# 極座標表示におけるラプラシアン (3次元)

におけるラプラシアン

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

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

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 \phi }{\partial x}$$\frac{{\partial }^{2}\phi }{\partial {x}^{2}}$$\frac{\partial r}{\partial y}$$\frac{{\partial }^{2}r}{\partial {y}^{2}}$$\frac{\partial \theta }{\partial y}$$\frac{{\partial }^{2}\theta }{\partial {y}^{2}}$$\frac{\partial \phi }{\partial y}$$\frac{{\partial }^{2}\phi }{\partial {y}^{2}}$$\frac{\partial r}{\partial z}$$\frac{{\partial }^{2}r}{\partial {z}^{2}}$$\frac{\partial \theta }{\partial z}$$\frac{{\partial }^{2}\theta }{\partial {z}^{2}}$$\frac{\partial \phi }{\partial z}$$\frac{{\partial }^{2}\phi }{\partial {z}^{2}}$

を用いて右辺へ式変形する．

## ■導出

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

$\frac{{\partial }^{2}\psi }{\partial {x}^{2}}$ $=\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}+\frac{\partial \psi }{\partial \phi }\frac{\partial \phi }{\partial x}\right)$

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

ここで

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

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

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

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

さらに

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

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

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

である．従って

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

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

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

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

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

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

を具体的に求めて(1)を式変形する．

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

の両辺を2乗して加えると

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

となり，これを整理すると　参考：三角関数の相互関係

${x}^{2}+{y}^{2}$ $={r}^{2}{\mathrm{sin}}^{2}\theta$

よって

$\sqrt{{x}^{2}+{y}^{2}}$ $=r\mathrm{sin}\theta$

となる．ここで $z=r\mathrm{cos}\theta$ の関係より，左辺を$z$ で，右辺を$r\mathrm{cos}\theta$ 割って

$\frac{\sqrt{{x}^{2}+{y}^{2}}}{z}$ $=\frac{r\mathrm{sin}\theta }{r\mathrm{cos}\theta }$

$\frac{\sqrt{{x}^{2}+{y}^{2}}}{z}$ $=\mathrm{tan}\theta$ 　　･･････(2)

とする．次に

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

の関係から

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

$\frac{y}{x}$ $=\frac{\mathrm{sin}\phi }{\mathrm{cos}\phi }$

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

が得られる．さらに

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

の辺々を2乗して足すことにより

${x}^{2}+{y}^{2}+{z}^{2}$ $={r}^{2}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\phi +{r}^{2}{\mathrm{sin}}^{2}{\theta {\mathrm{sin}}^{2}\phi +r}^{2}{\mathrm{cos}}^{2}\theta$

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

${x}^{2}+{y}^{2}+{z}^{2}$ $={r}^{2}{\mathrm{sin}}^{2}\theta +{{r}^{2}\mathrm{cos}}^{2}\theta$

${x}^{2}+{y}^{2}+{z}^{2}$ $={r}^{2}$ 　　･･････(4)

が得られる．上記の(2)，(3)，(4) を用いて

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

を求める．

(4)の両辺を$x$ で偏微分して

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

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

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

$\frac{\partial r}{\partial x}$ $=\mathrm{sin}\theta \mathrm{cos}\phi$ 　　･･････(5)

が得られる．(3)の両辺を$x$ で偏微分して

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

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

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

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

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

$\frac{\partial \phi }{\partial x}$ $=-\frac{y}{{r}^{2}-{z}^{2}}$$=-\frac{r\mathrm{sin}\theta \mathrm{sin}\phi }{{r}^{2}-{r}^{2}{\mathrm{cos}}^{2}\theta }$$=-\frac{r\mathrm{sin}\theta \mathrm{sin}\phi }{{r}^{2}\left(1-{\mathrm{cos}}^{2}\theta \right)}$$=-\frac{r\mathrm{sin}\theta \mathrm{sin}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$ $=-\frac{\mathrm{sin}\phi }{r\mathrm{sin}\theta }$ 　　･･････(6)

が得られる．(2)の両辺を$x$ で偏微分して

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

$\frac{\partial \theta }{\partial x}$ $=\frac{x{\mathrm{cos}}^{2}\theta }{z\sqrt{{x}^{2}+{y}^{2}}}$

$=\frac{r\mathrm{sin}\theta \mathrm{cos}\phi {\mathrm{cos}}^{2}\theta }{r\mathrm{cos}\theta \sqrt{{r}^{2}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\phi +{r}^{2}{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\phi }}$

$=\frac{r\mathrm{sin}\theta \mathrm{cos}\phi {\mathrm{cos}}^{2}\theta }{{r}^{2}\mathrm{cos}\theta \mathrm{sin}\theta \sqrt{{\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi }}$

$=\frac{\mathrm{cos}\theta \mathrm{cos}\phi }{r}$ 　　･･････(7)

が得られる．

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

(5)を代入して

$=\frac{\partial }{\partial x}\left(\mathrm{sin}\theta \mathrm{cos}\phi \right)$

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

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

(7)，(6)を代入して

$=\mathrm{cos}\theta \mathrm{cos}\phi \frac{\mathrm{cos}\theta \mathrm{cos}\phi }{r}-\mathrm{sin}\theta \mathrm{sin}\phi \left(-\frac{\mathrm{sin}\phi }{r\mathrm{sin}\theta }\right)$

