%<PLAINTEXT>
%
% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 8-3 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : g240947 $B0f>eF;M:(B
% $B:n@.G/7nF|(B : /02/10/15
% $BEE;R%a!<%k(B : micchan@bf7.so-net.ne.jp
%
%  \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%
\mondai
{g240947 $B0f>eF;M:(B}
{02/10/15}
{miccchan@bf7.so.net.ne.jp}
{8-3}
{$B#3<!85$N%i%W%i%7%"%s$N6K:BI8I=<((B(8.42$B!K$rF3$1(B}
{
$BHyJ,1i;;;R%i%W%i%7%"%s$r(B3$B<!856K:BI8$GI=$9$H$I$&$J$k$+$r9M$($k!#(B
3$B<!85D>8r:BI87O$G$O%i%W%i%7%"%s(B$\nabla^2$$B$O(B


\begin{eqnarray}
\nabla^{2}
=\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}
+\frac{\partial^2}{\partial z^2}
\label{eq:Q8-3-1}
\end{eqnarray}


$B$HI=$;$k!#$3$3$G(B$(x,y,z)$$B$H(B$(r,\theta,\varphi)$$B$N4V$NJQ?tJQ49$N<0$O!"(B
$(0<r<\infty,0<\theta<\pi,0<\varphi<2\pi)$$B$H$*$$$F(B

\begin{eqnarray}
x &=& r\sin\theta\cos\varphi \label{eq:Q8-3-2} \\
y &=& r\sin\theta\sin\varphi  \label{eq:Q8-3-3}\\
z &=& r\cos\theta         \label{eq:Q8-3-4}
\end{eqnarray}

$B$G$"$j!"$3$l$r5U$K2r$/$H!"(B

\begin{eqnarray}
r &=& \sqrt{x^{2} + y^{2} + z^{2}} \label{eq:Q8-3-5} \\
\cos\theta &=& \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \label{eq:Q8-3-6} \\
\tan\varphi &=& \frac{y}{x} \label{eq:Q8-3-7}
\end{eqnarray}

$B$H$J$k!#(B$(x,y,z)$$B$K$h$kJPHyJ,$r(B$(r,\theta,\varphi)$$B$K$h$kJPHyJ,$GI=$9$H<!$N(B
$B$h$&$K$J$k!#(B

\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 \varphi}{\partial
x}\frac{\partial}{\partial \varphi} \label{eq:Q8-3-8} \\
\frac{\partial}{\partial y} &=& \frac{\partial r}{\partial
y}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial
y}\frac{\partial}{\partial \theta} + \frac{\partial \varphi}{\partial
y}\frac{\partial}{\partial \varphi} \label{eq:Q8-3-9} \\
\frac{\partial}{\partial z} &=& \frac{\partial r}{\partial
z}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial
z}\frac{\partial}{\partial \theta} + \frac{\partial \varphi}{\partial
z}\frac{\partial}{\partial \varphi} \label{eq:Q8-3-10}
\end{eqnarray}



$B$D$^$j!"(B$\partial r/\partial x$,~$\partial \theta/\partial x$,~$\partial
\varphi/\partial x$,~$\partial r/\partial y$,~$\partial \theta/\partial
y$,~$\partial \varphi/\partial y$,~$\partial r/\partial z$,~$\partial
\theta/\partial z$,~$\partial \varphi/\partial z$
$B$,5a$^$l$P$h$$$H$o$+$k!#$3$3$G!"<0(B(\ref{eq:Q8-3-5})$B$r(B$x$$B$GJPHyJ,$9$k$H!"<!$N(B
$B$h$&$K$J$k!#(B

\begin{eqnarray}
\frac{\partial}{\partial x}r^2 &=& \frac{\partial}{\partial
x}\left(x^2+y^2+z^2\right) \nonumber \\
2r\frac{\partial r}{\partial x} &=& 2x \nonumber \\
\frac{\partial r}{\partial x} &=& \frac{x}{r} \nonumber \\
&=& \sin \theta \cos \varphi  \label{eq:Q8-3-11}
\end{eqnarray}

