ラプラシアンの極座標表示： 3次元

ラプラシアンの極座標表示を求めるシリーズ（目次）。　今回は3次元のラプラシアンの極座標表示。　ラプラシアンは

　　 ${ \displaystyle\begin{align*} \triangle = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \end{align*}}$

ですね。　3次元の極座標は次のように定義されてました（こちらを参照）：

　　 ${ \displaystyle\begin{align*} \begin{cases} x = r\sin\theta\cos\varphi \\ y = r\sin\theta\sin\varphi \\ z = r\cos\theta \end{cases} \quad \begin{pmatrix} 0 \le r < \infty \\ 0 \le \theta \le \pi \\ 0 \le \varphi \le 2\pi \end{pmatrix} \end{align*}}$

また、極座標を直交座標を用いて表すと以下のようになります：

　　 ${ \displaystyle\begin{align*} r &= \sqrt{x^2 + y^2 + z^2}, & \theta &= \tan^{-1}\frac{\sqrt{x^2+y^2}}{z}, & \varphi &= \tan^{-1}\frac{y}{x} \end{align*}}$

記事の流れは以下のようになります：

${ x, y, z }$ 微分の極座標表示
${ x }$ の2階微分の極座標表示
${ y }$ の2階微分の極座標表示
${ z }$ の2階微分の極座標表示
ラプラシアンの極座標表示

直交座標の微分を極座標とその微分で表す

第1段階として、直交座標 ${ x,\,y,\,z }$ による微分を、極座標 ${ r,\,\theta,\,\varphi }$ とその微分で表しましょう。

${ x }$ 微分

${ x }$ 微分に関して、連鎖律 より

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta} + \frac{\partial \varphi}{\partial x}\frac{\partial}{\partial \varphi} \end{align*}}$

が成り立ちます。　各項を地道に計算していきましょう。　まずは動径 ${ r }$ を ${ x }$ で微分：

　　 ${ \displaystyle\begin{align*} \frac{\partial r}{\partial x} &= \frac{\partial}{\partial x}\sqrt{x^2 + y^2 +z^2} \\ & = \frac{2x}{2\sqrt{x^2 + y^2 + z^2}} \\ &= \frac{x}{r} \\ &= \sin\theta\cos\varphi \end{align*}}$

次は ${ \theta }$ の ${ x }$ 微分。　 ${ \tan^{-1} x }$ の微分公式

　　 ${ \displaystyle\begin{align*} \left(\tan^{-1}x\right)' = \frac{1}{1+x^2} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} \frac{\partial \theta}{\partial x} &= \frac{1}{1+\frac{x^2+y^2}{z^2}} \cdot \frac{1}{z} \cdot \frac{x}{\sqrt{x^2+y^2}} \\[2mm] &= \frac{z}{x^2+y^2+z^2}\cdot\frac{x}{\sqrt{x^2+y^2}} \\[2mm] &= \frac{r\cos\theta}{r^2}\cdot\frac{r\sin\theta\cos\varphi}{r\sin\theta} \\[2mm] &= \frac{\cos\theta \cos\varphi}{r} \end{align*}}$

${ \varphi }$ の ${ x }$ 微分は ${ \theta }$ の場合と同様に計算できて

　　 ${ \displaystyle\begin{align*} \frac{\partial \varphi}{\partial x} &= \frac{1}{1+\frac{y^2}{x^2}} \cdot \left(-\frac{y}{x^2}\right) \\[2mm] &= -\frac{y}{x^2+y^2} \\[2mm] &= -\frac{r\sin\theta\sin\varphi}{r^2\sin^2\theta} \\[2mm] &= -\frac{\sin\varphi}{r\sin\theta} \end{align*}}$

よって

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial x} = \sin\theta\cos\varphi \frac{\partial}{\partial r} + \frac{\cos\theta \cos\varphi}{r}\frac{\partial}{\partial \theta} -\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

となり、 ${ x }$ 微分を極座標とその微分で表せました。

${ y }$ 微分

${ y }$ 座標に対しても、同様にして

　　 ${ \displaystyle\begin{align*} \frac{\partial r}{\partial y} &= \sin\theta\sin\varphi, & \frac{\partial \theta}{\partial y} &= \frac{\cos\theta\sin\varphi}{r}, & \frac{\partial \varphi}{\partial y} &= \frac{\cos\varphi}{r\sin\theta} \end{align*}}$

となるので

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial y} = \sin\theta\sin\varphi \frac{\partial}{\partial r} + \frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$
　　

${ z }$ 微分

　　 ${ \displaystyle\begin{align*} \frac{\partial r}{\partial z} &= \cos\theta, & \frac{\partial \theta}{\partial z} &= -\frac{\sin\theta}{r}, & \frac{\partial \varphi}{\partial z} &= 0 \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial z} = \cos\theta \frac{\partial}{\partial r} - \frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \end{align*}}$

