極座標ラプラシアンの導出
直交座標\((x,y,z)\)と極座標\((r,\theta,\psi)\)の関係式
\(\frac{\partial r}{\partial x}\)を求める
\(d\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}r^2&=&\frac{\partial }{\partial x}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial x}&=&2x \\\frac{\partial r}{\partial x} &=&\frac{\cancel{2}x}{\cancel{2}r}=\frac{x}{r}=\sin{\theta}\cos{\psi}\;\cdots\;aより \end{eqnarray}$$\(\frac{\partial \theta}{\partial x}\)を求める
\(e\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}\cos{\theta}&=&\frac{\partial }{\partial x}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial x}&=& z\frac{\partial }{\partial x}\frac{1}{r} \\&=&z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial x} \\&=&-\frac{z}{r^2}\frac{\partial r}{\partial x} \\&=&-\frac{z}{r^2}\;\sin{\theta}\cos{\psi}\;\cdots\;一つ前の\frac{\partial r}{\partial x}の結果 \\\frac{\partial \theta}{\partial x}&=&\frac{z\cos{\psi}}{r^2}=\frac{z}{r}\frac{\cos{\psi}}{r} \\&=&\frac{\cos{\theta}\cos{\psi}}{r}\;\cdots\;eより \end{eqnarray}$$\(\frac{\partial \psi}{\partial x}\)を求める
\(f\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}\tan{\psi}&=&\frac{\partial }{\partial x}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial x} &=&y\frac{\partial }{\partial x}\frac{1}{x} \\&=&y\left(-\frac{1}{x^2}\right) \\&=&-\frac{y}{x}\frac{1}{x} \\&=&-\tan{\psi}\left(\frac{1}{x}\right) \\&=&-\tan{\psi}\left(\frac{1}{r\sin{\theta}\cos{\psi}}\right) \\\frac{\partial \psi}{\partial x} &=&-\tan{\psi}\left(\frac{1}{r\sin{\theta}\cos{\psi}}\right)\cos^2{\psi} \\&=&-\frac{\sin{\psi}}{\cancel{\cos{\psi}}}\left(\frac{1}{r\sin{\theta}\cancel{\cos{\psi}}}\right)\cancel{\cos^2{\psi}} \\&=&-\frac{\sin{\psi}}{r\sin{\theta}} \end{eqnarray}$$\(\frac{\partial r}{\partial y}\)を求める
\(d\)の両辺を\(y\)の偏微分をとる． dの両辺をyの偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial y}r^2&=&\frac{\partial }{\partial y}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial y}&=&2y \\\frac{\partial r}{\partial y} &=&\frac{\cancel{2}y}{\cancel{2}r}=\frac{y}{r}=\sin{\theta}\sin{\psi}\;\cdots\;bより \end{eqnarray}$$\(\frac{\partial \theta}{\partial y}\)を求める
\(e\)の両辺を\(y\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial y}\cos{\theta}&=&\frac{\partial }{\partial y}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial y}&=& z\frac{\partial }{\partial y}\frac{1}{r} \\&=&z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial y} \\&=&-\frac{z}{r^2}\frac{\partial r}{\partial y} \\&=&-\frac{z}{r^2}\;\sin{\theta}\sin{\psi}\;\cdots\;一つ前の\frac{\partial r}{\partial y}の結果 \\\frac{\partial \theta}{\partial y}&=&\left(\cancel{-}\frac{1}{\cancel{\sin{\theta}}}\right)\left(\cancel{-}\frac{z}{r^2}\cancel{\sin{\theta}}\sin{\psi}\right) =\frac{z}{r}\frac{\sin{\psi}}{r} \\&=&\frac{\cos{\theta}\sin{\psi}}{r}\;\cdots\;eより \end{eqnarray}$$\(\frac{\partial \psi}{\partial y}\)を求める
\(f\)の両辺を\(y\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial y}\tan{\psi}&=&\frac{\partial }{\partial y}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial y} &=&\frac{1}{x} \\\frac{\partial \psi}{\partial y} &=&\left(\frac{1}{x}\right)\cos^2{\psi} \\&=&\left(\frac{1}{r\sin{\theta}\cancel{\cos{\psi}}}\right)\cos^\cancel{2}{\psi}\;\cdots\;aより \\&=&\frac{\cos{\psi}}{r\sin{\theta}} \end{eqnarray}$$\(\frac{\partial r}{\partial z}\)を求める
\(d\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}r^2&=&\frac{\partial }{\partial z}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial z}&=&2z \\\frac{\partial r}{\partial z} &=&\frac{\cancel{2}z}{\cancel{2}r}=\frac{z}{r}=\cos{\theta}\;\cdots\;eより \end{eqnarray}$$\(\frac{\partial \theta}{\partial z}\)を求める
\(e\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}\cos{\theta}&=&\frac{\partial }{\partial z}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial z}&=& \frac{1}{r}+z\frac{\partial }{\partial z}\frac{1}{r} \\&=&\frac{1}{r}+z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial z} \\&=&\frac{1}{r}-\frac{z}{r^2}\frac{\partial r}{\partial z} \\&=&\frac{1}{r}-\frac{z}{r^2}\cos{\theta}\;\cdots\;一つ前の\frac{\partial r}{\partial z}の結果 \\\frac{\partial \theta}{\partial y} &=&\left(-\frac{1}{\sin{\theta}}\right)\left(\frac{1}{r}-\frac{z}{r^2}\cos{\theta}\right) \\&=&-\frac{1}{r\sin{\theta}}+\frac{z}{r}\frac{\cos{\theta}}{r\sin{\theta}} \\&=&-\frac{1}{r\sin{\theta}}+\frac{\cos^2{\theta}}{r\sin{\theta}}\;\cdots\;eより \\&=&-\frac{1-\cos^2{\theta}}{r\sin{\theta}} \\&=&-\frac{\sin^\cancel{2}{\theta}}{r\cancel{\sin{\theta}}} \\&=&-\frac{\sin{\theta}}{r} \end{eqnarray}$$\(\frac{\partial \psi}{\partial z}\)を求める
\(f\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}\tan{\psi}&=&\frac{\partial }{\partial z}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial z}&=&0 \\\frac{\partial \psi}{\partial y}&=&0 \end{eqnarray}$$\(\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\)を求める
