極座標表示におけるラプラシアン (3次元)

関数 $ψ = ξ (x, y, z)$ において極座標表示

$\{\begin{cases} x = r \sin θ \cos φ \\ y = r \sin θ \sin φ \\ z = r \cos θ \end{cases}$

$Δ ψ = (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}) ψ$

$= {\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}} ψ$

で与えられる．

■導出手順

与式の左辺の $\frac{\partial^{2} ψ}{\partial x^{2}}$ ， $\frac{\partial^{2} ψ}{\partial y^{2}}$ ， $\frac{\partial^{2} ψ}{\partial z^{2}}$ を求め

$\frac{\partial r}{\partial x}$ ， $\frac{\partial^{2} r}{\partial x^{2}}$ ， $\frac{\partial θ}{\partial x}$ ， $\frac{\partial^{2} θ}{\partial x^{2}}$ ， $\frac{\partial φ}{\partial x}$ ， $\frac{\partial^{2} φ}{\partial x^{2}}$ ， $\frac{\partial r}{\partial y}$ ， $\frac{\partial^{2} r}{\partial y^{2}}$ ， $\frac{\partial θ}{\partial y}$ ， $\frac{\partial^{2} θ}{\partial y^{2}}$ ， $\frac{\partial φ}{\partial y}$ ， $\frac{\partial^{2} φ}{\partial y^{2}}$ ， $\frac{\partial r}{\partial z}$ ， $\frac{\partial^{2} r}{\partial z^{2}}$ ， $\frac{\partial θ}{\partial z}$ ， $\frac{\partial^{2} θ}{\partial z^{2}}$ ， $\frac{\partial φ}{\partial z}$ ， $\frac{\partial^{2} φ}{\partial z^{2}}$

を用いて右辺へ式変形する．

■導出

始めに，関数 $ψ$ を $x$ で偏微分する．参考:合成関数の偏導関数

$\frac{\partial ψ}{\partial x}$ $= \frac{\partial ψ}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial x} + \frac{\partial ψ}{\partial φ} \frac{\partial φ}{\partial x}$ 　　　　　

$\frac{\partial^{2} ψ}{\partial x^{2}}$ $= \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial x})$ $= \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial x} + \frac{\partial ψ}{\partial φ} \frac{\partial φ}{\partial x})$

$= \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r} \frac{\partial r}{\partial x}) + \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial x}) + \frac{\partial}{\partial x} (\frac{\partial ψ}{\partial φ} \frac{\partial φ}{\partial x})$

ここで

$\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r} \frac{\partial r}{\partial x})$ $= \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r})\} \frac{\partial r}{\partial x} + \frac{\partial ψ}{\partial r} \{\frac{\partial}{\partial x} (\frac{\partial r}{\partial x})\}$

$\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial x})$ $= \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial θ})\} \frac{\partial θ}{\partial x} + \frac{\partial ψ}{\partial θ} \{\frac{\partial}{\partial x} (\frac{\partial θ}{\partial x})\}$

$\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial φ} \frac{\partial φ}{\partial x})$ $= \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial φ})\} \frac{\partial φ}{\partial x} + \frac{\partial ψ}{\partial φ} \{\frac{\partial}{\partial x} (\frac{\partial φ}{\partial x})\}$

より

$= \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r})\} \frac{\partial r}{\partial x}$ $+ \frac{\partial ψ}{\partial r} \{\frac{\partial}{\partial x} (\frac{\partial r}{\partial x})\}$ $+ \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial θ})\} \frac{\partial θ}{\partial x}$ $+ \frac{\partial ψ}{\partial θ} \{\frac{\partial}{\partial x} (\frac{\partial θ}{\partial x})\}$ $+ \{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial φ})\} \frac{\partial φ}{\partial x}$ $+ \frac{\partial ψ}{\partial φ} \{\frac{\partial}{\partial x} (\frac{\partial φ}{\partial x})\}$

さらに

$\{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial r})\} \frac{\partial r}{\partial x}$ $= \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial r}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial r}) \frac{\partial θ}{\partial x}\} \frac{\partial r}{\partial x}$

$\{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial θ})\} \frac{\partial θ}{\partial x}$ $= \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial θ}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial θ}) \frac{\partial θ}{\partial x}\} \frac{\partial θ}{\partial x}$