$y$,~$z$$B$G$bF1MM$K$7$F!"(B
\begin{eqnarray}
\frac{\partial}{\partial y}r^2 &=& \frac{\partial}{\partial
y}\left(x^2+y^2+z^2\right) \nonumber \\
2r\frac{\partial r}{\partial y} &=& 2y \nonumber \\
\frac{\partial r}{\partial y} &=& \frac{y}{r} \nonumber \\
&=& \sin \theta \sin \varphi  \label{eq:Q8-3-12}
\end{eqnarray}

\begin{eqnarray}
\frac{\partial}{\partial z}r^2 &=& \frac{\partial}{\partial
z}\left(x^2+y^2+z^2\right) \nonumber \\
2r\frac{\partial r}{\partial z} &=& 2z \nonumber \\
\frac{\partial r}{\partial z} &=& \frac{z}{r} \nonumber \\
&=& \cos \theta  \label{eq:Q8-3-13}
\end{eqnarray}

$B$,F@$i$l$k!#$^$?!"<0(B(\ref{eq:Q8-3-6})$B$r(B$x$$B$GJPHyJ,$9$k$H!"<!$N$h$&$K$J$k!#(B

\begin{eqnarray}
\frac{\partial}{\partial x}\cos\theta &=& \frac{\partial}{\partial
x}\left(\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\right) \nonumber \\
-\sin\theta\frac{\partial \theta}{\partial x}
&=& -\frac{1}{2}\frac{2xz}{\sqrt{\left({x^{2} + y^{2} + z^{2}}\right)^3}}
\nonumber \\
\sin\theta\frac{\partial \theta}{\partial x} &=& \frac{xz}{r^3} \nonumber \\
&=& \frac{r^2\sin\theta\cos\varphi\cos\theta}{r^3} \nonumber \\
\frac{\partial \theta}{\partial x} &=& \frac{\cos\varphi\cos\theta}{r}
\label{eq:Q8-3-14}
\end{eqnarray}

$y$,~$z$$B$G$bF1MM$K$7$F!"(B
\begin{eqnarray}
\frac{\partial}{\partial y}\cos\theta &=& \frac{\partial}{\partial
y}\left(\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\right) \nonumber \\
-\sin\theta\frac{\partial \theta}{\partial y}
&=& -\frac{1}{2}\frac{2yz}{\sqrt{\left({x^{2} + y^{2} + z^{2}}\right)^3}}
\nonumber \\
\sin\theta\frac{\partial \theta}{\partial y} &=& \frac{yz}{r^3} \nonumber \\
&=& \frac{r^2\sin\theta\sin\varphi\cos\theta}{r^3} \nonumber \\
\frac{\partial \theta}{\partial y} &=& \frac{\sin\varphi\cos\theta}{r}
\label{eq:Q8-3-15}
\end{eqnarray}

\begin{eqnarray}
\frac{\partial}{\partial z}\cos\theta &=& \frac{\partial}{\partial
z}\left(\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}\right) \nonumber \\
-\sin\theta\frac{\partial \theta}{\partial z} &=& \frac{1}{\sqrt{x^{2} +
y^{2} + z^{2}}} -\frac{1}{2}\frac{2z^2}{\sqrt{\left({x^{2} + y^{2} +
z^{2}}\right)^3}} \nonumber \\
\sin\theta\frac{\partial \theta}{\partial z} &=& - \frac{1}{r} +
\frac{z^2}{r^3} \nonumber \\
&=& - \frac{1}{r} + \frac{r^2 \cos^2\theta}{r^3} \nonumber \\
&=& \frac{\cos^2\theta-1}{r} \nonumber \\
&=& - \frac{\sin^2\theta}{r} \nonumber \\
\frac{\partial \theta}{\partial z} &=& - \frac{\sin\theta}{r}
\label{eq:Q8-3-16}
\end{eqnarray}


$B$,F@$i$l$k!#<!$K!"<0(B(\ref{eq:Q8-3-7})$B$bF1MM$K(B$x$,~$y$,~$z$$B$=$l$>$l$GJPHyJ,$9(B
$B$k!#(B

