球座標系の3次元ラプラシアンの導出
計算力をつけるためにとTexでの数式入力に慣れるために3次元の球座標系のラプラシアンの導出をしようと思います.
\begin{align*} \end{align*}
直交座標系のラプラシアン
直交座標系のラプラシアンは \begin{align} \label{lap1} \Delta & =\frac{\partial ^{2}}{\partial x^{2}} +\frac{\partial ^{2}}{\partial y^{2}} +\frac{\partial ^{2}}{\partial z^{2}} \end{align} となります。
球座標系のラプラシアン
球座標系のラプラシアンは \begin{align} \label{lap2} \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 \varphi ^{2}} +\frac{2}{r}\frac{\partial }{\partial r} +\frac{\cos \theta }{r^{2}\sin \theta }\frac{\partial }{\partial \theta } \\ \label{lap3} & =\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}} \end{align} となります。
準備
(\ref{lap1})から(\ref{lap2})または(\ref{lap3})を求めるために座標変換をします。
直交座標\( (x,y,z ) \)から球座標 \( (r,\theta,\varphi) \)へ座標変換をするので
\begin{align*} \label{memo1} x & =r\sin \theta \cos \varphi \\ y & =r\sin \theta \sin \varphi \tag{4}\\ z & =r\cos \theta \end{align*}
であり、
\begin{align*} r & =\sqrt{x^{2} +y^{2} +z^{2}}\\ \label{memo2} \theta & =\tan^{-1}\frac{\sqrt{x^{2} +y^{2}}}{z} \tag{5}\\ \varphi & =\tan^{-1}\frac{y}{x} \end{align*} です。
1階偏微分を求める
1階偏微分である\( \nabla \)を求めていきます。
\( \nabla \)は
\begin{align*} \frac{\partial}{\partial x} +\frac{\partial}{\partial y} +\frac{\partial}{\partial z} \end{align*}
です。
では、\( \nabla \) の各項について求めます。
\( x \)について
連鎖律より
\begin{align*} \label{x1} \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 } \tag{6} \end{align*}
が得られます。
(\ref{memo2})を\( x \)で1階偏微分すると \begin{align*} \frac{\partial r}{\partial x} & =\sin \theta \cos \varphi \\ \frac{\partial \theta }{\partial x} & =\frac{\cos \theta \cos \varphi }{r}\\ \tag{7} \frac{\partial \varphi }{\partial x} & =\frac{\sin \varphi }{r\sin \theta } \end{align*}
なので、これらを(\ref{x1})に代入すると
\begin{align*} \frac{\partial }{\partial x} & =\sin \theta \cos \varphi \frac{\partial }{\partial r} \tag{8} +\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta } +\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi } \end{align*}
となります。
\( y \)について
連鎖律より
\begin{align*} \label{y1} \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 } \tag{9} \end{align*}
が得られます。
(\ref{memo2})を\( y \)で1階偏微分すると
\begin{align*} \frac{\partial r}{\partial y} & =\sin \theta \sin \varphi \\ \tag{10} \frac{\partial \theta }{\partial y} & =\frac{\cos \theta \sin \varphi }{r}\\ \frac{\partial \varphi }{\partial y} & =\frac{\cos \varphi }{r\sin \theta } \end{align*}
なので、これらを(\ref{y1})に代入すると
\begin{align*} \frac{\partial }{\partial y} & =\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 } \tag{11} \end{align*}
となります。
\( z \)について
連鎖律より
\begin{align*} \label{z1} \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 } \tag{12} \end{align*}
が得られます。
(\ref{memo2})を\( z \)で1階偏微分すると
\begin{align*} \frac{\partial r}{\partial z} & =\cos \theta \\ \tag{13} \frac{\partial \theta }{\partial z} & =-\frac{\sin \varphi }{r}\\ \frac{\partial \varphi }{\partial z} & =0 \end{align*}
なので、これらを (\ref{z1})に代入すると
\begin{align*} \tag{14} \frac{\partial }{\partial z} & =\cos \theta \frac{\partial }{\partial r} -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta } \end{align*}
が得られます。
2階偏微分を求める
ここから計算量がぐっと増えます。
\( x \) について
\begin{align*} \frac{\partial ^{2}}{\partial x^{2}} & =\frac{\partial }{\partial x}\frac{\partial }{\partial x}\\ & =\sin \theta \cos \varphi \frac{\partial }{\partial r}\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)\\ & \ + \frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\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)\\ & \ -\ \frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\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) \end{align*}
となるので,各項を求めていきます.
