計算過程
$$\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}$$
$$x=r\sin{\theta}\cos{\phi}$$
$$y=r\sin{\theta}\sin{\phi}$$
$$z=r\cos{\theta}$$
$(0\leq \theta\leq \pi,0\leq \phi\lt 2\pi,0\leq r)$
このとき、
$\dfrac{\partial}{\partial x}=\dfrac{\partial r}{\partial x}\dfrac{\partial }{\partial r}+\dfrac{\partial \phi}{\partial x}\dfrac{\partial }{\partial \phi}+\dfrac{\partial \theta}{\partial x}\dfrac{\partial }{\partial \theta}$
$\dfrac{\partial}{\partial y}=\dfrac{\partial r}{\partial y}\dfrac{\partial }{\partial r}+\dfrac{\partial \phi}{\partial y}\dfrac{\partial }{\partial \phi}+\dfrac{\partial \theta}{\partial y}\dfrac{\partial }{\partial \theta}$
$\dfrac{\partial}{\partial z}=\dfrac{\partial r}{\partial z}\dfrac{\partial }{\partial r}+\dfrac{\partial \phi}{\partial z}\dfrac{\partial }{\partial \phi}+\dfrac{\partial \theta}{\partial z}\dfrac{\partial }{\partial \theta}$
ここで、
$\dfrac{\partial r}{\partial x}=\dfrac{\partial}{\partial x}\sqrt{x^2+y^2+z^2}=\dfrac{x}{\sqrt{x^2+y^2+z^2}}=\dfrac{x}{r}=\sin{\theta}\cos{\phi}$
$\dfrac{\partial r}{\partial y}=\dfrac{\partial}{\partial y}\sqrt{x^2+y^2+z^2}=\dfrac{y}{\sqrt{x^2+y^2+z^2}}=\dfrac{y}{r}=\sin{\theta}\sin{\phi}$
$\dfrac{\partial r}{\partial z}=\dfrac{\partial}{\partial z}\sqrt{x^2+y^2+z^2}=\dfrac{z}{\sqrt{x^2+y^2+z^2}}=\dfrac{z}{r}=\cos{\theta}$
$\dfrac{\partial \phi}{\partial x}=\dfrac{1}{-\sin{\phi}}\dfrac{\partial }{\partial x}\dfrac{x}{\sqrt{x^2+y^2}}=\dfrac{1}{-\sin{\phi}}\dfrac{y^2}{(x^2+y^2)^{3/2}}=\dfrac{1}{-\sin{\phi}}\dfrac{r^2\sin^2{\theta}\sin^2{\phi}}{r^3\sin^3{\theta}}=-\dfrac{\sin{\phi}}{r\sin{\theta}}$
$\dfrac{\partial \phi}{\partial y}=\dfrac{1}{-\sin{\phi}}\dfrac{\partial }{\partial y}\dfrac{x}{\sqrt{x^2+y^2}}=\dfrac{1}{\sin{\phi}}\dfrac{xy}{(x^2+y^2)^{3/2}}=\dfrac{r^2\sin^2{\theta}\cos{\phi}\sin{\phi}}{r^3 \sin{\phi} \sin^3{\theta}}=\dfrac{\cos{\phi}}{r\sin{\theta}}$
$\dfrac{\partial \phi}{\partial z}=\dfrac{1}{-\sin{\phi}}\dfrac{\partial }{\partial z}\dfrac{x}{\sqrt{x^2+y^2}}=0$
$\dfrac{\partial \theta}{\partial x}=\dfrac{1}{-\sin{\theta}}\dfrac{\partial}{\partial x}\dfrac{z}{\sqrt{x^2+y^2+z^2}}=\dfrac{1}{\sin{\theta}}\dfrac{xz}{(x^2+y^2+z^2)^{3/2}}=\dfrac{r^2\sin{\theta}\cos{\theta}\cos{\phi}}{r^3\sin{\theta}}=\dfrac{\cos{\theta}\cos{\phi}}{r}$
$\dfrac{\partial \theta}{\partial y}=\dfrac{1}{-\sin{\theta}}\dfrac{\partial}{\partial y}\dfrac{z}{\sqrt{x^2+y^2+z^2}}=\dfrac{1}{\sin{\theta}}\dfrac{yz}{(x^2+y^2+z^2)^{3/2}}=\dfrac{r^2\sin{\theta}\cos{\theta}\sin{\phi}}{r^3\sin{\theta}}=\dfrac{\cos{\theta}\sin{\phi}}{r}$
$\dfrac{\partial \theta}{\partial z}=\dfrac{1}{-\sin{\theta}}\dfrac{\partial}{\partial z}\dfrac{z}{\sqrt{x^2+y^2+z^2}}=\dfrac{-1}{\sin{\theta}}\dfrac{x^2+y^2}{(x^2+y^2+z^2)^{3/2}}=-\dfrac{r^2\sin^2{\theta}}{r^3\sin{\theta}}=-\dfrac{\sin{\theta}}{r}$
よって、
$\dfrac{\partial }{\partial x}=\sin{\theta}\cos{\phi}\dfrac{\partial}{\partial r}-\dfrac{\sin{\phi}}{r\sin{\theta}}\dfrac{\partial }{\partial \phi}+\dfrac{\cos{\theta}\cos{\phi}}{r}\dfrac{\partial}{\partial \theta}$
$\dfrac{\partial }{\partial y}=\sin{\theta}\sin{\phi}\dfrac{\partial}{\partial r}+\dfrac{\cos{\phi}}{r\sin{\theta}}\dfrac{\partial }{\partial \phi}+\dfrac{\cos{\theta}\sin{\phi}}{r}\dfrac{\partial}{\partial \theta}$
$\dfrac{\partial }{\partial z}=\cos{\theta}\dfrac{\partial}{\partial r}-\dfrac{\sin{\theta}}{r}\dfrac{\partial}{\partial \theta}$
よって、
$$\begin{aligned}\frac{\partial^2}{\partial x^2}&=\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}\left(\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\cos{\phi}}{r}\frac{\partial}{\partial \theta}-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\cos{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\cos{\phi}}{r}\frac{\partial}{\partial \theta}-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\theta}\cos{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\cos{\phi}}{r}\frac{\partial}{\partial \theta}-\frac{\sin{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&=\sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2}+\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)-\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\cos^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)+\frac{\cos{\theta}\cos^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)-\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&-\frac{\sin{\phi}}{r}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial r}\right)-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \phi}\right)\\\end{aligned}$$