$$\begin{eqnarray} \frac{\partial }{\partial x}&=& \frac{\partial r}{\partial x}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial x}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial x}\frac{\partial }{\partial \psi} \\&=&\sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \\\; \frac{\partial }{\partial y}&=& \frac{\partial r}{\partial y}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial y}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial y}\frac{\partial }{\partial \psi} \\&=&\sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \\\; \\\frac{\partial }{\partial z}&=& \frac{\partial r}{\partial z}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial z}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial z}\frac{\partial }{\partial \psi} \\&=&\cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} +0\frac{\partial }{\partial \psi} \\&=&\cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \end{eqnarray}$$\(\frac{\partial^2 }{\partial x^2}\)を求める
$$\begin{eqnarray} \frac{\partial^2 }{\partial x^2}&=& \frac{\partial }{\partial x}\frac{\partial }{\partial x} \\&=& \left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=&\sin{\theta}\cos{\psi} \color{red}{ \frac{\partial }{\partial r}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\theta}\cos{\psi}}{r} \color{green}{ \frac{\partial }{\partial \theta}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) } \\&&-\frac{\sin{\psi}}{r\sin{\theta}} \color{blue}{ \frac{\partial }{\partial \psi}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&=&\sin{\theta}\cos{\psi}\cdot \color{red}{A_x}\color{black}{} +\frac{\cos{\theta}\cos{\psi}}{r}\cdot \color{green}{B_x}\color{black}{} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot \color{blue}{C_x}\color{black}{} \end{eqnarray}$$
$$\begin{eqnarray}
\\A_x&=&
\frac{\partial }{\partial r}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right)
+\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta}\right)
-\frac{\partial }{\partial r}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}\right)
\\&=&
\frac{\partial }{\partial r}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\cdot\frac{\partial }{\partial
\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial
\theta}\right)
\\&&-\frac{\partial }{\partial r}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial
\psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&
\left(0\right)\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2}
\\&&+\left(-\frac{\cos{\theta}\cos{\psi}}{r^2}\right)\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
\\&&-\left(-\frac{\sin{\psi}}{r^2\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r \partial \psi}
\\&=&
\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\cos{\psi}}{r^2}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\sin{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r \partial \psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\sin{\theta}\cos{\psi}\cdot A_x
&=&\sin{\theta}\cos{\psi}
\left\{
\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\cos{\psi}}{r^2}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\sin{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r \partial \psi}
\right\}
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r^2\cancel{\sin{\theta}}}\frac{\partial }{\partial \psi}
-\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial r
\partial \psi}
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi}
-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r
\partial \psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\\B_x&=&\frac{\partial }{\partial \theta}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right)
+\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial
\theta}\right)
-\frac{\partial }{\partial \theta}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial
\psi}\right)
\\&=&
\frac{\partial }{\partial \theta}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)
\cdot\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial
\theta}\right)
\\&&-\frac{\partial }{\partial \theta}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right)
\cdot\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&
\left(\cos{\theta}\cos{\psi}\right)\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \theta \partial r}
-\frac{\sin{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos{\theta}\sin{\psi}}{r\sin^2{\theta}}\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\cos{\theta}\cos{\psi}}{r}\cdot B_x
&=&\frac{\cos{\theta}\cos{\psi}}{r}\left\{
\left(\cos{\theta}\cos{\psi}\right)\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \theta \partial r}
