# 極座標表示におけるラプラシアン (2次元)

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

で与えられる．

## ■導出手順

$\frac{\partial r}{\partial x}$$\frac{{\partial }^{2}r}{\partial {x}^{2}}$$\frac{\partial \theta }{\partial x}$$\frac{{\partial }^{2}\theta }{\partial {x}^{2}}$$\frac{\partial r}{\partial y}$$\frac{{\partial }^{2}r}{\partial {y}^{2}}$$\frac{\partial \theta }{\partial y}$$\frac{{\partial }^{2}\theta }{\partial {y}^{2}}$ を用いて右辺へ式変形する．

## ■導出

$\frac{\partial \psi }{\partial x}$ $=\frac{\partial \psi }{\partial r}\frac{\partial r}{\partial x}+\frac{\partial \psi }{\partial \theta }\frac{\partial \theta }{\partial x}$

$\frac{{\partial }^{2}\psi }{\partial {x}^{2}}$ $=\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial x}\right)$

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

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

ここで

$\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial r}\frac{\partial r}{\partial x}\right)$$=\left\{\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial r}\right)\right\}\frac{\partial r}{\partial x}+\frac{\partial \psi }{\partial r}\left\{\frac{\partial }{\partial x}\left(\frac{\partial r}{\partial x}\right)\right\}$

$\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial \theta }\frac{\partial \theta }{\partial x}\right)$ $=\left\{\frac{\partial }{\partial x}\left(\frac{\partial \psi }{\partial \theta }\right)\right\}\frac{\partial \theta }{\partial x}+\frac{\partial \psi }{\partial \theta }\left\{\frac{\partial }{\partial x}\left(\frac{\partial \theta }{\partial x}\right)\right\}$

より

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

さらにここで

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

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

より

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

$=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\left(\frac{\partial r}{\partial x}\right)}^{2}+\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\partial \theta }{\partial x}\frac{\partial r}{\partial x}$$+\frac{\partial \psi }{\partial r}\frac{{\partial }^{2}r}{\partial {x}^{2}}$$+\frac{{\partial }^{2}\psi }{\partial r\partial \theta }\frac{\partial r}{\partial x}\frac{\partial \theta }{\partial x}$ $+\frac{{\partial }^{2}\psi }{\partial {\theta }^{2}}{\left(\frac{\partial \theta }{\partial x}\right)}^{2}$$+\frac{\partial \psi }{\partial \theta }\frac{{\partial }^{2}\theta }{\partial {x}^{2}}$
･･････(1)

$\frac{\partial r}{\partial x}$$\frac{{\partial }^{2}r}{\partial {x}^{2}}$$\frac{\partial \theta }{\partial x}$$\frac{{\partial }^{2}\theta }{\partial {x}^{2}}$

を用いて(1)を式変形する．

$x=r\mathrm{cos}\theta$$y=r\mathrm{sin}\theta$

の両辺を2乗して加えると

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

となり，これを整理すると

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

また，左辺同士、右辺同士で比をとることにより

$\frac{y}{x}=\frac{r\mathrm{sin}\theta }{r\mathrm{cos}\theta }$

となり，これを整理して

$\mathrm{tan}\theta =\frac{y}{x}$ 　　　　　･･････(3)

が得られる．上記の(2)，(3)を用いて

$\frac{\partial r}{\partial x}$$\frac{{\partial }^{2}r}{\partial {x}^{2}}$$\frac{\partial \theta }{\partial x}$$\frac{{\partial }^{2}\theta }{\partial {x}^{2}}$

を求める．

(2)の両辺を$x$ で微分して

$2x$ $=2r\frac{\partial r}{\partial x}$

$\frac{\partial r}{\partial x}$ $=\frac{x}{r}$ $=\frac{r\mathrm{cos}\theta }{r}$ $=\mathrm{cos}\theta$　　　･･････(4)

が得られる． (3)の両辺をxで微分して

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

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

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

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

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

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

が得られる．

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

(4)を代入して

$=\frac{\partial }{\partial x}\mathrm{cos}\theta$ $=\frac{\partial }{\partial r}\mathrm{cos}\theta \frac{\partial r}{\partial x}+\frac{\partial }{\partial \theta }\mathrm{cos}\theta \frac{\partial \theta }{\partial x}$ $=-\mathrm{sin}\theta \frac{\partial \theta }{\partial x}$