を得ます。

まとめ

結果をまとめると

　　 ${ \displaystyle\begin{align*} \nabla = \begin{pmatrix} \dfrac{\partial}{\partial x} \\[4mm] \dfrac{\partial}{\partial y} \\[4mm] \dfrac{\partial}{\partial z} \end{pmatrix} = \begin{pmatrix} \sin\theta\cos\varphi \dfrac{\partial}{\partial r} + \dfrac{\cos\theta \cos\varphi}{r}\dfrac{\partial}{\partial \theta} -\dfrac{\sin\varphi}{r\sin\theta}\dfrac{\partial}{\partial \varphi} \\[4mm] \sin\theta\sin\varphi \dfrac{\partial}{\partial r} + \dfrac{\cos\theta\sin\varphi}{r}\dfrac{\partial}{\partial \theta} + \dfrac{\cos\varphi}{r\sin\theta}\dfrac{\partial}{\partial \varphi} \\[4mm] \cos\theta \dfrac{\partial}{\partial r} - \dfrac{\sin\theta}{r}\dfrac{\partial}{\partial \theta} \end{pmatrix} \end{align*}}$

行列の形で書くとこうなります：

　　 ${ \displaystyle\begin{align*} \begin{pmatrix} \dfrac{\partial}{\partial x} \\[4mm] \dfrac{\partial}{\partial y} \\[4mm] \dfrac{\partial}{\partial z} \end{pmatrix} = \begin{pmatrix} \sin\theta\cos\varphi & \dfrac{\cos\theta \cos\varphi}{r} & -\dfrac{\sin\varphi}{r\sin\theta} \\[4mm] \sin\theta\sin\varphi & \dfrac{\cos\theta\sin\varphi}{r} & \dfrac{\cos\varphi}{r\sin\theta} \\[4mm] \cos\theta & - \dfrac{\sin\theta}{r} & 0 \end{pmatrix} \begin{pmatrix} \dfrac{\partial}{\partial r} \\[4mm] \dfrac{\partial}{\partial \theta} \\[4mm] \dfrac{\partial}{\partial \varphi} \end{pmatrix} \end{align*}}$

${ x }$ の2階微分の極座標表示

上記で得られた ${ x }$ 微分の極座標表示

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial x} = \sin\theta\cos\varphi\frac{\partial}{\partial r} + \frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} - \frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

を用いて、 ${ x }$ の2階微分 ${ \frac{\partial^2}{\partial x^2} }$ の極座標表示を計算します。

${ r }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial r} \frac{\partial}{\partial x} &= \sin\theta\cos\varphi\frac{\partial^2}{\partial r^2} + \frac{\cos\theta\cos\varphi}{r}\frac{\partial^2}{\partial r \partial \theta} -\frac{\sin\varphi}{r\sin\theta}\frac{\partial^2}{\partial r \partial\varphi} \\[2mm] &\qquad\qquad -\frac{\cos\theta\cos\varphi}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin\varphi}{r^2 \sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} &\sin\theta\cos\varphi\frac{\partial}{\partial r} \frac{\partial}{\partial x} \\[2mm] &= \sin^2\theta\cos^2\varphi\frac{\partial^2}{\partial r^2} + \frac{\sin\theta\cos\theta\cos^2\varphi}{r}\frac{\partial^2}{\partial r \partial \theta} -\frac{\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial\varphi} \\[2mm] &\qquad\qquad -\frac{\sin\theta\cos\theta\cos^2\varphi}{r^2}\frac{\partial}{\partial \theta} + \frac{\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial \varphi} \end{align*}}$

${ \theta }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial \theta} \frac{\partial}{\partial x} &= \sin\theta\cos\varphi\frac{\partial^2}{\partial r \partial\theta} + \frac{\cos\theta\cos\varphi}{r}\frac{\partial^2}{\partial \theta^2} - \frac{\sin\varphi}{r\sin\theta}\frac{\partial^2}{\partial \theta \partial\varphi} \\[2mm] &\qquad\qquad +\cos\theta\cos\varphi\frac{\partial}{\partial r} -\frac{\sin\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\theta\sin\varphi}{r \sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} & \frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} \frac{\partial}{\partial x} \\[2mm] &= \frac{\sin\theta\cos\theta\cos^2\varphi}{r}\frac{\partial^2}{\partial r \partial\theta} + \frac{\cos^2\theta\cos^2\varphi}{r^2}\frac{\partial^2}{\partial \theta^2} -\frac{\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta \partial\varphi} \\[2mm] &\qquad\qquad +\frac{\cos^2\theta\cos^2\varphi}{r}\frac{\partial}{\partial r} -\frac{\sin\theta\cos\theta\cos^2\varphi}{r^2}\frac{\partial}{\partial \theta} + \frac{\cos^2\theta\sin\varphi\cos\varphi}{r^2 \sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