-\frac{\sin{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos{\theta}\sin{\psi}}{r\sin^2{\theta}}\frac{\partial }{\partial \psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi}
\right\}
\\&=&\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\\C_x&=&
\frac{\partial }{\partial \psi}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right)
+\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial
\theta}\right)
-\frac{\partial }{\partial \psi}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}\right)
\\&=&
\frac{\partial }{\partial \psi}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\cdot\frac{\partial
}{\partial
\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial
\theta}\right)
\\&&-\frac{\partial }{\partial \psi}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial
}{\partial
\psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&
\left(-\sin{\theta}\sin{\psi}\right)\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r}
\\&&+\left(-\frac{\cos{\theta}\sin{\psi}}{r}\right)\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta}
\\&&-\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\frac{\partial }{\partial\psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial^2 }{\partial \psi^2}
\\&=&
-\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r}
-\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta}
-\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial\psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2}
\end{eqnarray}$$
$$\begin{eqnarray}
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot C_x
&=&-\frac{\sin{\psi}}{r\sin{\theta}}\left\{
-\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r}
-\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta}
-\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial\psi}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2}
\right\}
\\&=&\frac{\cancel{\sin{\theta}}\sin^2{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial }{\partial r}
-\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial
\psi\partial r}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta}
+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r}
-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial
\psi\partial r}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta}
+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\partial^2 }{\partial x^2}&=&
\sin{\theta}\cos{\psi}\cdot A_x
+\frac{\cos{\theta}\cos{\psi}}{r}\cdot B_x
-\frac{\sin{\psi}}{r\sin{\theta}}\cdot C_x
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi}
-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi}
\\&&+\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi}
\\&&+\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r}
-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial \psi\partial r}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta}
+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta}
+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r}
\\&&-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi}
-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r}
\\&&-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial
\psi}
-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta}
\\&&
+\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r}
\\&&
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta}
-\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
\\&&+\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi}
+\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi}
+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi}
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}
\frac{\partial^2}{\partial r \partial \theta}
\;\cdots\;\frac{\partial^2 }{\partial r \partial \theta}=\frac{\partial^2 }{\partial \theta \partial r}を仮定
\\&&-2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi}
\;\cdots\;\frac{\partial^2 }{\partial r \partial \psi}=\frac{\partial^2 }{\partial \psi \partial r}を仮定
\\&&-2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\;\cdots\;\frac{\partial^2 }{\partial\theta\partial\psi}=\frac{\partial^2 }{\partial\psi\partial\theta}を仮定
\\&&+\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r}
\\&&
+\left(
-2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}
\right)\frac{\partial }{\partial\theta}
\\&&+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\left(
\sin^2{\theta}
+\cos^2{\theta}
+1
\right)\frac{\partial }{\partial\psi}
\;\cdots\;\sin^2{\theta}+\cos^2{\theta}=1
\\&=&
\sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}
\frac{\partial^2}{\partial r \partial \theta}
-2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi}
-2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\\&&+\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r}
+\left(