$$\begin{aligned}\frac{\partial^2}{\partial y^2}&=\sin{\theta}\sin{\phi}\frac{\partial}{\partial r}\left(\sin{\theta}\sin{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\sin{\phi}}{r}\frac{\partial}{\partial \theta}+\frac{\cos{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\sin{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\sin{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\sin{\phi}}{r}\frac{\partial}{\partial \theta}+\frac{\cos{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\theta}\sin{\phi}\frac{\partial}{\partial r}+\frac{\cos{\theta}\sin{\phi}}{r}\frac{\partial}{\partial \theta}+\frac{\cos{\phi}}{r\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&=\sin^2{\theta}\sin^2{\phi}\frac{\partial}{\partial r^2}+\sin{\theta}\cos{\theta}\sin^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)+\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\sin^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)+\frac{\cos{\theta}\sin^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\phi}}{r}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial r}\right)+\frac{\cos{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \phi}\right)\\\end{aligned}$$
$$\begin{aligned}\frac{\partial^2}{\partial z^2}&=\cos{\theta}\frac{\partial}{\partial r}\left(\cos{\theta}\frac{\partial}{\partial r}-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\right)-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial r}-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\right)\\&=\cos^2{\theta}\frac{\partial^2}{\partial r^2}-\sin{\theta}\cos{\theta}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial r}\right)+\frac{\sin{\theta}}{r^2}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial \theta}\right)\\\end{aligned}$$
よって、
$$\begin{aligned}\Delta&=\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}+\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)-\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\cos^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)+\frac{\cos{\theta}\cos^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)-\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&-\frac{\sin{\phi}}{r}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial r}\right)-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \phi}\right)\\&+\sin^2{\theta}\sin^2{\phi}\frac{\partial}{\partial r^2}+\sin{\theta}\cos{\theta}\sin^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)+\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\sin^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)+\frac{\cos{\theta}\sin^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\phi}}{r}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial r}\right)+\frac{\cos{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \phi}\right)\\&+\cos^2{\theta}\frac{\partial^2}{\partial r^2}-\sin{\theta}\cos{\theta}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)\\&-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial r}\right)+\frac{\sin{\theta}}{r^2}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial \theta}\right)\\&=\sin^2{\theta}\cos^2{\phi}\frac{\partial^2}{\partial r^2}+\sin^2{\theta}\sin^2{\phi}\frac{\partial}{\partial r^2}+\cos^2{\theta}\frac{\partial^2}{\partial r^2}\\&+\sin{\theta}\cos{\theta}\cos^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)+\sin{\theta}\cos{\theta}\sin^2{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)-\sin{\theta}\cos{\theta}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)\\&-\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)+\sin{\phi}\cos{\phi}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)-\frac{\cos{\theta}\sin{\phi}\cos{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi}\right)\\&+\frac{\cos{\theta}\cos^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)+\frac{\cos{\theta}\sin^2{\phi}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial r}\right)\\&+\frac{\cos{\theta}\cos^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\theta}\sin^2{\phi}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)+\frac{\sin{\theta}}{r^2}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial \theta}\right)\\&-\frac{\sin{\phi}}{r}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial r}\right)+\frac{\cos{\phi}}{r}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial r}\right)\\&+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \phi}\right)+\frac{\cos{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \phi}\right)\\&-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \theta}\right)\\&=\frac{\partial^2}{\partial