$=\frac{{\mathrm{cos}}^{2}\theta {\mathrm{cos}}^{2}\phi }{r}+\frac{{\mathrm{sin}}^{2}\phi }{r}$

$=\frac{{\mathrm{cos}}^{2}\theta {\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi }{r}$ 　　･･････(8)

が得られる．

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

(7)を代入して

$=\frac{\partial }{\partial x}\left(\frac{\mathrm{cos}\theta \mathrm{cos}\phi }{r}\right)$

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

$=-\frac{\mathrm{cos}\theta \mathrm{cos}\phi }{{r}^{2}}\frac{\partial r}{\partial x}-\frac{\mathrm{sin}\theta \mathrm{cos}\phi }{r}\frac{\partial \theta }{\partial x}-\frac{\mathrm{cos}\theta \mathrm{sin}\phi }{r}\frac{\partial \phi }{\partial x}$

(5)，(7)，(6)を代入して

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

$=-\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi }{{r}^{2}}-\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi }{{r}^{2}}+\frac{\mathrm{cos}\theta {\mathrm{sin}}^{2}\phi }{{r}^{2}\mathrm{sin}\theta }$

$=-\frac{\mathrm{2sin}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi }{{r}^{2}}+\frac{\mathrm{cos}\theta {\mathrm{sin}}^{2}\phi }{{r}^{2}\mathrm{sin}\theta }$

$=\frac{-2{\mathrm{sin}}^{2}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi +\mathrm{cos}\theta {\mathrm{sin}}^{2}\phi }{{r}^{2}\mathrm{sin}\theta }$

$=\frac{\mathrm{cos}\theta \left(-2{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi \right)}{{r}^{2}\mathrm{sin}\theta }$　･･････(9)

が得られる．

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

(6)を代入して

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

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

$=\frac{\mathrm{sin}\phi }{{r}^{2}\mathrm{sin}\theta }\frac{\partial r}{\partial x}+\frac{\mathrm{cos}\theta \mathrm{sin}\phi }{r{\mathrm{sin}}^{2}\theta }\frac{\partial \theta }{\partial x}-\frac{\mathrm{cos}\phi }{r\mathrm{sin}\theta }\frac{\partial \phi }{\partial x}$

(5)，(7)，(6)を代入して

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

$=\frac{\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}}+\frac{{\mathrm{cos}}^{2}\theta \mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }+\frac{\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$

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

$=\frac{2\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$　･･････(10)

ここで，(2)〜(10)を用いて(1)を式変形すると

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

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

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

$=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\phi +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi }{r}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta {\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi }{r}$

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

$+\frac{{\partial }^{2}\psi }{\partial r\partial \phi }\left(-\frac{\mathrm{sin}\phi \mathrm{cos}\phi }{r}\right)+\frac{{\partial }^{2}\psi }{\partial {\phi }^{2}}\frac{{\mathrm{sin}}^{2}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }+\frac{\partial \psi }{\partial \phi }\frac{2\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$　･･････(11)

が得られる．

$\frac{{\partial }^{2}\psi }{\partial {y}^{2}}$ $=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\phi +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{sin}}^{2}\phi }{r}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta {\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi }{r}$

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

$+\frac{{\partial }^{2}\psi }{\partial r\partial \phi }\frac{\mathrm{sin}\phi \mathrm{cos}\phi }{r}+\frac{{\partial }^{2}\psi }{\partial {\phi }^{2}}\frac{{\mathrm{cos}}^{2}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }-\frac{\partial \psi }{\partial \phi }\frac{2\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$　･･････(12)

が得られる．

$\frac{{\partial }^{2}\psi }{\partial {z}^{2}}$ $=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{cos}}^{2}\theta -\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\mathrm{sin}\theta \mathrm{cos}\theta }{r}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{sin}}^{2}\theta }{r}$

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

が得られる．

(11)，(12)，(13)を足し合わせると

$\frac{{\partial }^{2}\psi }{\partial {x}^{2}}+\frac{{\partial }^{2}\psi }{\partial {y}^{2}}+\frac{{\partial }^{2}\psi }{\partial {z}^{2}}$ $=\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\phi +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{cos}}^{2}\phi }{r}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta {\mathrm{cos}}^{2}\phi +{\mathrm{sin}}^{2}\phi }{r}$

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

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

$+\frac{{\partial }^{2}\psi }{\partial {r}^{2}}{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\phi +\frac{{\partial }^{2}\psi }{\partial \theta \partial r}\frac{\mathrm{sin}\theta \mathrm{cos}\theta {\mathrm{sin}}^{2}\phi }{r}+\frac{\partial \psi }{\partial r}\frac{{\mathrm{cos}}^{2}\theta {\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi }{r}$

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

$+\frac{{\partial }^{2}\psi }{\partial r\partial \phi }\frac{\mathrm{sin}\phi \mathrm{cos}\phi }{r}+\frac{{\partial }^{2}\psi }{\partial {\phi }^{2}}\frac{{\mathrm{cos}}^{2}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }-\frac{\partial \psi }{\partial \phi }\frac{2\mathrm{sin}\phi \mathrm{cos}\phi }{{r}^{2}{\mathrm{sin}}^{2}\theta }$

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

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

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

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

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

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

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

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

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

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

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

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

となり，示された．

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