{\partial r}{\partial x} \\ &= -\frac{r\cos\theta}{r^2} \sin\theta\cos\phi \\ \therefore \frac{\partial \theta}{\partial x} &= \frac{\cos\theta\cos\phi}{r} \end{align}
\]
③\(\tan\phi = \frac{y}{x} \)より
\[
\begin{align} \frac{\partial}{\partial x} \tan\phi &= \frac{\partial}{\partial x}\frac{y}{x} \\ \frac{1}{\cos^2\phi}\frac{\partial \phi}{\partial x} &= -\frac{y}{x^2} \\ &= -\frac{y}{x}\frac{1}{x} \\ &= -\tan\phi \frac{1}{r\sin\theta\cos\phi} \\ \therefore \frac{\partial \phi}{\partial x} &= -\frac{\sin\phi}{r\sin\theta} \end{align}
\]
①②③の結果より
\[
\begin{align} \frac{\partial}{\partial x} &= \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi} \\ &= \sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi} \end{align}
\]
次に
\[
\frac{\partial}{\partial y} = \frac{\partial r}{\partial y}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial y}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi}
\]
①\(r^2 = x^2 + y^2 + z^2\)より
\[
\begin{align} \frac{\partial}{\partial y}r^2 &= \frac{\partial}{\partial y}(x^2 + y^2 +z^2) \\ 2r \frac{\partial r}{\partial y} &= 2y \\ \therefore \frac{\partial r}{\partial y} &= \frac{y}{r} \\ &= \frac{r\sin\theta\sin\phi}{r} \\ &= \sin\theta\sin\phi \end{align}
\]
②\(\cos\theta = \frac{z}{r}\)より
\[
\begin{align} \frac{\partial}{\partial y} \cos\theta &= \frac{\partial}{\partial y}\frac{z}{r} \\ -\sin\theta \frac{\partial \theta}{\partial y} &= -\frac{z}{r^2}\frac{\partial r}{\partial y} \\ &= -\frac{r\cos\theta}{r^2} \sin\theta\sin\phi \\ \therefore \frac{\partial \theta}{\partial y} &= \frac{\cos\theta\sin\phi}{r} \end{align}
\]
③\(\tan\phi = \frac{y}{x} \)より
\[
\begin{align} \frac{\partial}{\partial y} \tan\phi &= \frac{\partial}{\partial y}\frac{y}{x} \\ \frac{1}{\cos^2\phi}\frac{\partial \phi}{\partial y} &= \frac{1}{x} \\ &= \frac{1}{r\sin\theta\cos\phi} \\ \therefore \frac{\partial \phi}{\partial y} &= \frac{\cos\phi}{r\sin\theta} \end{align}
\]
①②③の結果より
\[
\begin{align} \frac{\partial}{\partial y} &= \frac{\partial r}{\partial y}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial y}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi} \\ &= \sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi} \end{align}
\]
さらに
\[
\frac{\partial}{\partial z} = \frac{\partial r}{\partial z}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial z}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial z}\frac{\partial}{\partial \phi}
\]
①\(r^2 = x^2 + y^2 + z^2\)より
\[
\begin{align} \frac{\partial}{\partial z}r^2 &= \frac{\partial}{\partial z}(x^2 + y^2 +z^2) \\ 2r \frac{\partial r}{\partial z} &= 2z \\ \therefore \frac{\partial r}{\partial z} &= \frac{z}{r} \\ &= \frac{r\cos\theta}{r} \\ &= \cos\theta \end{align}
\]
②\(\cos\theta = \frac{z}{r}\)より
\[
\begin{align} \frac{\partial}{\partial z} \cos\theta &= \frac{\partial}{\partial z}\frac{z}{r} \\ -\sin\theta \frac{\partial \theta}{\partial z} &= \frac{r – z\frac{\partial r}{\partial z}}{r^2} \\ &= \frac{r – (r\cos\theta)(\cos\theta)}{r^2} \\ &= \frac{1 – \cos^2\theta}{r} \\ &= \frac{\sin^2\theta}{r} \\ \therefore \frac{\partial \theta}{\partial z} &= -\frac{\sin\theta}{r} \end{align}
\]
③\(\tan\phi = \frac{y}{x} \)より
\[
\begin{align} \frac{\partial}{\partial z} \tan\phi &= \frac{\partial}{\partial z}\frac{y}{x} \\ \frac{1}{\cos^2\phi}\frac{\partial \phi}{\partial z} &= 0 \\ \therefore \frac{\partial \phi}{\partial z} &= 0 \end{align}
\]
①②③の結果より
\[
\begin{align} \frac{\partial}{\partial z} &= \frac{\partial r}{\partial z}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial z}\frac{\partial}{\partial \theta} + \frac{\partial \phi}{\partial z}\frac{\partial}{\partial \phi} \\ &= \cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \end{align}
\]
したがって
\[
\begin{align} \nabla &= \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \\ &= \sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi} \\ &+ \sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi} \\ &+ \cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \end{align}
\]
ここから2階微分を計算する
\[
\Delta = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
\]
まず，
\[
\begin{align} \frac{\partial^2}{\partial x^2} = (\sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ (\sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \end{align}
