【微積分】ラプラシアンの極座標表示

　(\ref{2pt})の両辺を\(r\)で割り、右辺の形を(\ref{2pr})に揃える。

\begin{align}
&\frac{\partial}{\partial r}=\cos\theta\frac{\partial}{\partial x}+\sin\theta\frac{\partial}{\partial y} \tag{a1}\label{2pr-2}\\
&\frac{1}{r}\frac{\partial}{\partial \theta}=-\sin\theta\frac{\partial}{\partial x}+\cos\theta\frac{\partial}{\partial y} \tag{a2}\label{2pt-2}
\end{align}

　まず\(\displaystyle{\frac{\partial}{\partial x}}\)を求める。(\ref{2pr-2})の両辺に\(\cos\theta\)、(\ref{2pt-2})の両辺に\(\sin\theta\)を掛けて

\begin{align}
&\cos\theta\frac{\partial}{\partial r}=\cos^{2}\theta\frac{\partial}{\partial x}+\sin\theta\cos\theta\frac{\partial}{\partial y} \tag{a3}\label{2pr-3}\\
&\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}=-\sin^{2}\theta\frac{\partial}{\partial x}+\sin\theta\cos\theta\frac{\partial}{\partial y} \tag{a4}\label{2pt-3}
\end{align}

とし、(\ref{2pr-3})\(-\)(\ref{2pt-3})を計算して整理すると

\begin{align}
\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}&=\cos^{2}\theta\frac{\partial}{\partial x}+\sin\theta\cos\theta\frac{\partial}{\partial y}-\left(-\sin^{2}\theta\frac{\partial}{\partial x}+\sin\theta\cos\theta\frac{\partial}{\partial y}\right)\notag \\
&=(\cos^{2}\theta+\sin^{2}\theta)\frac{\partial}{\partial x}=\frac{\partial}{\partial x}
\end{align}

となり、(\ref{2px})が求められる。

　続いて\(\displaystyle{\frac{\partial}{\partial y}}\)を求める。(\ref{2pr-2})の両辺に\(\sin\theta\)、(\ref{2pt-2})の両辺に\(\cos\theta\)を掛けて

\begin{align}
&\sin\theta\frac{\partial}{\partial r}=\sin\theta\cos\theta\frac{\partial}{\partial x}+\sin^{2}\theta\frac{\partial}{\partial y} \tag{a5}\label{2pr-4}\\
&\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}=-\sin\theta\cos\theta\frac{\partial}{\partial x}+\cos^{2}\theta\frac{\partial}{\partial y} \tag{a6}\label{2pt-4}
\end{align}

とし、両辺を足し上げて整理すると

\begin{align}
\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}&=\sin\theta\cos\theta\frac{\partial}{\partial x}+\sin^{2}\theta\frac{\partial}{\partial y}-\sin\theta\cos\theta\frac{\partial}{\partial x}+\cos^{2}\theta\frac{\partial}{\partial y} \notag \\
&=(\cos^{2}\theta+\sin^{2}\theta)\frac{\partial}{\partial y}=\frac{\partial}{\partial y}
\end{align}

となり、(\ref{2py})が求められる。

　まず\(\displaystyle{\frac{\partial^{2}}{\partial x^{2}}}\)は(\ref{2px})の両辺を2乗して整理すれば

\begin{align}
\frac{\partial^{2}}{\partial x^{2}}&=\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) \notag \\
&=\cos\theta\frac{\partial}{\partial r}\left(\cos\theta\frac{\partial}{\partial r}\right)-\cos\theta\frac{\partial}{\partial r}\left(\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)\notag \\
&\quad-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\left(\cos\theta\frac{\partial}{\partial r}\right)+\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\left(\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right) \notag \\
&=\cos^{2}\theta\frac{\partial^{2}}{\partial r^{2}}-\cos\theta\left(-\frac{\sin\theta}{r^{2}}\frac{\partial}{\partial \theta}+\frac{\sin\theta}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\right) \notag \\
&\quad-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\left(-\sin\theta\frac{\partial}{\partial r}+\cos\theta\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right)+\frac{\sin\theta}{r^{2}}\left(\cos\theta\frac{\partial}{\partial \theta}+\sin\theta\frac{\partial^{2}}{\partial \theta^{2}}\right) \notag \\
&=\cos^{2}\theta\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin\theta\cos\theta}{r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}-\frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\right) \notag \\
&\quad+\frac{\sin\theta}{r}\left(\sin\theta\frac{\partial}{\partial r}-\cos\theta\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right)+\frac{\sin\theta}{r^{2}}\left(\cos\theta\frac{\partial}{\partial \theta}+\sin\theta\frac{\partial^{2}}{\partial \theta^{2}}\right)
\end{align}