-2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}
+\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}
\right)\frac{\partial }{\partial\theta}
+2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
\end{eqnarray}$$
\(\frac{\partial^2 }{\partial y^2}\)を求める
$$\begin{eqnarray} \frac{\partial^2 }{\partial y^2}&=& \frac{\partial }{\partial y}\frac{\partial }{\partial y} \\&=& \left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=& \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} \color{red}{\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\theta}\sin{\psi}}{r} \color{green}{\frac{\partial }{\partial \theta}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\psi}}{r\sin{\theta}} \color{blue}{\frac{\partial }{\partial \psi}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&=& \sin{\theta}\sin{\psi}\frac{\partial }{\partial r}\color{red}{A_y}\color{black}{} +\frac{\cos{\theta}\sin{\psi}}{r}\color{green}{B_y}\color{black}{} +\frac{\cos{\psi}}{r\sin{\theta}}\color{blue}{C_y}\color{black}{} \end{eqnarray}$$
$$\begin{eqnarray}
A_y&=&\frac{\partial }{\partial r}
\left(
\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}
\right)
\\&=&\frac{\partial }{\partial r}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right)
\cdot\frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \cdot
\frac{\partial }{\partial r}\left( \frac{\partial }{\partial\theta} \right)
\\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial
\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&\left(0\right)\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2}
\\&&+\left(-\frac{\cos{\theta}\sin{\psi}}{r^2}\right)\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
\\&&+\left(-\frac{\cos{\psi}}{r^2\sin{\theta}}\right)\frac{\partial }{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi}
\\&=&\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\psi}}{r^2}\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
-\frac{\cos{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\sin{\theta}\sin{\psi}\cdot A_y
&=&
\sin{\theta}\sin{\psi}\left(
\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\psi}}{r^2}\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
-\frac{\cos{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi}
\right)
\\&=&\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
-\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r^2\cancel{\sin{\theta}}}\frac{\partial }{\partial\psi}
+\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}
\frac{\partial^2 }{\partial r\partial\psi}
\\&=&\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
-\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi}
+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}
\end{eqnarray}$$
$$\begin{eqnarray}
B_y&=&\frac{\partial }{\partial \theta}\left(
\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}
\right)
\\&=&\frac{\partial }{\partial \theta}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right)\cdot\frac{\partial
}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \cdot
\frac{\partial }{\partial \theta}\left( \frac{\partial }{\partial\theta} \right)
\\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial
}{\partial
\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&\cos{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\sin{\theta}\sin{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial \theta^2}
-\frac{\cos{\theta}\cos{\psi}}{r\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial\psi}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\cos{\theta}\sin{\psi}}{r}\cdot B_y
&=&\frac{\cos{\theta}\sin{\psi}}{r}\left(
\cos{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\sin{\theta}\sin{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial \theta^2}
-\frac{\cos{\theta}\cos{\psi}}{r\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial\psi}
\right)
\\&=&
\frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
-\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\end{eqnarray}$$
$$\begin{eqnarray}
C_y&=&\frac{\partial }{\partial \psi}\left(
\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}
\right)
\\&=&\frac{\partial }{\partial \psi}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right)\cdot\frac{\partial
}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \cdot
\frac{\partial }{\partial \psi}\left( \frac{\partial }{\partial\theta} \right)
\\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial
}{\partial
\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial
\psi}\right)
\\&=&\left(\sin{\theta}\cos{\psi}\right)\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r}
\\&&+\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta}