$\{\frac{\partial}{\partial x} (\frac{\partial ψ}{\partial φ})\} \frac{\partial φ}{\partial x}$ $= \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial φ}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial φ} (\frac{\partial ψ}{\partial φ}) \frac{\partial φ}{\partial x}\} \frac{\partial φ}{\partial x}$

である．従って

$\frac{\partial^{2} ψ}{\partial x^{2}}$ $= \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial r}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial r}) \frac{\partial θ}{\partial x}\} \frac{\partial r}{\partial x} + \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial x^{2}}$

$+ \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial θ}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial θ}) \frac{\partial θ}{\partial x}\} \frac{\partial θ}{\partial x} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial x^{2}}$

$+ \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial φ}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial φ} (\frac{\partial ψ}{\partial φ}) \frac{\partial φ}{\partial x}\} \frac{\partial φ}{\partial x} + \frac{\partial ψ}{\partial φ} \frac{\partial^{2} φ}{\partial x^{2}}$

$= \frac{\partial^{2} ψ}{\partial r^{2}} {(\frac{\partial r}{\partial x})}^{2} + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\partial θ}{\partial x} \frac{\partial r}{\partial x} + \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial x^{2}} + \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\partial r}{\partial x} \frac{\partial θ}{\partial x}$

$+ \frac{\partial^{2} ψ}{\partial θ^{2}} {(\frac{\partial θ}{\partial x})}^{2} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial x^{2}} + \frac{\partial^{2} ψ}{\partial r \partial φ} \frac{\partial r}{\partial x} \frac{\partial φ}{\partial x} + \frac{\partial^{2} ψ}{\partial φ^{2}} {(\frac{\partial φ}{\partial x})}^{2} + \frac{\partial ψ}{\partial φ} \frac{\partial^{2} φ}{\partial x^{2}}$ 　･･････(1)

次に

を具体的に求めて(1)を式変形する．

始めに

$x = r \sin θ \cos φ$ ， $y = r \sin θ \sin φ$

の両辺を2乗して加えると

$x^{2} + y^{2}$ $= {(r \sin θ \cos φ)}^{2} + {(r \sin θ \sin φ)}^{2}$

となり，これを整理すると　参考：三角関数の相互関係

$x^{2} + y^{2}$ $= r^{2} \sin^{2} θ$

よって

$\sqrt{x^{2} + y^{2}}$ $= r \sin θ$

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

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

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

とする．次に

$x = r \sin θ \cos φ$ ， $y = r \sin θ \sin φ$

の関係から

$\frac{y}{x}$ $= \frac{r \sin θ \sin φ}{r \sin θ \cos φ}$

$\frac{y}{x}$ $= \frac{\sin φ}{\cos φ}$

$\frac{y}{x}$ $= \tan φ$ 　　･･････(3)

が得られる．さらに

$x = r \sin θ \cos φ$ ， $y = r \sin θ \sin φ$ ， $z = r \cos θ$

の辺々を2乗して足すことにより

$x^{2} + y^{2} + z^{2}$ $= r^{2} \sin^{2} θ \cos^{2} φ + r^{2} \sin^{2} {θ \sin^{2} φ + r}^{2} \cos^{2} θ$

$x^{2} + y^{2} + z^{2}$ $= r^{2} \sin^{2} θ {(\cos}^{2} φ + {\sin^{2} φ) + r}^{2} \cos^{2} θ$

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

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

が得られる．上記の(2)，(3)，(4) を用いて

を求める．

(4)の両辺を $x$ で偏微分して

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

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

$\frac{\partial r}{\partial x}$ $= \frac{r \sin θ \cos φ}{r}$

$\frac{\partial r}{\partial x}$ $= \sin θ \cos φ$ 　　･･････(5)

が得られる．(3)の両辺を $x$ で偏微分して

$y (- \frac{1}{x^{2}})$ $= \frac{1}{\cos^{2} φ} \frac{\partial φ}{\partial x}$

$(\tan^{2} φ + 1) \frac{\partial φ}{\partial x}$ $= - \frac{y}{x^{2}}$

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

$(\frac{y^{2} + x^{2}}{x^{2}}) \frac{\partial φ}{\partial x}$ $= - \frac{y}{x^{2}}$

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

$\frac{\partial φ}{\partial x}$ $= - \frac{y}{r^{2} - z^{2}}$ $= - \frac{r \sin θ \sin φ}{r^{2} - r^{2} \cos^{2} θ}$ $= - \frac{r \sin θ \sin φ}{r^{2} (1 - \cos^{2} θ)}$ $= - \frac{r \sin θ \sin φ}{r^{2} \sin^{2} θ}$ $= - \frac{\sin φ}{r \sin θ}$ 　　･･････(6)

