極座標表示におけるラプラシアン (2次元)

関数 $3$\displaystyle{ {\psi}={\xi}$x,y$ }$ において極座標表示 $3$\displaystyle{ x=r\cos {\theta} }$ ， $3$\displaystyle{\displaystyle{\displaystyle{y=r\sin {\theta}}}}$ におけるラプラシアンは

$3$\displaystyle{ {\Delta}{\psi}=\left({ \frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}+ \frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }\right) {\psi}}$ $3$\displaystyle{=\left({ \frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}} }\right){\psi}}$

で与えられる．

■導出手順

与式の左辺の $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ を求め，
$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ を用いて右辺へ式変形する．

■導出

始めに，関数 $3$\displaystyle{{\psi}}$ を $3$\displaystyle{x}$ で偏微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{=\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$

参考:合成関数の偏導関数

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }$

$3$\displaystyle{\displaystyle{\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}\left({\displaystyle{\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}}\right)}}$

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right)+ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }$

ここで

$3$\displaystyle{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }$ $3$\displaystyle{ =\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} }\right) }\right}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }\right} }$

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right)}$ $3$\displaystyle{=\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} }\right) }\right}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }\right}}$

より

$3$\displaystyle{=\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} }\right) }\right}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }\right}}$ $3$\displaystyle{+\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} }\right) }\right}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right) }\right}}$

さらにここで

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} }\right)}$ $3$\displaystyle{=\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} }\right}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} }\right}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$

$3$\displaystyle{\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} }\right)}$ $3$\displaystyle{=\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} }\right}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\left{{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} }\right}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$

より

$3$\displaystyle{=\left({ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right)\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ $3$\displaystyle{+\left({ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{ $ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{2}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}{ $ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{2}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$
･･････(1)

次に

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$

を用いて(1)を式変形する．

始めに

$3$\displaystyle{ x=r\cos {\theta} }$ ， $3$\displaystyle{\displaystyle{\displaystyle{y=r\sin {\theta}}}}$

の両辺を2乗して加えると

$3$\displaystyle{ {x}^{2}+{y}^{2}={\left({ r\cos {\theta} }\right)}^{2}+{\left({ r\sin {\theta} }\right)}^{2} }$

となり，これを整理すると

$3$\displaystyle{ {x}^{2}+{y}^{2}={r}^{2} }$ ･･････(2)

また，左辺同士、右辺同士で比をとることにより

$3$\displaystyle{ \frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}}=\frac{\hspace{2} r\sin {\theta} \hspace{2}}{\hspace{2} r\cos {\theta} \hspace{2}} }$

となり，これを整理して

$3$\displaystyle{ \tan {\theta}=\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$ ･･････(3)

参考：三角関数の相互関係

が得られる．上記の(2)，(3)を用いて

を求める．

(2)の両辺を $3$\displaystyle{x}$ で微分して

$3$\displaystyle{ 2x }$ $3$\displaystyle{ =2r\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2}x\hspace{2}}{\hspace{2}r\hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} r\cos {\theta} \hspace{2}}{\hspace{2} r \hspace{2}} }$ $3$\displaystyle{ =\cos {\theta} }$ ･･････(4)

が得られる． (3)の両辺を $3$\displaystyle{x}$ で微分して

$3$\displaystyle{ \frac{\hspace{2} 1 \hspace{2}}{\hspace{2} {\cos }^{2}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =y\left({ - \frac{\hspace{2} 1 \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }\right) }$

$3$\displaystyle{ \left({ {\tan }^{2}{\theta}+1 }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} y \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }$

$3$\displaystyle{ \left({ \frac{\hspace{2}{y}^{2}\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}}+1 }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} y \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }$

$3$\displaystyle{ \left({\frac{\hspace{2} {y}^{2}+{x}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}}}\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} y \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }$

$3$\displaystyle{ \left({\frac{\hspace{2}{r}^{2}\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}}}\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} y \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} y \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} }$ $3$\displaystyle{=- \frac{\hspace{2} r\sin {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}}$ $3$\displaystyle{=- \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}$ ･･････(5)

が得られる．

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}} }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\left({ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }\right)}$

(4)を代入して

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\cos {\theta} }$ $3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\cos {\theta}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\cos {\theta}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{=- \sin {\theta}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$

(5)を代入して

$3$\displaystyle{=- \sin {\theta} $ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ }$ $3$\displaystyle{=\frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}$ ･･････(6)