$x$$B$G$NJPHyJ,!#(B
\begin{eqnarray}
\frac{\partial}{\partial x}\tan\varphi &=& \frac{\partial}{\partial
x}\left(\frac{y}{x}\right) \nonumber \\
\frac{1}{\cos^2\varphi}\frac{\partial \varphi}{\partial x} &=& -
\frac{y}{x^2} \nonumber \\
\frac{\partial \varphi}{\partial x} &=& - \frac{\sin\varphi}{r\sin\theta}
\label{eq:Q8-3-17}
\end{eqnarray}

$y$$B$G$NJPHyJ,!#(B
\begin{eqnarray}
\frac{\partial}{\partial y}\tan\varphi &=& \frac{\partial}{\partial
y}\left(\frac{y}{x}\right) \nonumber \\
\frac{1}{\cos^2\varphi}\frac{\partial \varphi}{\partial y} &=& \frac{1}{x}
\nonumber \\
&=& \frac{1}{r\sin\theta\cos\varphi} \nonumber \\
\frac{\partial \varphi}{\partial y} &=& \frac{\cos\varphi}{r\sin\theta}
\label{eq:Q8-3-18}
\end{eqnarray}

$z$$B$G$NJPHyJ,!#(B
\begin{eqnarray}
\frac{\partial}{\partial z}\tan\varphi &=& \frac{\partial}{\partial
z}\left(\frac{y}{x}\right) \nonumber \\
\frac{1}{\cos^2\varphi}\frac{\partial \varphi}{\partial y} &=& 0 \nonumber
\\
\frac{\partial \varphi}{\partial y} &=& 0  \label{eq:Q8-3-19}
\end{eqnarray}

$B$3$l$GI,MW$J$b$N$O5a$^$C$?!#<!$K!"(B$\partial/\partial x$,~$\partial/\partial
y$,~$\partial/\partial x$$B$r7W;;$9$k!#(B


$B<0(B(\ref{eq:Q8-3-8}),$B<0(B(\ref{eq:Q8-3-11}),$B<0(B(\ref{eq:Q8-3-14}),$B<0(B
(\ref{eq:Q8-3-17})$B$h$j!"(B
\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 \varphi}{\partial
x}\frac{\partial}{\partial \varphi} \nonumber\\
&=& \sin\theta\cos\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}
\label{eq:Q8-3-20}
\end{eqnarray}


$B$,F@$i$l$k!#<!$K<0(B(\ref{eq:Q8-3-9}),$B<0(B(\ref{eq:Q8-3-12}),$B<0(B
(\ref{eq:Q8-3-15}),$B<0(B(\ref{eq:Q8-3-18})$B$h$j!"(B
\begin{eqnarray}
\frac{\partial}{\partial y} &=& \frac{\partial r}{\partial
y}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial
y}\frac{\partial}{\partial \theta} + \frac{\partial \varphi}{\partial
y}\frac{\partial}{\partial \varphi} \nonumber\\
&=& \sin\theta\sin\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}
\label{eq:Q8-3-21}
\end{eqnarray}


$B$,F@$i$l$k!#<!$K<0(B(\ref{eq:Q8-3-10}),$B<0(B(\ref{eq:Q8-3-13}),$B<0(B
(\ref{eq:Q8-3-16}),$B<0(B(\ref{eq:Q8-3-19})$B$h$j!"(B
\begin{eqnarray}
\frac{\partial}{\partial z} &=& \frac{\partial r}{\partial
z}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial
z}\frac{\partial}{\partial \theta} + \frac{\partial \varphi}{\partial
z}\frac{\partial}{\partial \varphi} \nonumber\\
&=& \cos\theta\frac{\partial}{\partial r} -
\frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \label{eq:Q8-3-22}
\end{eqnarray}