となり(\ref{2px2})が求められる。

　続いて\(\displaystyle{\frac{\partial^{2}}{\partial y^{2}}}\)は(\ref{2py})の両辺を2乗して整理すれば

\begin{align}
\frac{\partial^{2}}{\partial y^{2}}&=\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right) \notag \\
&=\sin\theta\frac{\partial}{\partial r}\left(\sin\theta\frac{\partial}{\partial r}\right)+\sin\theta\frac{\partial}{\partial r}\left(\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)\notag \\
&\quad+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial r}\right)+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\left(\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right) \notag \\
&=\sin^{2}\theta\frac{\partial^{2}}{\partial r^{2}}+\sin\theta\left(-\frac{\cos\theta}{r^{2}}\frac{\partial}{\partial \theta}+\frac{\cos\theta}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\right) \notag \\
&\quad+\frac{\cos\theta}{r}\left(\cos\theta\frac{\partial}{\partial r}+\sin\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) \notag \\
&=\sin^{2}\theta\frac{\partial^{2}}{\partial r^{2}}-\frac{\sin\theta\cos\theta}{r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}-\frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\right) \notag \\
&\quad+\frac{\cos\theta}{r}\left(\cos\theta\frac{\partial}{\partial r}+\sin\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)
\end{align}

となり(\ref{2py2})が求められる。

　(\ref{2px2})と(\ref{2py2})の右辺の第2項同士が相殺されるため、残りの項を足し上げればよい。

\begin{align}
\triangle&=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} \notag \\
&=\cos^{2}\theta\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin\theta}{r}\left(\sin\theta\frac{\partial}{\partial r}-\cos\theta\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right)+\frac{\sin\theta}{r^{2}}\left(\cos\theta\frac{\partial}{\partial \theta}+\sin\theta\frac{\partial^{2}}{\partial \theta^{2}}\right) \notag \\
&+\sin^{2}\theta\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{\cos\theta}{r^{2}}\left(\sin\theta\frac{\partial}{\partial \theta}-\cos\theta\frac{\partial^{2}}{\partial \theta^{2}}\right) \notag \\
&=(\sin^{2}\theta+\cos^{2}\theta)\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin^{2}\theta+\cos^{2}\theta}{r}\frac{\partial}{\partial r}+\frac{\sin^{2}\theta+\cos^{2}\theta}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}} \notag \\
&=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}}
\end{align}

　(\ref{3pt})と(\ref{3pp})の両辺を\(r\)で割り、右辺の形を(\ref{3pr})に揃える。

\begin{align}
&\frac{\partial}{\partial r}=\sin\theta\cos\phi\frac{\partial}{\partial x}+\sin\theta\sin\phi\frac{\partial}{\partial y}+\cos\theta\frac{\partial}{\partial z} \tag{b1}\label{3pr-2}\\
&\frac{1}{r}\frac{\partial}{\partial \theta}=\cos\theta\cos\phi\frac{\partial}{\partial x}+\cos\theta\sin\phi\frac{\partial}{\partial y}-\sin\theta\frac{\partial}{\partial z} \tag{b2}\label{3pt-2}\\
&\frac{1}{r}\frac{\partial}{\partial \phi}=-\sin\theta\sin\phi\frac{\partial}{\partial x}+\sin\theta\cos\phi\frac{\partial}{\partial y}\tag{b3}\label{3pp-2}
\end{align}