\( r \)微分の項
\begin{align*} \sin \theta \cos \varphi \frac{\partial }{\partial r}\left(\sin \theta \cos \varphi \frac{\partial }{\partial r}\right) & = \sin \theta \cos \varphi \frac{\partial }{\partial r}(\sin \theta \cos \varphi )\frac{\partial }{\partial r} +\ \sin^{2} \theta \cos^{2} \varphi \frac{\partial ^{2}}{\partial r^{2}}\\ & = \sin^{2} \theta \cos^{2} \varphi \frac{\partial ^{2}}{\partial r^{2}} \label{15} \tag{15} \end{align*}
\begin{align*} \sin \theta \cos \varphi \frac{\partial }{\partial r}\left(\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\right) & = \sin \theta \cos \varphi \frac{\partial }{\partial r}\left(\frac{\cos \theta \cos \varphi }{r}\right)\frac{\partial }{\partial \theta } +\frac{ \sin \theta \cos \theta \cos^{2} \varphi }{r}\frac{\partial }{\partial r\partial \theta }\\ & =-\frac{\sin \theta \cos \theta \cos^{2} \varphi }{r^{2}} \ + \frac{ \sin \theta \cos \theta \cos^{2} \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta }\right) \tag{16} \end{align*}
\begin{align*} \sin \theta \cos \varphi \frac{\partial }{\partial r}\left( -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & = \sin \theta \cos \varphi \frac{\partial }{\partial r}\left( -\frac{\sin \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } -\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \varphi }\right)\\ & =\frac{\sin \varphi \cos \varphi }{r^{2}}\frac{\partial }{\partial \varphi } -\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \varphi }\right) \tag{17} \end{align*}
\( \theta \)微分の項
\begin{align*} \frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\left(\sin \theta \cos \varphi \frac{\partial }{\partial r}\right) & =\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }(\sin \theta \cos \varphi )\frac{\partial }{\partial r} +\frac{\sin \theta \cos \theta \cos^{2} \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right)\\ & =\frac{\cos^{2} \theta \cos^{2} \varphi }{r}\frac{\partial }{\partial r} +\frac{\sin \theta \cos \theta \cos^{2} \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right) \tag{18} \end{align*}
\begin{align*} \frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\right) & =\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \theta \cos \varphi }{r}\right)\frac{\partial }{\partial \theta } +\frac{\cos^{2} \theta \cos^{2} \varphi }{r^{2}}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \theta }\right)\\ & =-\frac{\sin \theta \cos \theta \cos^{2} \varphi }{r^{2}}\frac{\partial }{\partial \theta } +\frac{\cos^{2} \theta \cos^{2} \varphi }{r^{2}}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \theta }\right) \tag{19} \end{align*}
\begin{align*} \frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\left( -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & =\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\left( -\frac{\sin \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } -\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \varphi }\right)\\ & =\frac{\cos^{2} \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \varphi } -\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \varphi }\right) \tag{20} \end{align*}
\( \varphi \)微分の項
\begin{align*} -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\sin \theta \cos \varphi \frac{\partial }{\partial r}\right) & =-\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }(\sin \theta \cos \varphi )\frac{\partial }{\partial r} -\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial r}\right)\\ & =\frac{\sin^{2} \varphi }{r}\frac{\partial }{\partial r} -\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial r}\right) \tag{21} \end{align*}
\begin{align*} -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \theta \cos \varphi }{r}\frac{\partial }{\partial \theta }\right) & =-\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \theta \cos \varphi }{r}\right)\frac{\partial }{\partial \theta } -\frac{\cos \theta \sin^{2} \varphi }{r^{2}}\frac{\partial }{\partial \varphi }\frac{\partial }{\partial \theta }\\ & =\frac{\cos \theta \sin^{2} \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \theta } -\frac{\cos \theta \sin^{2} \varphi }{r^{2}}\frac{\partial }{\partial \varphi }\frac{\partial }{\partial \theta } \tag{22} \end{align*}
\begin{align*} -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left( -\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & =-\frac{\sin \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left( -\frac{\sin \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } +\frac{\sin^{2} \varphi }{r^{2}\sin^{2} \theta }\frac{\partial ^{2}}{\partial \varphi ^{2}}\\ & =\dfrac{\sin \varphi \cos \varphi }{r\sin^{2} \theta }\frac{\partial }{\partial \varphi } +\frac{\sin^{2} \varphi }{r^{2}\sin^{2} \theta }\frac{\partial ^{2}}{\partial \varphi ^{2}} \tag{23} \end{align*}
\( y \) について
\begin{align*} \frac{\partial }{\partial y^{2}} & =\frac{\partial }{y}\frac{\partial }{y}\\ & =\sin \theta \sin \varphi \frac{\partial }{\partial r}\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)\\ & \ +\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\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)\\ & \ +\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\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) \end{align*}
となるので,各項を求めていきます.