$B$,F@$i$l$k!#<0(B(\ref{eq:Q8-3-1})$B$K<0(B(\ref{eq:Q8-3-20}),$B<0(B(\ref{eq:Q8-3-21}),
$B<0(B(\ref{eq:Q8-3-22})$B$rBeF~$9$l$P!"#3<!85$N%i%W%i%7%"%s$N6K:BI8I=<((B(8.42$B!K$r(B
$BF3$1$k!#$=$NA0$K(B$\partial^2/\partial x^2$,~$\partial^2/\partial
y^2$,~$\partial^2/\partial z^2$$B$r7W;;$7$F$*$/!#(B

\begin{eqnarray}
&&\frac{\partial^2}{\partial x^2} \nonumber \\
&=& \left(\sin\theta\cos\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial
\varphi}\right)\left(\sin\theta\cos\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&=& \sin^2\theta\cos^2\varphi\frac{\partial^2}{\partial r^2}
+ \sin\theta\cos\theta\cos^2\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)
- \sin\varphi\cos\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \varphi}\right) \nonumber \\
&+& \frac{\cos\theta\cos^2\varphi}{r}\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)
+ \frac{\cos\theta\cos^2\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\cos\theta\frac{\partial}{\partial \theta}\right)
- \frac{\cos\theta\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&-& \frac{\sin\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial r}\right)
- \frac{\cos\theta\sin\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \theta}\right)
+ \frac{\sin\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \varphi}\right)
\label{eq:Q8-3-23}
\end{eqnarray}

\begin{eqnarray}
&&\frac{\partial^2}{\partial y^2} \nonumber \\
&=& \left(\sin\theta\sin\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial
\varphi}\right)\left(\sin\theta\sin\varphi\frac{\partial}{\partial r} +
\frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} -
\frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&=& \sin^2\theta\sin^2\varphi\frac{\partial^2}{\partial r^2}
+ \sin\theta\cos\theta\sin^2\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)
+ \sin\varphi\cos\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \varphi}\right) \nonumber \\
&+& \frac{\cos\theta\sin^2\varphi}{r}\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)
+ \frac{\cos\theta\sin^2\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\cos\theta\frac{\partial}{\partial \theta}\right)
+ \frac{\cos\theta\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&+& \frac{\cos\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial r}\right)
+ \frac{\cos\theta\cos\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \theta}\right)
+ \frac{\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \varphi}\right)
\label{eq:Q8-3-24}
\end{eqnarray}

\begin{eqnarray}
&& \frac{\partial^2}{\partial z^2} \nonumber \\
&=& \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) \nonumber \\
&=& \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) \nonumber \\
&-& \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)
\label{eq:Q8-3-25}
\end{eqnarray}


$B$h$C$F!"<0(B(\ref{eq:Q8-3-1})$B$K<0(B(\ref{eq:Q8-3-23}),$B<0(B(\ref{eq:Q8-3-24}),$B<0(B
(\ref{eq:Q8-3-25})$B$rBeF~$9$k$H!"(B

\begin{eqnarray}
\nabla^{2} &=& \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial
y^2} + \frac{\partial^2}{\partial z^2} \nonumber \\
&=& \sin^2\theta\cos^2\varphi\frac{\partial^2}{\partial r^2}
+ \sin\theta\cos\theta\cos^2\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)
- \sin\varphi\cos\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \varphi}\right) \nonumber \\
&+& \frac{\cos\theta\cos^2\varphi}{r}\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)
+ \frac{\cos\theta\cos^2\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\cos\theta\frac{\partial}{\partial \theta}\right)
- \frac{\cos\theta\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&-& \frac{\sin\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial r}\right)
- \frac{\cos\theta\sin\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \theta}\right)
+ \frac{\sin\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \varphi}\right) \nonumber
\\
&+& \sin^2\theta\sin^2\varphi\frac{\partial^2}{\partial r^2}
+ \sin\theta\cos\theta\sin^2\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)
+ \sin\varphi\cos\varphi\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial \varphi}\right) \nonumber \\
&+& \frac{\cos\theta\sin^2\varphi}{r}\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)
+ \frac{\cos\theta\sin^2\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\cos\theta\frac{\partial}{\partial \theta}\right)
+ \frac{\cos\theta\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial
\theta}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \varphi}\right)
\nonumber \\
&+& \frac{\cos\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial r}\right)
- \frac{\cos\theta\cos\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \theta}\right)
+ \frac{\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \varphi}\right) \nonumber
\\
&+& \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) \nonumber \\
&-& \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) \nonumber
\end{eqnarray}
\\