が得られる．(2)の両辺を $x$ で偏微分して

$\frac{{x (x^{2} + y^{2})}^{- \frac{1}{2}}}{z}$ $= \frac{1}{\cos^{2} θ} \frac{\partial θ}{\partial x}$

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

$= \frac{r \sin θ \cos φ \cos^{2} θ}{r \cos θ \sqrt{r^{2} \sin^{2} θ \cos^{2} φ + r^{2} \sin^{2} θ \sin^{2} φ}}$

$= \frac{r \sin θ \cos φ \cos^{2} θ}{r^{2} \cos θ \sin θ \sqrt{\cos^{2} φ + \sin^{2} φ}}$

$= \frac{\cos θ \cos φ}{r}$ 　　･･････(7)

が得られる．

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

(5)を代入して

$= \frac{\partial}{\partial x} (\sin θ \cos φ)$

$= \frac{\partial}{\partial r} (\sin θ \cos φ) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\sin θ \cos φ) \frac{\partial θ}{\partial x} + \frac{\partial}{\partial φ} (\sin θ \cos φ) \frac{\partial φ}{\partial x}$

$= \cos θ \cos φ \frac{\partial θ}{\partial x} - \sin θ \sin φ \frac{\partial φ}{\partial x}$

(7)，(6)を代入して

$= \cos θ \cos φ \frac{\cos θ \cos φ}{r} - \sin θ \sin φ (- \frac{\sin φ}{r \sin θ})$

$= \frac{\cos^{2} θ \cos^{2} φ}{r} + \frac{\sin^{2} φ}{r}$

$= \frac{\cos^{2} θ \cos^{2} φ + \sin^{2} φ}{r}$ 　　･･････(8)

が得られる．

$\frac{\partial^{2} θ}{\partial x^{2}}$ $= \frac{\partial}{\partial x} (\frac{\partial θ}{\partial x})$

(7)を代入して

$= \frac{\partial}{\partial x} (\frac{\cos θ \cos φ}{r})$

$= \frac{\partial}{\partial r} (\frac{\cos θ \cos φ}{r}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (\frac{\cos θ \cos φ}{r}) \frac{\partial θ}{\partial x} + \frac{\partial}{\partial φ} (\frac{\cos θ \cos φ}{r}) \frac{\partial φ}{\partial x}$

$= - \frac{\cos θ \cos φ}{r^{2}} \frac{\partial r}{\partial x} - \frac{\sin θ \cos φ}{r} \frac{\partial θ}{\partial x} - \frac{\cos θ \sin φ}{r} \frac{\partial φ}{\partial x}$

(5)，(7)，(6)を代入して

$= - \frac{\cos θ \cos φ}{r^{2}} \sin θ \cos φ - \frac{\sin θ \cos φ}{r} \frac{\cos θ \cos φ}{r} - \frac{\cos θ \sin φ}{r} (- \frac{\sin φ}{r \sin θ})$

$= - \frac{\sin θ \cos θ \cos^{2} φ}{r^{2}} - \frac{\sin θ \cos θ \cos^{2} φ}{r^{2}} + \frac{\cos θ \sin^{2} φ}{r^{2} \sin θ}$

$= - \frac{2sin θ \cos θ \cos^{2} φ}{r^{2}} + \frac{\cos θ \sin^{2} φ}{r^{2} \sin θ}$

$= \frac{- 2 \sin^{2} θ \cos θ \cos^{2} φ + \cos θ \sin^{2} φ}{r^{2} \sin θ}$

$= \frac{\cos θ (- 2 \sin^{2} θ \cos^{2} φ + \sin^{2} φ)}{r^{2} \sin θ}$ 　･･････(9)

が得られる．

$\frac{\partial^{2} φ}{\partial x^{2}}$ $= \frac{\partial}{\partial x} (\frac{\partial φ}{\partial x})$

(6)を代入して

$= \frac{\partial}{\partial x} (- \frac{\sin φ}{r \sin θ})$

$= \frac{\partial}{\partial r} (- \frac{\sin φ}{r \sin θ}) \frac{\partial r}{\partial x} + \frac{\partial}{\partial θ} (- \frac{\sin φ}{r \sin θ}) \frac{\partial θ}{\partial x} + \frac{\partial}{\partial φ} (- \frac{\sin φ}{r \sin θ}) \frac{\partial φ}{\partial x}$