　さらに(\ref{3pr-2})の両辺に\(\sin\theta\)、(\ref{3pt-2})の両辺に\(\cos\theta\)を掛けて

\begin{align}
&\sin\theta\frac{\partial}{\partial r}=\sin^{2}\theta\cos\phi\frac{\partial}{\partial x}+\sin^{2}\theta\sin\phi\frac{\partial}{\partial y}+\sin\theta\cos\theta\frac{\partial}{\partial z} \tag{b4}\label{3pr-3}\\
&\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}=\cos^{2}\theta\cos\phi\frac{\partial}{\partial x}+\cos^{2}\theta\sin\phi\frac{\partial}{\partial y}-\sin\theta\cos\theta\frac{\partial}{\partial z} \tag{b5}\label{3pt-3}
\end{align}

とし、両辺を足し上げて整理すると

\begin{align}
&\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\notag \\
=&(\sin^{2}\theta+\cos^{2}\theta)\cos\phi\frac{\partial}{\partial x}+(\sin^{2}\theta+\cos^{2}\theta)\sin\phi\frac{\partial}{\partial y} \notag \\
=&\cos\phi\frac{\partial}{\partial x}+\sin\phi\frac{\partial}{\partial y} \tag{b6}\label{rpt}
\end{align}

となるため、(\ref{rpt})の最左辺と最右辺の両辺に\(\sin\theta\)を掛ければ

\begin{align}
\sin\theta\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)=\sin\theta\cos\phi\frac{\partial}{\partial x}+\sin\theta\sin\phi\frac{\partial}{\partial y} \tag{b7}\label{rpt2}
\end{align}

となり、(\ref{3pp-2})と(\ref{rpt2})で\(\displaystyle{\frac{\partial}{\partial x},\frac{\partial}{\partial y}}\)の連立方程式になる。

　まず\(\displaystyle{\frac{\partial}{\partial x}}\)を求める。(\ref{3pp-2})の両辺に\(\sin\phi\)、(\ref{rpt2})の両辺に\(\cos\phi\)を掛けて

\begin{align}
&\frac{\sin\phi}{r}\frac{\partial}{\partial \phi}=-\sin\theta\sin^{2}\phi\frac{\partial}{\partial x}+\sin\theta\sin\phi\cos\phi\frac{\partial}{\partial y}\tag{b8}\label{rtpxy1}\\
&\sin\theta\cos\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)=\sin\theta\cos^{2}\phi\frac{\partial}{\partial x}+\sin\theta\sin\phi\cos\phi\frac{\partial}{\partial y}\tag{b9}\label{rtpxy2}
\end{align}

とし、(\ref{rtpxy2})\(-\)(\ref{rtpxy1})を計算して整理すると

\begin{align}
&\sin\theta\cos\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)-\frac{\sin\phi}{r}\frac{\partial}{\partial \phi} \notag \\
=&\sin\theta\cos^{2}\phi\frac{\partial}{\partial x}-\left(-\sin\theta\sin^{2}\phi\frac{\partial}{\partial x}\right) \notag \\
=&\sin\theta(\sin^{2}\phi+\cos^{2}\phi)\frac{\partial}{\partial x}=\sin\theta\frac{\partial}{\partial x} \tag{b10}\label{rtpxx}
\end{align}

となるため、(\ref{rtpxx})の最左辺と最右辺を\(\sin\theta\)で割って

\begin{align}
\frac{\partial}{\partial x}&=\cos\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi} \notag \\
&=\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}

となり、(\ref{3px})が求められる。

　続いて\(\displaystyle{\frac{\partial}{\partial y}}\)を求める。(\ref{3pp-2})の両辺に\(\cos\phi\)、(\ref{rpt2})の両辺に\(\sin\phi\)を掛けて

\begin{align}
&\frac{\cos\phi}{r}\frac{\partial}{\partial \phi}=-\sin\theta\sin\phi\cos\phi\frac{\partial}{\partial x}+\sin\theta\cos^{2}\phi\frac{\partial}{\partial y}\tag{b11}\label{rtpxy3}\\
&\sin\theta\sin\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)=\sin\theta\sin\phi\cos\phi\frac{\partial}{\partial x}+\sin\theta\sin^{2}\phi\frac{\partial}{\partial y}\tag{b12}\label{rtpxy4}
\end{align}