$B$3$3$G!"Bh(B1$B9`!"Bh(B10$B9`!"Bh(B19$B9`(B~($B6&DL9`!'(B$\frac{\partial^2}{\partial r^2}$)~$BBh(B
4$B9`!"Bh(B13$B9`(B~($B6&DL9`!'(B$\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)$)~
$BBh(B5$B9`!"Bh(B14$B9`(B~($B6&DL9`!'(B$\frac{\partial}{\partial
\theta}\left(\sin\theta\frac{\partial}{\partial r}\right)$)~$B$r$=$l$>$l$^$H$a(B
$B$k!#$^$?Bh(B2$B9`$HBh(B11$B9`$HBh(B20$B9`!"Bh(B3$B9`$HBh(B12$B9`!"Bh(B6$B9`$HBh(B15$B9`$G>C$($k!#(B
$\cos^2\varphi+\sin^2\varphi=1$$B$rMxMQ$9$k!#(B


\begin{eqnarray}
&=& \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) \nonumber \\
&-& \frac{\sin\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial r}\right) +
\frac{\cos\varphi}{r}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial r}\right) \nonumber \\
&+& \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) \nonumber \\
&-& \frac{\cos\theta\sin\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \theta}\right) +
\frac{\cos\theta\cos\varphi}{r^2\sin\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \theta}\right) \nonumber
\\
&+& \frac{\sin\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\sin\varphi\frac{\partial}{\partial \varphi}\right) +
\frac{\cos\theta}{r^2\sin^2\theta}\frac{\partial}{\partial
\varphi}\left(\cos\varphi\frac{\partial}{\partial \varphi}\right) \nonumber
\end{eqnarray}
\\
$B$3$3$G!"(B
\begin{eqnarray}
\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial
r}\right) &=& \cos\theta +
\sin\theta\frac{\partial^2}{\partial\theta\partial r}$B!!(B\nonumber
\end{eqnarray}
$B0J2<<!$N(B
\begin{eqnarray*}
\frac{\partial}{\partial \theta}\left(\cos\theta\frac{\partial}{\partial
r}\right),~
\frac{\partial}{\partial \varphi}\left(\cos\varphi\frac{\partial}{\partial
r}\right),~
\frac{\partial}{\partial \varphi}\left(\sin\varphi\frac{\partial}{\partial
r}\right),~
\frac{\partial}{\partial \theta}\left(\cos\theta\frac{\partial}{\partial
\theta}\right),~
\frac{\partial}{\partial \theta}\left(\cos\theta\frac{\partial}{\partial
\theta}\right)\\
\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial
\theta}\right),~
\frac{\partial}{\partial \varphi}\left(\cos\varphi\frac{\partial}{\partial
\theta}\right),~
\frac{\partial}{\partial \varphi}\left(\sin\varphi\frac{\partial}{\partial
\theta}\right),~
\frac{\partial}{\partial \varphi}\left(\sin\varphi\frac{\partial}{\partial
\varphi}\right),~
\frac{\partial}{\partial \varphi}\left(\cos\varphi\frac{\partial}{\partial
\varphi}\right)
\end{eqnarray*}
\\$B$G$bF1MM$KJQ7A$7!"6&DL9`$r8+$D$1$^$H$a$k$H!"(B

\begin{eqnarray}
&=& \frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta +
\sin^2\theta}{r}\frac{\partial}{\partial r} + \frac{\cos^2\varphi +
\sin^2\varphi}{r}\frac{\partial}{\partial r} \nonumber \\
&+& \frac{\cos^2\theta + \sin^2\theta}{r^2}\frac{\partial^2}{\partial
\theta^2} + \frac{\cos\theta\left(\sin^2\varphi +
\cos^2\varphi\right)}{r^2\sin\theta}\frac{\partial}{\partial \theta}$B!!(B
\nonumber \\
&+& \frac{\sin^2\varphi +
\cos^2\varphi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2}
\nonumber
\end{eqnarray}

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\begin{eqnarray}
&=& \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 \varphi^2} \nonumber \\
&=& \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 \varphi^2}
\label{eq:Q8-3-26}
\end{eqnarray}

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