\\&&+\left(-\frac{\sin{\psi}}{r\sin{\theta}}\right)\frac{\partial}{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\cos{\psi}}{r\sin{\theta}}\cdot C_y
&=&\frac{\cos{\psi}}{r\sin{\theta}} \left(
\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}
+\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r}
+\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta}
-\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2}
\right)
\\&=&
\frac{\cancel{\sin{\theta}}\cos^2{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial }{\partial r}
+\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial
\psi\partial r}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta}
-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&
\frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta}
-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\partial^2 }{\partial y^2}
&=&
\sin{\theta}\sin{\psi}\frac{\partial }{\partial r}A_y
+\frac{\cos{\theta}\sin{\psi}}{r}B_y
+\frac{\cos{\psi}}{r\sin{\theta}}C_y
\\&=&
\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
-\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi}
+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}
\\&&+\frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
-\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\\&&+\frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta}
-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&
\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r}
\\&&+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}
+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r}
\\&&+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta}
\\&&
+\frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r}
+\frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r}
\\&&-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
\\&&-\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi}
-\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
\\&=&
\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
\;\cdots\;\frac{\partial^2 }{\partial r \partial \theta}=\frac{\partial^2 }{\partial \theta \partial r}を仮定
\\&&+2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}
\;\cdots\;\frac{\partial^2 }{\partial r \partial \psi}=\frac{\partial^2 }{\partial \psi \partial r}を仮定
\\&&+2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\;\cdots\;\frac{\partial^2 }{\partial \theta \partial \psi}=\frac{\partial^2 }{\partial \psi \partial
\theta}を仮定
\\&&
+\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r}
\\&&+\left(
-2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}
\right)\frac{\partial}{\partial \theta}
\\&&-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\left(
\sin^2{\theta}
+\cos^2{\theta}
+1
\right)\frac{\partial}{\partial\psi}
\;\cdots\;\sin^2{\theta}+\cos^2{\theta}=1
\\&=&
\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2}
+\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&&+2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta}
+2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}
+2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi}
\\&&+\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r}
+\left(
-2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}
+\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}
\right)\frac{\partial}{\partial \theta}
-2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}
\end{eqnarray}$$
\(\frac{\partial^2 }{\partial z^2}\)を求める
$$\begin{eqnarray} \frac{\partial^2 }{\partial z^2}&=& \frac{\partial }{\partial z}\frac{\partial }{\partial z} \\&=&\left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \\&=& \cos{\theta}\color{red}{\frac{\partial }{\partial r} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right)}\color{black}{} -\frac{\sin{\theta}}{r}\color{green}{\frac{\partial }{\partial \theta} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right)} \\&=&\cos{\theta}\color{red}{A_z}\color{black}{}-\frac{\sin{\theta}}{r}\color{green}{B_z}\color{black}{} \end{eqnarray}$$
$$\begin{eqnarray}
A_z&=&\frac{\partial }{\partial r}
\left(
\cos{\theta}\frac{\partial }{\partial r}
-\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta}
\right)
\\&=&
\frac{\partial }{\partial r}
\left(\cos{\theta}\right)\cdot\frac{\partial }{\partial r}
+\cos{\theta}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial r}\left(-\frac{\sin{\theta}}{r}\right)\cdot\frac{\partial }{\partial \theta}
-\frac{\sin{\theta}}{r}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta}\right)
\\&=&
\left(0\right)\frac{\partial }{\partial r}
+\cos{\theta}\frac{\partial^2 }{\partial r^2}
\\&&+\left(\frac{\sin{\theta}}{r^2}\right)\frac{\partial }{\partial \theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\\&=&