とし、両辺を足し上げて整理すると

\begin{align}
&\frac{\cos\phi}{r}\frac{\partial}{\partial \phi}+\sin\theta\sin\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)\notag \\
=&\sin\theta\cos^{2}\phi\frac{\partial}{\partial y}+\sin\theta\sin^{2}\phi\frac{\partial}{\partial y} \notag \\
=&\sin\theta(\sin^{2}\phi+\cos^{2}\phi)\frac{\partial}{\partial y}=\sin\theta\frac{\partial}{\partial y} \tag{b13}\label{rtpyy}
\end{align}

となるため、(\ref{rtpyy})の最左辺と最右辺を\(\sin\theta\)で割って

\begin{align}
\frac{\partial}{\partial y}&=\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}+\sin\phi\left(\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right) \notag \\
&=\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}

となり、(\ref{3py})が求められる。\\

　最後に\(\displaystyle{\frac{\partial}{\partial z}}\)を求める。(\ref{3pr-2})の両辺に\(\cos\theta\)、(\ref{3pt-2})の両辺に\(\sin\theta\)を掛けて

\begin{align}
&\cos\theta\frac{\partial}{\partial r}=\sin\theta\cos\theta\cos\phi\frac{\partial}{\partial x}+\sin\theta\cos\theta\sin\phi\frac{\partial}{\partial y}+\cos^{2}\theta\frac{\partial}{\partial z} \tag{b14}\label{3pr-2-2}\\
&\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}=\sin\theta\cos\theta\cos\phi\frac{\partial}{\partial x}+\sin\theta\cos\theta\sin\phi\frac{\partial}{\partial y}-\sin^{2}\theta\frac{\partial}{\partial z} \tag{b15}\label{3pt-2-2}
\end{align}

とし(\ref{3pr-2-2})\(-\)(\ref{3pt-2-2})を計算して整理すると

\begin{align}
\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}&=\cos^{2}\theta\frac{\partial}{\partial z}-\left(-\sin^{2}\theta\frac{\partial}{\partial z}\right) \notag \\
&=(\sin^{2}\theta+\cos^{2}\theta)\frac{\partial}{\partial z}=\frac{\partial}{\partial z}
\end{align}

となり、(\ref{3pz})が求められる。

\begin{align}
&\left(\sin\theta\cos\phi\frac{\partial}{\partial r}\right)^{2}+\left(\sin\theta\sin\phi\frac{\partial}{\partial r}\right)^{2}+\left(\cos\theta\frac{\partial}{\partial r}\right)^{2} \notag \\
=&\sin^{2}\theta\cos^{2}\phi\frac{\partial^{2}}{\partial r^{2}}+\sin^{2}\theta\sin^{2}\phi\frac{\partial^{2}}{\partial r^{2}}+\cos^{2}\theta\frac{\partial^{2}}{\partial r^{2}} \notag \\
=&(\sin^{2}\phi+\cos^{2}\phi)\sin^{2}\theta\frac{\partial^{2}}{\partial r^{2}}+\cos^{2}\theta\frac{\partial^{2}}{\partial r^{2}} \notag \\
=&(\sin^{2}\theta+\cos^{2}\theta)\frac{\partial^{2}}{\partial r^{2}}=\frac{\partial^{2}}{\partial r^{2}}
\end{align}

\begin{align}
&\left(\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta}\right)^{2}+\left(\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta}\right)^{2}+\left(-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)^{2}\notag \\
=&\frac{\sin^{2}\phi+\cos^{2}\phi}{r^{2}}\left(\cos\theta\frac{\partial}{\partial \theta}\right)^{2}+\frac{1}{r^{2}}\left(\sin\theta\frac{\partial}{\partial \theta}\right)^{2} \notag \\
=&\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) \notag \\
=&\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) \notag \\
=&\frac{\sin^{2}\theta+\cos^{2}\theta}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}}=\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \theta^{2}}
\end{align}

\begin{align}
&\left(-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\right)^{2}+\left(\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\right)^{2}\notag \\
=&\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) \notag \\
=&\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) \notag \\
=&\frac{\sin^{2}\phi+\cos^{2}\phi}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial \phi^{2}}=\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial \phi^{2}}
\end{align}