$= \frac{\sin φ}{r^{2} \sin θ} \frac{\partial r}{\partial x} + \frac{\cos θ \sin φ}{r \sin^{2} θ} \frac{\partial θ}{\partial x} - \frac{\cos φ}{r \sin θ} \frac{\partial φ}{\partial x}$

参考：基本となる関数の導関数（微分）

(5)，(7)，(6)を代入して

$= \frac{\sin φ}{r^{2} \sin θ} \sin θ \cos φ + \frac{\cos θ \sin φ}{r \sin^{2} θ} \frac{\cos θ \cos φ}{r} - \frac{\cos φ}{r \sin θ} (- \frac{\sin φ}{r \sin θ})$

$= \frac{\sin φ \cos φ}{r^{2}} + \frac{\cos^{2} θ \sin φ \cos φ}{r^{2} \sin^{2} θ} + \frac{\sin φ \cos φ}{r^{2} \sin^{2} θ}$

$= \frac{\sin φ \cos φ (\sin^{2} θ + \cos^{2} θ) + \sin φ \cos φ}{r^{2} \sin^{2} θ}$

$= \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ}$ 　･･････(10)

ここで，(2)～(10)を用いて(1)を式変形すると

$\frac{\partial^{2} ψ}{\partial x^{2}}$ $= \frac{\partial^{2} ψ}{\partial r^{2}} {(\sin θ \cos φ)}^{2} + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\cos θ \cos φ}{r} \sin θ \cos φ + \frac{\partial ψ}{\partial r} \frac{\cos^{2} θ \cos^{2} φ + \sin^{2} φ}{r}$

$+ \frac{\partial^{2} ψ}{\partial r \partial θ} \sin θ \cos φ \frac{\cos θ \cos φ}{r} + \frac{\partial^{2} ψ}{\partial θ^{2}} {(\frac{\cos θ \cos φ}{r})}^{2} + \frac{\partial ψ}{\partial θ} \frac{\cos θ (- 2 \sin^{2} θ \cos^{2} φ + \sin^{2} φ)}{r^{2} \sin θ}$

$+ \frac{\partial^{2} ψ}{\partial r \partial φ} \sin θ \cos φ (- \frac{\sin φ}{r \sin θ}) + \frac{\partial^{2} ψ}{\partial φ^{2}} {(- \frac{\sin φ}{r \sin θ})}^{2} + \frac{\partial ψ}{\partial φ} \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ}$

$= \frac{\partial^{2} ψ}{\partial r^{2}} \sin^{2} θ \cos^{2} φ + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ \cos^{2} φ}{r} + \frac{\partial ψ}{\partial r} \frac{\cos^{2} θ \cos^{2} φ + \sin^{2} φ}{r}$

$+ \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\sin θ \cos θ \cos^{2} φ}{r} + \frac{\partial^{2} ψ}{\partial θ^{2}} \frac{\cos^{2} θ \cos^{2} φ}{r^{2}} + \frac{\partial ψ}{\partial θ} \frac{\cos θ (- 2 \sin^{2} θ \cos^{2} φ + \sin^{2} φ)}{r^{2} \sin θ}$

$+ \frac{\partial^{2} ψ}{\partial r \partial φ} (- \frac{\sin φ \cos φ}{r}) + \frac{\partial^{2} ψ}{\partial φ^{2}} \frac{\sin^{2} φ}{r^{2} \sin^{2} θ} + \frac{\partial ψ}{\partial φ} \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ}$ 　･･････(11)

が得られる．

同様にして(詳しくはこちら)

$\frac{\partial^{2} ψ}{\partial y^{2}}$ $= \frac{\partial^{2} ψ}{\partial r^{2}} \sin^{2} θ \sin^{2} φ + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ \sin^{2} φ}{r} + \frac{\partial ψ}{\partial r} \frac{\cos^{2} θ \sin^{2} φ + \cos^{2} φ}{r}$

$+ \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\sin θ \cos θ \sin^{2} φ}{r} + \frac{\partial^{2} ψ}{\partial θ^{2}} \frac{\cos^{2} θ \sin^{2} φ}{r^{2}} + \frac{\partial ψ}{\partial θ} \frac{\cos θ (- 2 \sin^{2} θ \sin^{2} φ + \cos^{2} φ)}{r^{2} \sin θ}$

$+ \frac{\partial^{2} ψ}{\partial r \partial φ} \frac{\sin φ \cos φ}{r} + \frac{\partial^{2} ψ}{\partial φ^{2}} \frac{\cos^{2} φ}{r^{2} \sin^{2} θ} - \frac{\partial ψ}{\partial φ} \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ}$ 　･･････(12)