\cos{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\sin{\theta}}{r^2}\frac{\partial }{\partial \theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\end{eqnarray}$$
$$\begin{eqnarray}
\cos{\theta}\cdot A_z&=&\cos{\theta}\left(
\cos{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\sin{\theta}}{r^2}\frac{\partial }{\partial \theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\right)
\\&=&\cos^2{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\cos{\theta}\sin{\theta}}{r^2}\frac{\partial }{\partial \theta}
-\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\end{eqnarray}$$
$$\begin{eqnarray}
B_z&=&\frac{\partial }{\partial \theta}
\left(
\cos{\theta}\frac{\partial }{\partial r}
-\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta}
\right)
\\&=&
\frac{\partial }{\partial \theta}
\left(\cos{\theta}\right)\cdot\frac{\partial }{\partial r}
+\cos{\theta}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right)
\\&&+\frac{\partial }{\partial \theta}
\left(-\frac{\sin{\theta}}{r}\right)\cdot\frac{\partial }{\partial\theta}
-\frac{\sin{\theta}}{r}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial \theta}\right)
\\&=&
\left(-\sin{\theta}\right)\frac{\partial }{\partial r}
+\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r}
\\&&+\left(-\frac{\cos{\theta}}{r}\right)\frac{\partial }{\partial\theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2}
\\&=&
-\sin{\theta}\frac{\partial }{\partial r}
+\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\cos{\theta}}{r}\frac{\partial }{\partial\theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2}
\end{eqnarray}$$
$$\begin{eqnarray}
-\frac{\sin{\theta}}{r}B_z
&=&-\frac{\sin{\theta}}{r}\left(
-\sin{\theta}\frac{\partial }{\partial r}
+\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r}
-\frac{\cos{\theta}}{r}\frac{\partial }{\partial\theta}
-\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2}
\right)
\\&=&
\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r}
-\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r}
+\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
+\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\partial^2 }{\partial z^2}
&=&
\cos{\theta}\cdot A_z-\frac{\sin{\theta}}{r}\cdot B_z
\\&=&
\cos^2{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\cos{\theta}\sin{\theta}}{r^2}\frac{\partial }{\partial \theta}
-\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r}
-\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r}
+\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
+\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2}
\\&=&
\cos^2{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2}
\\&&-\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
-\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r}
\\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r}
\\&&+\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
+\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
\\&=&
\cos^2{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2}
\\&&-2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
\;\cdots\;\frac{\partial^2 }{\partial r\partial \theta}=\frac{\partial^2 }{\partial \theta\partial r}を仮定
\\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r}
\\&&+2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
\\&=&
\cos^2{\theta}\frac{\partial^2 }{\partial r^2}
+\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2}
-2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta}
+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r}
+2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta}
\end{eqnarray}$$
\(\Delta\left(=\laplacian\right)\)を求める
$$\begin{eqnarray} \Delta&=& \frac{\partial^2 }{\partial x^2} +\frac{\partial^2 }{\partial y^2} +\frac{\partial^2 }{\partial z^2} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2}{\partial r \partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} -2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} +\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \\&&+\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} +2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} +\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \\&&+\cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} +\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} +2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\cos^2{\theta}\frac{\partial^2 }{\partial r^2} \\&& +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} \\&& +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&& +2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2}{\partial r\partial\theta} +2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2}{\partial