(5)を代入して

$=-\mathrm{sin}\theta \left(-\frac{\mathrm{sin}\theta }{r}\right)$ $=\frac{{\mathrm{sin}}^{2}\theta }{r}$ 　　　･･････(6)

が得られる．

$\frac{{\partial }^{2}\theta }{\partial {x}^{2}}$ $=\frac{\partial }{\partial x}\left(\frac{\partial \theta }{\partial x}\right)$

(5)を代入して

$=\frac{\partial }{\partial x}\left(-\frac{\mathrm{sin}\theta }{r}\right)$ $=\frac{\partial }{\partial r}\left(-\frac{\mathrm{sin}\theta }{r}\right)\frac{\partial r}{\partial x}$$+\frac{\partial }{\partial \theta }\left(-\frac{\mathrm{sin}\theta }{r}\right)\frac{\partial \theta }{\partial x}$

(4)，(5)より

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

(4)〜(7)を用いて(1)を整理すると

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

$\frac{\partial \psi }{\partial y}$ $=\frac{\partial \psi }{\partial r}\frac{\partial r}{\partial y}+\frac{\partial \psi }{\partial \theta }\frac{\partial \theta }{\partial y}$

$y$ に関しても $x$ と同様に計算して

$\frac{{\partial }^{2}\psi }{\partial {y}^{2}}$ $=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\left(\frac{\partial r}{\partial y}\right)}^{2}+\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\partial \theta }{\partial y}\frac{\partial r}{\partial y}$$+\frac{\partial \psi }{\partial r}\frac{{\partial }^{2}r}{\partial {y}^{2}}+\frac{{\partial }^{2}\psi }{\partial r\partial \theta }\frac{\partial r}{\partial y}\frac{\partial \theta }{\partial y}$$+\frac{{\partial }^{2}\psi }{\partial {\theta }^{2}}{\left(\frac{\partial \theta }{\partial y}\right)}^{2}+\frac{\partial \psi }{\partial \theta }\frac{{\partial }^{2}\theta }{\partial {y}^{2}}$　･･････(9)

また，(2)，(3)を用いて

$\frac{\partial r}{\partial y}$$\frac{{\partial }^{2}r}{\partial {y}^{2}}$$\frac{\partial \theta }{\partial y}$$\frac{{\partial }^{2}\theta }{\partial {y}^{2}}$

を求める．

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

$\frac{\partial r}{\partial y}$ $=\mathrm{sin}\theta$ 　　･･････(10)

(3)より

$\mathrm{tan}\theta =\frac{y}{x}$

これを $y$ で微分して

$\frac{1}{{\mathrm{cos}}^{2}\theta }\frac{\partial \theta }{\partial y}$ $=\frac{1}{x}$

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

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

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

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

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

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

(10)を代入して

$=\frac{\partial }{\partial y}\mathrm{sin}\theta$ $=\mathrm{cos}\theta \frac{\partial \theta }{\partial y}$

(11)を代入して

$=\frac{{\mathrm{cos}}^{2}\theta }{r}$ 　　･･････(12)

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

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

(10)〜(13)を用いて(9)を整理すると

$\frac{{\partial }^{2}\psi }{\partial {y}^{2}}$$=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\left(\frac{\mathrm{cos}\theta }{r}\right)\mathrm{sin}\theta \frac{{\partial }^{2}\psi }{\partial {y}^{2}}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta }{r}+\frac{{\partial }^{2}\psi }{\partial r\partial \theta }\mathrm{sin}\theta \left(\frac{\mathrm{cos}\theta }{r}\right)$ $+\frac{{\partial }^{2}\psi }{\partial {\theta }^{2}}\frac{{\mathrm{cos}}^{2}\theta }{{r}^{2}}+\frac{\partial \psi }{\partial \theta }\left(-\frac{2\mathrm{sin}\theta \mathrm{cos}\theta }{{r}^{2}}\right)$
･･････(14)

ここで(8)，(14)より，

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

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

$+\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\left(\frac{\mathrm{cos}\theta }{r}\right)\mathrm{sin}\theta +\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta }{r}$

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

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

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

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

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

となる．よって示された．

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