\]
各要素を計算する
\[
\begin{align} \frac{\partial}{\partial r}(\sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = \sin\theta\cos\phi\frac{\partial^2}{\partial r^2} – \frac{\cos\theta\cos\phi}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos\theta\cos\phi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\phi}{r^2\sin\theta}\frac{\partial}{\partial \phi} – \frac{\sin\phi}{r\sin\theta}\frac{\partial^2}{\partial r\partial \phi} \end{align}
\]
\[
\begin{align} \frac{\partial}{\partial \theta}(\sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = \cos\theta\cos\phi\frac{\partial}{\partial r} + \sin\theta\cos\phi\frac{\partial^2}{\partial \theta\partial r} – \frac{\sin\theta\cos\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\theta\cos\phi}{r}\frac{\partial ^2}{\partial \theta^2} + \frac{\cos\theta\sin\phi}{r\sin^2\theta}\frac{\partial}{\partial \phi} – \frac{\sin\phi}{r\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} \end{align}
\]
\[
\begin{align} \frac{\partial}{\partial \phi}(\sin\theta\cos\phi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = -\sin\theta\sin\phi\frac{\partial}{\partial r} + \sin\theta\cos\phi\frac{\partial^2}{\partial\phi\partial r} – \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\theta\cos\phi}{r}\frac{\partial ^2}{\partial\phi\partial \theta} – \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi} – \frac{\sin\phi}{r\sin\theta}\frac{\partial^2}{\partial \phi^2} \end{align}
\]
したがって
\[
\begin{align} \frac{\partial^2}{\partial x^2} &= \sin^2\theta\cos^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\cos^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\sin^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{\sin\theta\cos\theta\cos^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\theta\cos\theta\cos^2\phi}{r}\frac{\partial^2}{\partial \theta\partial r} \\ &- \frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} – \frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial ^2}{\partial\phi\partial \theta} \\ &- \frac{\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} – \frac{\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial r\partial \phi} \\ &+ \frac{\cos^2\theta\cos^2\phi}{r}\frac{\partial}{\partial r} + \frac{\sin^2\phi}{r}\frac{\partial}{\partial r} \\ &- \frac{\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\partial}{\partial \theta} – \frac{\sin\theta\cos\theta\cos^2\phi}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos\theta\sin^2\phi}{r^2 \sin\theta}\frac{\partial}{\partial \theta} \\ &+ \frac{\sin\phi\cos\phi}{r^2}\frac{\partial}{\partial \phi} + \frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} – \frac{\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} \\ &= \sin^2\theta\cos^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\cos^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\sin^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{2\sin\theta\cos\theta\cos^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} – \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} – \frac{2\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} \\ &+ \frac{\cos^2\theta\cos^2\phi + \sin^2\phi}{r}\frac{\partial}{\partial r} – (\frac{2\sin\theta\cos\theta\cos^2\phi}{r^2} – \frac{\cos\theta\sin^2\phi}{r^2 \sin\theta})\frac{\partial}{\partial \theta} + \frac{2\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} \end{align}
\]
同様にして
\[
\begin{align} \frac{\partial^2}{\partial y^2} = (\sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ (\sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \end{align}
\]
各要素を計算する
\[
\begin{align} \frac{\partial}{\partial r}(\sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = \sin\theta\sin\phi\frac{\partial^2}{\partial r^2} – \frac{\cos\theta\sin\phi}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos\theta\sin\phi}{r}\frac{\partial^2}{\partial r \partial \theta} – \frac{\cos\phi}{r^2\sin\theta}\frac{\partial}{\partial \phi} + \frac{\cos\phi}{r\sin\theta}\frac{\partial^2}{\partial r\partial \phi} \end{align}
\]
\[