が得られる．

同様にして（詳しくはこちら）

$\frac{\partial^{2} ψ}{\partial z^{2}}$ $= \frac{\partial^{2} ψ}{\partial r^{2}} \cos^{2} θ - \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ}{r} + \frac{\partial ψ}{\partial r} \frac{\sin^{2} θ}{r}$

$- \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\sin θ \cos θ}{r} + \frac{\partial^{2} ψ}{\partial θ^{2}} \frac{\sin^{2} θ}{r^{2}} + \frac{\partial ψ}{\partial θ} \frac{2 \sin θ \cos θ}{r^{2}}$ 　　･･････(13)

が得られる．

(11)，(12)，(13)を足し合わせると

$\frac{\partial^{2} ψ}{\partial x^{2}} + \frac{\partial^{2} ψ}{\partial y^{2}} + \frac{\partial^{2} ψ}{\partial z^{2}}$ $= \frac{\partial^{2} ψ}{\partial r^{2}} \sin^{2} θ \cos^{2} φ + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ \cos^{2} φ}{r} + \frac{\partial ψ}{\partial r} \frac{\cos^{2} θ \cos^{2} φ + \sin^{2} φ}{r}$

$+ \frac{\partial^{2} ψ}{\partial r^{2}} \sin^{2} θ \sin^{2} φ + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ \sin^{2} φ}{r} + \frac{\partial ψ}{\partial r} \frac{\cos^{2} θ \sin^{2} φ + \cos^{2} φ}{r}$

$+ \frac{\partial^{2} ψ}{\partial r^{2}} \cos^{2} θ - \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\sin θ \cos θ}{r} + \frac{\partial ψ}{\partial r} \frac{\sin^{2} θ}{r}$

$= \frac{\partial^{2} ψ}{\partial r^{2}} (\sin^{2} θ \cos^{2} φ + \sin^{2} θ \sin^{2} φ + \cos^{2} θ)$

$+ \frac{\partial^{2} ψ}{\partial θ \partial r} (\frac{\sin θ \cos θ \cos^{2} φ}{r} + \frac{\sin θ \cos θ \sin^{2} φ}{r} - \frac{\sin θ \cos θ}{r})$

$+ \frac{\partial ψ}{\partial r} (\frac{\cos^{2} θ \cos^{2} φ + \sin^{2} φ}{r} + \frac{\cos^{2} θ \sin^{2} φ + \cos^{2} φ}{r} + \frac{\sin^{2} θ}{r})$

$+ \frac{\partial^{2} ψ}{\partial r \partial θ} (\frac{\sin θ \cos θ \cos^{2} φ}{r} + \frac{\sin θ \cos θ \sin^{2} φ}{r} - \frac{\sin θ \cos θ}{r})$

$+ \frac{\partial^{2} ψ}{\partial θ^{2}} (\frac{\cos^{2} θ \cos^{2} φ}{r^{2}} + \frac{\cos^{2} θ \sin^{2} φ}{r^{2}} + \frac{\sin^{2} θ}{r^{2}})$

$+ \frac{\partial ψ}{\partial θ} (\frac{\cos θ (- 2 \sin^{2} θ \cos^{2} φ + \sin^{2} φ)}{r^{2} \sin θ} + \frac{\cos θ (- 2 \sin^{2} θ \sin^{2} φ + \cos^{2} φ)}{r^{2} \sin θ} + \frac{2 \sin θ \cos θ}{r^{2}})$

$+ \frac{\partial^{2} ψ}{\partial r \partial φ} (- \frac{\sin φ \cos φ}{r} + \frac{\sin φ \cos φ}{r})$

$+ \frac{\partial^{2} ψ}{\partial φ^{2}} (\frac{\sin^{2} φ}{r^{2} \sin^{2} θ} + \frac{\cos^{2} φ}{r^{2} \sin^{2} θ})$

$+ \frac{\partial ψ}{\partial φ} (\frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ} - \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ})$

$= \frac{\partial^{2} ψ}{\partial r^{2}} + \frac{2}{r} \frac{\partial ψ}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} ψ}{\partial θ^{2}} + \frac{\cos θ}{r^{2} \sin θ} \frac{\partial ψ}{\partial θ} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2} ψ}{\partial φ^{2}}$ $= {\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}} ψ$

となり，示された．

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最終更新日：2023年1月17日