${ \varphi }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial \varphi} \frac{\partial}{\partial x} &= \sin\theta\cos\varphi\frac{\partial^2}{\partial r \partial \varphi} + \frac{\cos\theta\cos\varphi}{r}\frac{\partial^2}{\partial \theta \partial \varphi} -\frac{\sin\varphi}{r\sin\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad\qquad -\sin\theta\sin\varphi\frac{\partial}{\partial r} -\frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} - \frac{\cos\varphi}{r \sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} & -\frac{\sin\varphi}{r\sin\theta} \frac{\partial}{\partial \varphi} \frac{\partial}{\partial x} \\[2mm] &= -\frac{\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial \varphi} - \frac{\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta \partial \varphi} +\frac{\sin^2\varphi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad\qquad + \frac{\sin^2\varphi}{r}\frac{\partial}{\partial r} + \frac{\cos\theta\sin^2\varphi}{r^2\sin\theta}\frac{\partial}{\partial \theta} + \frac{\sin\varphi\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

${ x }$ の2回微分

よって、 ${ x }$ の2階微分は以下のようになります：

　　 ${ \displaystyle\begin{align*} \frac{\partial^2}{\partial x^2} &= \sin^2\theta\cos^2\varphi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\cos^2\varphi}{r^2}\frac{\partial^2}{\partial \theta^2} +\frac{\sin^2\varphi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad + \frac{2\sin\theta\cos\theta\cos^2\varphi}{r}\frac{\partial^2}{\partial r \partial\theta} \\ &\qquad - \frac{2\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial \varphi} \\ & \qquad - \frac{2\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta} \frac{\partial^2}{\partial \theta \partial \varphi} \\[2mm] &\qquad\qquad +\frac{\cos^2\theta\cos^2\varphi+\sin^2\varphi}{r}\frac{\partial}{\partial r} \\ &\qquad\qquad - \left(\frac{2\sin\theta\cos\theta\cos^2\varphi}{r^2} +\frac{\cos\theta\sin^2\varphi}{r^2\sin\theta}\right)\frac{\partial}{\partial \theta} \\ &\qquad\qquad +\frac{2\sin\varphi\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

となります。　ホントに綺麗にまとまるのか不安・・・

${ y }$ の2階微分の極座標表示

${ x }$ の場合と同様にして ${ y }$ 微分の極座標表示

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial y} = \sin\theta\sin\varphi\frac{\partial}{\partial r} + \frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} + \frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

を用いて、 ${ y }$ の2階微分 ${ \frac{\partial^2}{\partial y^2} }$ の極座標表示を計算します。

${r }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial r} \frac{\partial}{\partial y} &= \sin\theta\sin\varphi\frac{\partial^2}{\partial r^2} + \frac{\cos\theta\sin\varphi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\cos\varphi}{r\sin\theta}\frac{\partial^2}{\partial r \partial\varphi} \\[2mm] &\qquad\qquad -\frac{\cos\theta\sin\varphi}{r^2}\frac{\partial}{\partial \theta} - \frac{\cos\varphi}{r^2 \sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} & \sin\theta\sin\varphi\frac{\partial}{\partial r} \frac{\partial}{\partial y} \\[2mm] &= \sin^2\theta\sin^2\varphi\frac{\partial^2}{\partial r^2} + \frac{\sin\theta\cos\theta\sin^2\varphi}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial\varphi} \\[2mm] &\qquad\qquad - \frac{\sin\theta\cos\theta\sin^2\varphi}{r^2}\frac{\partial}{\partial \theta} - \frac{\sin\varphi\cos\varphi}{r^2}\frac{\partial}{\partial \varphi} \end{align*}}$

${ \theta }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial \theta} \frac{\partial}{\partial y} &= \sin\theta\sin\varphi\frac{\partial^2}{\partial r \partial\theta} + \frac{\cos\theta\sin\varphi}{r}\frac{\partial^2}{\partial \theta^2} + \frac{\cos\varphi}{r\sin\theta}\frac{\partial^2}{\partial \theta \partial\varphi} \\[2mm] &\qquad\qquad +\cos\theta\sin\varphi\frac{\partial}{\partial r} - \frac{\sin\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} - \frac{\cos\theta\cos\varphi}{r \sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} & \frac{\cos\theta\sin\varphi}{r}\frac{\partial}{\partial \theta} \frac{\partial}{\partial y} \\[2mm] &= \frac{\sin\theta\cos\theta\sin^2\varphi}{r}\frac{\partial^2}{\partial r \partial\theta} + \frac{\cos^2\theta\sin^2\varphi}{r^2}\frac{\partial^2}{\partial \theta^2} + \frac{\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta \partial\varphi} \\[2mm] &\qquad\qquad +\frac{\cos^2\theta\sin^2\varphi}{r}\frac{\partial}{\partial r} - \frac{\sin\theta\cos\theta\sin^2\varphi}{r^2}\frac{\partial}{\partial \theta} - \frac{\cos^2\theta\sin\varphi\cos\varphi}{r^2 \sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