r\partial\theta} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta} \\&& \cancel{-2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}} \cancel{+2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}} \\&& \cancel{ -2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} } \cancel{ +2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} } \\&& +\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} \\&& +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} +\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} +2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&& \cancel{+2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}} \cancel{-2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}} \\&=& \sin^2{\theta}\left(\cos^2{\psi}+\sin^2{\psi}\right)\frac{\partial^2 }{\partial r^2} +\cos^2{\theta}\frac{\partial^2 }{\partial r^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{\cos^2{\theta}}{r^2}\left( \cos^2{\psi} +\sin^2{\psi} \right)\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{1}{r^2\sin^2{\theta}}\left(\sin^2{\psi}+\cos^2{\psi}\right)\frac{\partial^2 }{\partial \psi^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +2\frac{\cos{\theta}\sin{\theta}}{r}\left( \cos^2{\psi}+\sin^2{\psi} \right)\frac{\partial^2}{\partial r\partial\theta} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{ \cos^2{\theta}\left(\cos^2{\psi}+\sin^2{\psi}\right)+\left(\cos^2{\psi}+\sin^2{\psi}\right) +\sin^2{\theta} }{r}\frac{\partial }{\partial r} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\left\{ -2\frac{\cos{\theta}\sin{\theta}}{r^2}\left(\cos^2{\psi}+\sin^2{\psi}\right) +2\frac{\cos{\theta}\sin{\theta}}{r^2} +\frac{\cos{\theta}}{r^2\sin{\theta}}\left(\cos^2{\psi}+\sin^2{\psi}\right) \right\}\frac{\partial }{\partial\theta} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&=& \left(\cos^2{\theta}+\sin^2{\theta}\right)\frac{\partial^2 }{\partial r^2} \\&& +\frac{1}{r^2}\left( \cos^2{\theta} +\sin^2{\theta} \right)\frac{\partial^2 }{\partial \theta^2} \;\cdots\;\cos^2{\theta}+\sin^2{\theta}=1 \\&& +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&& \cancel{+2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta}} \cancel{-2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta}} \\&& +\frac{\cos^2{\theta}+\sin^2{\theta}+1}{r}\frac{\partial }{\partial r} \;\cdots\;\cos^2{\theta}+\sin^2{\theta}=1 \\&& +\left\{ \cancel{-2\frac{\cos{\theta}\sin{\theta}}{r^2}} \cancel{+2\frac{\cos{\theta}\sin{\theta}}{r^2}} +\frac{\cos{\theta}}{r^2\sin{\theta}} \right\}\frac{\partial }{\partial\theta} \\&=& \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 \psi^2} +\frac{2}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \end{eqnarray}$$
$$\begin{eqnarray}
\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)
&=&
\frac{1}{r^2}\left\{
\frac{\partial }{\partial r}\left(r^2\right)\cdot\frac{\partial }{\partial r}
+r^2\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right)
\right\}
\\&=&
\frac{1}{r^2}\left\{
2r\frac{\partial }{\partial r}
+r^2\frac{\partial^2}{\partial r^2}
\right\}
\\&=&
\frac{2}{r}\frac{\partial }{\partial r}
+\frac{\cancel{r^2}}{\cancel{r^2}}\frac{\partial^2}{\partial r^2}
\\&=&
\frac{\partial^2}{\partial r^2}
+\frac{2}{r}\frac{\partial }{\partial r}
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
\left(
\sin{\theta}\frac{\partial }{\partial\theta}
\right)
&=&
\frac{1}{r^2\sin{\theta}}
\left\{
\frac{\partial }{\partial\theta}\left(\sin{\theta}\right)\cdot\frac{\partial }{\partial\theta}
+\sin{\theta}\cdot\frac{\partial }{\partial\theta}\left(\frac{\partial }{\partial\theta}\right)
\right\}
\\&=&
\frac{1}{r^2\sin{\theta}}
\left\{
\cos{\theta}\frac{\partial }{\partial\theta}
+\sin{\theta}\frac{\partial^2 }{\partial\theta^2}
\right\}
\\&=&
\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
+\frac{\cancel{\sin{\theta}}}{r^2\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial\theta^2}
\\&=&
\frac{1}{r^2}\frac{\partial^2 }{\partial\theta^2}
+\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
\end{eqnarray}$$
$$\begin{eqnarray}
\Delta&=&
\frac{\partial^2 }{\partial r^2}
+\frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
+\frac{2}{r}\frac{\partial }{\partial r}
+\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
\\&=&
\left\{
\frac{\partial^2 }{\partial r^2}
+\frac{2}{r}\frac{\partial }{\partial r}
\right\}
+\left\{
\frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2}
+\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta}
\right\}
+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\\&=&
\frac{1}{r^2}
\frac{\partial }{\partial r}
\left(
r^2\frac{\partial }{\partial r}
\right)
+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}
\left(
\sin{\theta}\frac{\partial }{\partial\theta}
\right)
+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2}
\end{eqnarray}$$