\( r \)微分の項
\begin{align*} \sin \theta \sin \varphi \frac{\partial }{\partial r}\left(\sin \theta \sin \varphi \frac{\partial }{\partial r}\right) & =\sin \theta \sin \varphi \frac{\partial }{\partial r}(\sin \theta \sin \varphi )\frac{\partial }{\partial r} +\sin^{2} \theta \sin^{2} \varphi \frac{\partial ^{2}}{\partial r^{2}}\\ & =\sin^{2} \theta \sin^{2} \varphi \frac{\partial ^{2}}{\partial r^{2}} \tag{24} \end{align*}
\begin{align*} \sin \theta \sin \varphi \frac{\partial }{\partial r}\left(\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\right) & =\sin \theta \sin \varphi \frac{\partial }{\partial r}\left(\frac{\cos \theta \sin \varphi }{r}\right)\frac{\partial }{\partial \theta } +\frac{\sin \theta \cos \theta \sin^{2} \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta }\right)\\ & =-\dfrac{\sin \theta \cos \theta \sin^{2} \varphi }{r^{2}}\frac{\partial }{\partial \theta } +\frac{\sin \theta \cos \theta \sin^{2} \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta }\right) \tag{25} \end{align*}
\begin{align*} \sin \theta \sin \varphi \frac{\partial }{\partial r}\left(\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & =\sin \theta \sin \varphi \frac{\partial }{\partial r}\left(\frac{\cos \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } +\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \varphi }\right)\\ & =-\frac{\sin \varphi \cos \varphi }{r^{2}}\frac{\partial }{\partial \varphi } +\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \varphi }\right) \tag{26} \end{align*}
\( \theta \)微分の項
\begin{align*} \frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\left(\sin \theta \sin \varphi \frac{\partial }{\partial r}\right) & =\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }(\sin \theta \sin \varphi )\frac{\partial }{\partial r} +\frac{\sin \theta \cos \theta \sin^{2} \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right)\\ & =\frac{\cos^{2} \theta \sin^{2} \varphi }{r}\frac{\partial }{\partial r} +\frac{\sin \theta \cos \theta \sin^{2} \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right) \tag{27} \end{align*}
\begin{align*} \frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\right) & =\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \theta \sin \varphi }{r}\right)\frac{\partial }{\partial \theta } +\frac{\cos^{2} \theta \sin^{2} \varphi }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}\\ & =-\frac{\sin \theta \cos \theta \sin^{2} \varphi }{r^{2}}\frac{\partial }{\partial \theta } +\frac{\cos^{2} \theta \sin^{2} \varphi }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}} \tag{28} \end{align*}
\begin{align*} \frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & =\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\left(\frac{\cos \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } +\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \varphi }\right)\\ & =\frac{\cos^{2} \theta \sin \varphi \cos \varphi }{r^{2}\sin^{2} \theta }\frac{\partial }{\partial \varphi } +\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial \varphi }\right) \tag{29} \end{align*}
\( \varphi \)微分の項
\begin{align*} \frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\sin \theta \sin \varphi \frac{\partial }{\partial r}\right) & =\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }(\sin \theta \sin \varphi )\frac{\partial }{\partial r} +\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial r}\right)\\ & =\frac{\cos^{2} \varphi }{r}\frac{\partial }{\partial r} +\frac{\sin \varphi \cos \varphi }{r}\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial r}\right) \tag{30} \end{align*}