\begin{align} \frac{\partial}{\partial \theta}(\sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = \cos\theta\sin\phi\frac{\partial}{\partial r} + \sin\theta\sin\phi\frac{\partial^2}{\partial \theta\partial r} – \frac{\sin\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\theta\sin\phi}{r}\frac{\partial ^2}{\partial \theta^2} – \frac{\cos\theta\cos\phi}{r\sin^2\theta}\frac{\partial}{\partial \phi} + \frac{\cos\phi}{r\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} \end{align}
\]
\[
\begin{align} \frac{\partial}{\partial \phi}(\sin\theta\sin\phi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}) \\ = \sin\theta\cos\phi\frac{\partial}{\partial r} + \sin\theta\sin\phi\frac{\partial^2}{\partial\phi\partial r} + \frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\theta\sin\phi}{r}\frac{\partial ^2}{\partial\phi\partial \theta} – \frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi} + \frac{\cos\phi}{r\sin\theta}\frac{\partial^2}{\partial \phi^2} \end{align}
\]
したがって
\[
\begin{align} \frac{\partial^2}{\partial y^2} &= \sin^2\theta\sin^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\sin^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\cos^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{\sin\theta\cos\theta\sin^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\theta\cos\theta\sin^2\phi}{r}\frac{\partial^2}{\partial \theta\partial r} \\ &+ \frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} + \frac{\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial ^2}{\partial\phi\partial \theta} \\ &+ \frac{\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} + \frac{\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial r\partial \phi} \\ &+ \frac{\cos^2\theta\sin^2\phi}{r}\frac{\partial}{\partial r} + \frac{\cos^2\phi}{r}\frac{\partial}{\partial r} \\ &- \frac{\sin\theta\cos\theta\sin^2\phi}{r^2}\frac{\partial}{\partial \theta} – \frac{\sin\theta\cos\theta\sin^2\phi}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos\theta\cos^2\phi}{r^2 \sin\theta}\frac{\partial}{\partial \theta} \\ &- \frac{\sin\phi\cos\phi}{r^2}\frac{\partial}{\partial \phi} – \frac{\cos^2\theta\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} – \frac{\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} \\ &= \sin^2\theta\sin^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\sin^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\cos^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{2\sin\theta\cos\theta\sin^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} + \frac{2\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} \\ &+ \frac{\cos^2\theta\sin^2\phi + \cos^2\phi}{r}\frac{\partial}{\partial r} – (\frac{2\sin\theta\cos\theta\sin^2\phi}{r^2} – \frac{\cos\theta\cos^2\phi}{r^2 \sin\theta})\frac{\partial}{\partial \theta} – \frac{2\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi} \end{align}
\]
さらに
\[
\begin{align} \frac{\partial^2}{\partial z^2} = (\cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta}) \\ (\cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta}) \end{align}
\]
各要素を計算する
\[
\begin{align} \frac{\partial}{\partial r}(\cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta}) \\ = \cos\theta\frac{\partial^2}{\partial r^2} + \frac{\sin\theta}{r^2}\frac{\partial}{\partial \theta} – \frac{\sin\theta}{r}\frac{\partial^2}{\partial r \partial \theta} \end{align}
\]
\[
\begin{align} \frac{\partial}{\partial \theta}(\cos\theta\frac{\partial}{\partial r} – \frac{\sin\theta}{r}\frac{\partial}{\partial \theta}) \\ = -\sin\theta\frac{\partial}{\partial r} + \cos\theta\frac{\partial^2}{\partial \theta\partial r} – \frac{\cos\theta}{r}\frac{\partial}{\partial \theta} – \frac{\sin\theta}{r}\frac{\partial ^2}{\partial \theta^2} \end{align}
\]
したがって
\[
\begin{align} \frac{\partial^2}{\partial z^2} &= \cos^2\theta\frac{\partial^2}{\partial r^2} + \frac{\sin^2\theta}{r^2}\frac{\partial ^2}{\partial \theta^2} \\ &- \frac{\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial \theta} – \frac{\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial \theta\partial r} \\ &+ \frac{\sin^2\theta}{r}\frac{\partial}{\partial r} + \frac{\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} \\ &= \cos^2\theta\frac{\partial^2}{\partial r^2} + \frac{\sin^2\theta}{r^2}\frac{\partial ^2}{\partial \theta^2} \\ &- \frac{2\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin^2\theta}{r}\frac{\partial}{\partial r} + \frac{2\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} \end{align}