\begin{align}
&\sin\theta\cos\phi\frac{\partial}{\partial r}\left(\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta}\right)+\sin\theta\sin\phi\frac{\partial}{\partial r}\left(\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta}\right)\notag \\
=&\sin\theta\cos\theta(\cos^{2}\phi+\sin^{2}\phi)\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right) \notag \\
=&\sin\theta\cos\theta\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right) \notag \\
=&\sin\theta\cos\theta\left(-\frac{1}{r^{2}}\frac{\partial}{\partial \theta}+\frac{1}{r}\frac{\partial}{\partial r}\frac{\partial}{\partial \theta}\right)
\end{align}

\begin{align}
&\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta}\left(\sin\theta\cos\phi\frac{\partial}{\partial r}\right)+\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta}\left(\sin\theta\sin\phi\frac{\partial}{\partial r}\right) \notag \\
=&\frac{\cos\theta}{r}(\cos^{2}\phi+\sin^{2}\phi)\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial r}\right) \notag \\
=&\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial r}\right) \notag \\
=&\frac{\cos\theta}{r}\left(\cos\theta\frac{\partial}{\partial r}+\sin\theta\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}\right)
\end{align}

\begin{align}
D_{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) \notag \\
&=\frac{\cos^{2}\theta+\sin^{2}\theta}{r}\frac{\partial}{\partial r}+\frac{\sin\theta\cos\theta}{r}\frac{\partial}{\partial \theta}\frac{\partial}{\partial r}-\frac{\sin\theta\cos\theta}{r}\frac{\partial}{\partial \theta}\frac{\partial}{\partial r} \notag \\
&=\frac{1}{r}\frac{\partial}{\partial r}
\end{align}

\begin{align}
D_{3}&=\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\left(\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial \theta}\right)-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\left(\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial \theta}\right) \notag \\
&=\frac{\cos\theta\cos\phi}{r^{2}\sin\theta}\frac{\partial}{\partial \phi}\left(\sin\phi\frac{\partial}{\partial \theta}\right)-\frac{\cos\theta\sin\phi}{r^{2}\sin\theta}\frac{\partial}{\partial \phi}\left(\cos\phi\frac{\partial}{\partial \theta}\right) \notag \\
&=\frac{\cos\theta\cos\phi}{r^{2}\sin\theta}\left(\cos\phi\frac{\partial}{\partial \theta}+\sin\phi\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta}\right)-\frac{\cos\theta\sin\phi}{r^{2}\sin\theta}\left(-\sin\phi\frac{\partial}{\partial \theta}+\cos\phi\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta}\right) \notag \\
&=\frac{\cos\theta(\cos^{2}\phi+\sin^{2}\phi)}{r^{2}\sin\theta}\frac{\partial}{\partial \theta}+\frac{\cos\theta\sin\phi\cos\phi}{r^{2}\sin\theta}\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta}-\frac{\cos\theta\sin\phi\cos\phi}{r^{2}\sin\theta}\frac{\partial}{\partial \phi}\frac{\partial}{\partial \theta} \notag \\
&=\frac{\cos\theta}{r^{2}\sin\theta}\frac{\partial}{\partial \theta}
\end{align}

\begin{align}
D_{4}&=-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\left(\sin\theta\cos\phi\frac{\partial}{\partial r}\right)+\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial \phi}\left(\sin\theta\sin\phi\frac{\partial}{\partial r}\right) \notag \\
&=-\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) \notag \\
&=-\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) \notag \\
&=\frac{\sin^{2}\phi+\cos^{2}\phi}{r}\frac{\partial}{\partial r}-\frac{\sin\phi\cos\phi}{r}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r}+\frac{\sin\phi\cos\phi}{r}\frac{\partial}{\partial \phi}\frac{\partial}{\partial r} \notag \\
&=\frac{1}{r}\frac{\partial}{\partial r}
\end{align}

\begin{align}
\triangle&=D_{1}+D_{2}+D_{3}+D_{4} \notag \\
&=\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{1}{r}\frac{\partial}{\partial r}+\frac{\cos\theta}{r^{2}\sin\theta}\frac{\partial}{\partial \theta}+\frac{1}{r}\frac{\partial}{\partial r} \notag \\
&=\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right)
\end{align}