が得られる．

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }$

(5)を代入して

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} $ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$

(4)，(5)より

$3$\displaystyle{=\frac{\hspace{2} \sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} \cos {\theta}\sin {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}}$ ･･････(7)

(4)～(7)を用いて(1)を整理すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}}$ $3$\displaystyle{ =\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{ \cos }^{2}{\theta}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\cos {\theta} }$ $3$\displaystyle{ +\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\cos {\theta}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right) }$ $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}}$ ･･････(8)

次に，関数 $3$\displaystyle{{\psi}}$ を $3$\displaystyle{y}$ で偏微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$

参考:合成関数の偏導関数

$3$\displaystyle{y}$ に関しても $3$\displaystyle{x}$ と同様に計算して

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{ $ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{2}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}{ $ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{2}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ ･･････(9)

また，(2)，(3)を用いて

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ ， $3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$

を求める．

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ は (4) と同様に計算して

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\sin {\theta} }$ ･･････(10)

(3)より

$3$\displaystyle{ \tan {\theta}=\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$

これを $3$\displaystyle{ y }$ で微分して

$3$\displaystyle{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} {\cos }^{2}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 1 \hspace{2}}{\hspace{2} x \hspace{2}} }$

$3$\displaystyle{ \left({ {\tan }^{2}{\theta}+ 1 }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 1 \hspace{2}}{\hspace{2} x \hspace{2}} }$

$3$\displaystyle{ \left({ \frac{\hspace{2}{y}^{2}\hspace{2}}{\hspace{2}{x}^{2}\hspace{2}}+ 1 }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 1 \hspace{2}}{\hspace{2} x \hspace{2}} }$

$3$\displaystyle{ \left({\frac{\hspace{2} {y}^{2}+ {x}^{2} \hspace{2}}{\hspace{2}{x}^{2}\hspace{2}}}\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 1 \hspace{2}}{\hspace{2} x \hspace{2}} }$

$3$\displaystyle{ \left({\frac{\hspace{2}{r}^{2}\hspace{2}}{\hspace{2}{x}^{2}\hspace{2}}}\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} 1 \hspace{2}}{\hspace{2} x \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} x \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} r\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2} r \hspace{2}} }$ ･･････(11)

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\left({ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }\right)}$

(10)を代入して

$3$\displaystyle{ =\frac{\hspace{2} {\partial} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\sin {\theta} }$ $3$\displaystyle{ =\cos {\theta}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$

(11)を代入して

$3$\displaystyle{ =\frac{\hspace{2} {\cos }^{2}{\theta} \hspace{2}}{\hspace{2} r \hspace{2}} }$ ･･････(12)

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ は(7)と同様に計算して

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}} }$ ･･････(13)

(10)～(13)を用いて(9)を整理すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$ $3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\sin }^{2}{\theta}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}} $ \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ \sin {\theta}\displaystyle{\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}}$ + ∂ ψ ∂ r cos 2 θ r + ∂ 2 ψ ∂ r ∂ θ sin θ ( cos θ r ) $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} $ - \frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} $ }$
･･････(14)

ここで(8)，(14)より，

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}+ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}} }$

$3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\cos }^{2}{\theta}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\cos {\theta}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}$ $3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\cos {\theta}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}\frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}}$

$3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\sin }^{2}{\theta}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}} $ \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ \sin {\theta}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}$

$3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\sin {\theta} $ \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} $ +\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} $ - \frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} $ }$

$3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}\left({ { \cos }^{2}{\theta}+{ \sin }^{2}{\theta} }\right)+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\left{{ \left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\cos {\theta}- \left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\cos {\theta} }\right}}$

$3$\displaystyle{+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\left({ \frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}+\frac{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)++\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\left{{ \cos {\theta}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)+\sin {\theta}\left({ \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right) }\right}}$

$3$\displaystyle{+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}\left({ \frac{\hspace{2} { \sin }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}+\frac{\hspace{2} { \cos }^{2}{\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} }\right)+\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\left{{ \left({ \frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}- \frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} }\right) }\right}}$

$3$\displaystyle{ = \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}{r}^{2}\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}} }$ $3$\displaystyle{=\left({ \frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}} }\right){\psi}}$

となる．よって示された．

ホーム>>カテゴリー別分類>>微分>>偏微分>>極座標表示におけるラプラシアン(2次元)

最終更新日：2026年3月12日