\begin{align*} \frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \theta \sin \varphi }{r}\frac{\partial }{\partial \theta }\right) & =\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \theta \sin \varphi }{r}\right)\frac{\partial }{\partial \theta } +\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial \theta }\right)\\ & =\frac{\cos \theta \cos^{2} \varphi }{r\sin \theta }\frac{\partial }{\partial \theta } +\frac{\cos \theta \sin \varphi \cos \varphi }{r^{2}\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\partial }{\partial \theta }\right) \tag{31} \end{align*}
\begin{align*} \frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\right) & =\frac{\cos \varphi }{r\sin \theta }\frac{\partial }{\partial \varphi }\left(\frac{\cos \varphi }{r\sin \theta }\right)\frac{\partial }{\partial \varphi } +\frac{\cos^{2} \varphi }{r^{2}\sin^{2} \theta }\frac{\partial ^{2}}{\partial \varphi ^{2}}\\ & =-\frac{\sin \varphi \cos \varphi }{r^{2}\sin^{2} \theta }\frac{\partial }{\partial \varphi } +\frac{\cos^{2} \varphi }{r^{2}\sin^{2} \theta }\frac{\partial ^{2}}{\partial \varphi ^{2}} \tag{32} \end{align*}
\( z \) について
\begin{align*} \frac{\partial ^{2}}{\partial z^{2}} & =\frac{\partial }{\partial z}\frac{\partial }{\partial z}\\ & \ +\cos \theta \frac{\partial }{\partial r}\left(\cos \theta \frac{\partial }{\partial r} -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\right)\\ & \ -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\left(\cos \theta \frac{\partial }{\partial r} -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\right) \end{align*}
となるので,各項を求めていきます.
\( r \)微分の項
\begin{align*} \cos \theta \frac{\partial }{\partial r}\left(\cos \theta \frac{\partial }{\partial r}\right) & =\cos \theta \frac{\partial }{\partial r}(\cos \theta )\frac{\partial }{\partial r} +\cos^{2} \theta \frac{\partial ^{2}}{\partial r^{2}}\\ & =\cos^{2} \theta \frac{\partial ^{2}}{\partial r^{2}} \tag{33} \end{align*}
\begin{align*} \cos \theta \frac{\partial }{\partial r}\left( -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\right) & =\cos \theta \frac{\partial }{\partial r}\left( -\frac{\sin \theta }{r}\right)\frac{\partial }{\partial \theta } \ -\frac{\sin \theta \cos \theta }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta }\right)\\ & =\frac{\sin \theta \cos \theta }{r^{2}}\frac{\partial }{\partial \theta } -\frac{\sin \theta \cos \theta }{r}\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta }\right) \tag{34} \end{align*}
\( \theta \)微分の項
\begin{align*} -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\left(\cos \theta \frac{\partial }{\partial r}\right) & =-\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }(\cos \theta )\frac{\partial }{\partial r} -\frac{\sin \theta \cos \theta }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right)\\ & =\frac{\sin^{2} \theta }{r}\frac{\partial }{\partial r} -\frac{\sin \theta \cos \theta }{r}\frac{\partial }{\partial \theta }\left(\frac{\partial }{\partial r}\right) \tag{35} \end{align*}
\begin{align*} -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\left( -\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\right) & =-\frac{\sin \theta }{r}\frac{\partial }{\partial \theta }\left( -\frac{\sin \theta }{r}\right)\frac{\partial }{\partial \theta } +\frac{\sin^{2} \theta }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}\\ & =\frac{\sin \theta \cos \theta }{r^{2}}\frac{\partial }{\partial \theta } +\frac{\sin^{2} \theta }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}} \label{36} \tag{36} \end{align*}
\( \varphi \)微分の項
この項は0です.
結果
以上,(\ref{15})から(\ref{36})をまとめると,
\begin{align*} \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 \varphi ^{2}} +\frac{2}{r}\frac{\partial }{\partial r} +\frac{\cos \theta }{r^{2}\sin \theta }\frac{\partial }{\partial \theta } \\ & =\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}} \end{align*}
を求めることができます.
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