r^2}\\&+\frac{\cos{\theta}}{r}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial r}\right)-\frac{\sin{\theta}}{r}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial r}\right)\\&+\frac{\cos{\theta}}{r^2}\frac{\partial}{\partial \theta}\left(\cos{\theta}\frac{\partial}{\partial \theta}\right)+\frac{\sin{\theta}}{r^2}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial \theta}\right)\\&-\frac{\sin{\phi}}{r}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial r}\right)+\frac{\cos{\phi}}{r}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial r}\right)\\&+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \phi}\right)+\frac{\cos{\phi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \phi}\right)\\&-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\cos{\phi}\frac{\partial}{\partial \theta}\right)+\frac{\cos{\phi}\cos{\theta}}{r^2\sin{\theta}}\frac{\partial}{\partial \phi}\left(\sin{\phi}\frac{\partial}{\partial \theta}\right)\\&=\frac{\partial^2}{\partial r^2}\\&+\frac{\cos{\theta}}{r}\left[\cos{\theta}\frac{\partial}{\partial r}+\sin{\theta}\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right]-\frac{\sin{\theta}}{r}\left[-\sin{\theta}\frac{\partial}{\partial r}+\cos{\theta}\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right]\\&+\frac{\cos{\theta}}{r^2}\left[-\sin{\theta}\frac{\partial}{\partial \theta}+\cos{\theta}\frac{\partial^2}{\partial \theta^2}\right]+\frac{\sin{\theta}}{r^2}\left[\cos{\theta}\frac{\partial}{\partial \theta}+\sin{\theta}\frac{\partial^2}{\partial \theta^2}\right]\\&-\frac{\sin{\phi}}{r}\left[-\sin{\phi}\frac{\partial}{\partial r}+\cos{\phi}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r}\right]+\frac{\cos{\phi}}{r}\left[\cos{\phi}\frac{\partial}{\partial r}+\sin{\phi}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r}\right]\\&+\frac{\sin{\phi}}{r^2\sin^2{\theta}}\left[\cos{\phi}\frac{\partial}{\partial \phi}+\sin{\phi}\frac{\partial^2}{\partial \phi^2}\right]+\frac{\cos{\phi}}{r^2\sin^2{\theta}}\left[-\sin{\phi}\frac{\partial}{\partial \phi}+\cos{\phi}\frac{\partial^2}{\partial \phi^2}\right]\\&-\frac{\sin{\phi}\cos{\theta}}{r^2\sin{\theta}}\left[-\sin{\phi}\frac{\partial}{\partial \theta}+\cos{\phi}\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta}\right]+\frac{\cos{\phi}\cos{\theta}}{r^2\sin{\theta}}\left[\cos{\phi}\frac{\partial}{\partial \theta}+\sin{\phi}\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta}\right]\\&=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2}{\partial \phi^2}-\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{r^2}{r^2}\frac{\partial^2}{\partial r^2}+\frac{2r}{r^2}\frac{\partial}{\partial r}+\frac{\sin{\theta}}{r^2\sin{\theta}}\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}\left(r^2\frac{\partial}{\partial r}\right)+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial}{\partial \theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2}{\partial \phi^2}\\\end{aligned}$$
初心者向けの罠（ここが本編）
と、ここで
$$\dfrac{\partial x}{\partial r}=\dfrac{\partial r}{\partial x}=\dfrac{x}{r}$$
という式が成り立っている。これは逆関数の微分公式$dx/dy=(dy/dx)^{-1}$を連想すると違和感を感じてしまうが、これは問題ない。
なぜかと言うと、逆関数の微分公式は1変数でしか成り立たないのである。
では、多変数の場合ではどうなるかというと、ヤコビ行列
$$\begin{pmatrix}\dfrac{\partial x}{\partial r}&\dfrac{\partial x}{\partial \phi}&\dfrac{\partial x}{\partial \theta}\\\dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial \phi}&\dfrac{\partial y}{\partial \theta}\\\dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial \phi}&\dfrac{\partial z}{\partial \theta}\\\end{pmatrix}$$
について、逆変換においてはヤコビ行列が逆行列となる。つまり、
$$\begin{pmatrix}\dfrac{\partial r}{\partial x}&\dfrac{\partial \phi}{\partial x}&\dfrac{\partial \theta}{\partial x}\\\dfrac{\partial r}{\partial y}&\dfrac{\partial \phi}{\partial y}&\dfrac{\partial \theta}{\partial y}\\\dfrac{\partial r}{\partial z}&\dfrac{\partial \phi}{\partial z}&\dfrac{\partial \theta}{\partial z}\\\end{pmatrix}^{-1}=\begin{pmatrix}\dfrac{\partial x}{\partial r}&\dfrac{\partial x}{\partial \phi}&\dfrac{\partial x}{\partial \theta}\\\dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial \phi}&\dfrac{\partial y}{\partial \theta}\\\dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial \phi}&\dfrac{\partial z}{\partial \theta}\\\end{pmatrix}$$
となっているのである。これが多変数の場合における逆関数の微分公式に対応する。よって
$$\dfrac{\partial x}{\partial r}=\dfrac{\partial r}{\partial x}=\dfrac{x}{r}$$
については、逆行列の関係の2つの行列において、単にそれぞれの1成分が等しいという状況に過ぎないので、おかしくもなんともないのである。
この罠は筆者がB1の頃に躓いたものなので備忘録として残しておくことにした次第である。