\]
以上より
\[
\begin{align} \Delta = \nabla^2 &= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \\ &= \sin^2\theta\cos^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\cos^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\sin^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{2\sin\theta\cos\theta\cos^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} – \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} – \frac{2\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} \\ &+ \frac{\cos^2\theta\cos^2\phi + \sin^2\phi}{r}\frac{\partial}{\partial r} – (\frac{2\sin\theta\cos\theta\cos^2\phi}{r^2} – \frac{\cos\theta\sin^2\phi}{r^2 \sin\theta})\frac{\partial}{\partial \theta} + \frac{2\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi}\\ &+ \sin^2\theta\sin^2\phi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\sin^2\phi}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\cos^2\phi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &+ \frac{2\sin\theta\cos\theta\sin^2\phi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta\partial \phi} + \frac{2\sin\phi\cos\phi}{r}\frac{\partial^2}{\partial\phi\partial r} \\ &+ \frac{\cos^2\theta\sin^2\phi + \cos^2\phi}{r}\frac{\partial}{\partial r} – (\frac{2\sin\theta\cos\theta\sin^2\phi}{r^2} – \frac{\cos\theta\cos^2\phi}{r^2 \sin\theta})\frac{\partial}{\partial \theta} – \frac{2\sin\phi\cos\phi}{r^2\sin^2\theta}\frac{\partial}{\partial \phi}\\ &+ \cos^2\theta\frac{\partial^2}{\partial r^2} + \frac{\sin^2\theta}{r^2}\frac{\partial ^2}{\partial \theta^2} \\ &- \frac{2\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin^2\theta}{r}\frac{\partial}{\partial r} + \frac{2\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta}\\ &= (\sin^2\theta\cos^2\phi + \sin^2\theta\sin^2\phi + \cos^2\theta)\frac{\partial^2}{\partial r^2} \\ &+ (\frac{\cos^2\theta\cos^2\phi}{r^2} + \frac{\cos^2\theta\sin^2\phi}{r^2} + \frac{\sin^2\theta}{r^2})\frac{\partial ^2}{\partial \theta^2} \\ &+ (\frac{\sin^2\phi}{r^2\sin^2\theta} + \frac{\cos^2\phi}{r^2\sin^2\theta})\frac{\partial^2}{\partial \phi^2} \\ &+ (\frac{2\sin\theta\cos\theta\cos^2\phi}{r} + \frac{2\sin\theta\cos\theta\sin^2\phi}{r} – \frac{2\sin\theta\cos\theta}{r})\frac{\partial^2}{\partial r \partial \theta} \\ &+ (\frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta} – \frac{2\cos\theta\sin\phi\cos\phi}{r^2\sin\theta})\frac{\partial^2}{\partial \theta\partial \phi} \\ &+ (\frac{2\sin\phi\cos\phi}{r} – \frac{2\sin\phi\cos\phi}{r})\frac{\partial^2}{\partial\phi\partial r} \\ &+ (\frac{\cos^2\theta\cos^2\phi + \sin^2\phi}{r} + \frac{\cos^2\theta\sin^2\phi + \cos^2\phi}{r} + \frac{\sin^2\theta}{r})\frac{\partial}{\partial r} \\ &- [(\frac{2\sin\theta\cos\theta\cos^2\phi}{r^2} – \frac{\cos\theta\sin^2\phi}{r^2 \sin\theta}) + (\frac{2\sin\theta\cos\theta\sin^2\phi}{r^2} – \frac{\cos\theta\cos^2\phi}{r^2 \sin\theta}) – \frac{2\sin\theta\cos\theta}{r^2}]\frac{\partial}{\partial \theta} \\ &+ (\frac{2\sin\phi\cos\phi}{r^2\sin^2\theta} – \frac{2\sin\phi\cos\phi}{r^2\sin^2\theta})\frac{\partial}{\partial \phi} \\ &= \frac{\partial^2}{\partial r^2} + \frac{1}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} + \frac{2}{r}\frac{\partial}{\partial r} + \frac{\cos\theta}{r^2 \sin\theta}\frac{\partial}{\partial \theta} \\ &= \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial ^2}{\partial \theta^2} + \frac{\cos\theta}{r^2 \sin\theta}\frac{\partial}{\partial \theta} + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ &= \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial \theta}(\sin\theta\frac{\partial}{\partial \theta}) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ \end{align}
\]
\[
\begin{align} \therefore\Delta = \nabla^2 &= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \\ &= \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial \theta}(\sin\theta\frac{\partial}{\partial \theta}) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \\ \end{align}
\]
というわけで，一生に一度はやってみました．解説はぜひ動画をみてください．