${ \varphi }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial \varphi} \frac{\partial}{\partial y} &= \sin\theta\sin\varphi\frac{\partial^2}{\partial r \partial \varphi} + \frac{\cos\theta\sin\varphi}{r}\frac{\partial^2}{\partial \theta \partial \varphi} + \frac{\cos\varphi}{r\sin\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad\qquad -\sin\theta\cos\varphi\frac{\partial}{\partial r} -\frac{\cos\theta\cos\varphi}{r}\frac{\partial}{\partial \theta} - \frac{\sin\varphi}{r \sin\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} & \frac{\cos\varphi}{r\sin\theta} \frac{\partial}{\partial \varphi} \frac{\partial}{\partial y} \\[2mm] &= \frac{\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial \varphi} + \frac{\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta}\frac{\partial^2}{\partial \theta \partial \varphi} + \frac{\cos^2\varphi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad\qquad + \frac{\cos^2\varphi}{r}\frac{\partial}{\partial r} + \frac{\cos\theta\cos^2\varphi}{r^2\sin\theta}\frac{\partial}{\partial \theta} - \frac{\sin\varphi\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

${ y }$ の2回微分

以上をまとめると、 ${ y }$ の2階微分は以下のようになります：

　　 ${ \displaystyle\begin{align*} \frac{\partial^2}{\partial y^2} &= \sin^2\theta\sin^2\varphi\frac{\partial^2}{\partial r^2} + \frac{\cos^2\theta\sin^2\varphi}{r^2}\frac{\partial^2}{\partial \theta^2} +\frac{\cos^2\varphi}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &\qquad + \frac{2\sin\theta\cos\theta\sin^2\varphi}{r}\frac{\partial^2}{\partial r \partial\theta} \\ &\qquad + \frac{2\sin\varphi\cos\varphi}{r}\frac{\partial^2}{\partial r \partial \varphi} \\ & \qquad + \frac{2\cos\theta\sin\varphi\cos\varphi}{r^2\sin\theta} \frac{\partial^2}{\partial \theta \partial \varphi} \\[2mm] &\qquad\qquad +\frac{\cos^2\theta\sin^2\varphi+\cos^2\varphi}{r}\frac{\partial}{\partial r} \\ &\qquad\qquad + \left(-\frac{2\sin\theta\cos\theta\sin^2\varphi}{r^2} +\frac{\cos\theta\cos^2\varphi}{r^2\sin\theta}\right)\frac{\partial}{\partial \theta} \\ &\qquad\qquad - \frac{2\sin\varphi\cos\varphi}{r^2\sin^2\theta}\frac{\partial}{\partial \varphi} \end{align*}}$

${ z }$ の2階微分の極座標表示

最後は ${ z }$ 微分の極座標表示

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial z} = \cos\theta\frac{\partial}{\partial r} - \frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \end{align*}}$

を用いて、 ${ z }$ の2階微分 ${ \frac{\partial^2}{\partial z^2} }$ の極座標表示を計算します。　やることは同じ。

${ r }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial r} \frac{\partial}{\partial z} = \cos\theta\frac{\partial^2}{\partial r^2} - \frac{\sin\theta}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\theta}{r^2}\frac{\partial}{\partial \theta} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} \cos\theta\frac{\partial}{\partial r} \frac{\partial}{\partial z} = \cos^2\theta\frac{\partial^2}{\partial r^2} - \frac{\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial \theta} + \frac{\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} \end{align*}}$

${ \theta }$ 微分の項

　　 ${ \displaystyle\begin{align*} \frac{\partial}{\partial \theta} \frac{\partial}{\partial z} = \cos\theta\frac{\partial^2}{\partial r \partial\theta} - \frac{\sin\theta}{r}\frac{\partial^2}{\partial \theta^2} - \sin\theta\frac{\partial}{\partial r} - \frac{\cos\theta}{r}\frac{\partial}{\partial \theta} \end{align*}}$

より

　　 ${ \displaystyle\begin{align*} -\frac{\sin\theta}{r}\frac{\partial}{\partial \theta} \frac{\partial}{\partial z} = - \frac{\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial\theta} + \frac{\sin^2\theta}{r^2}\frac{\partial^2}{\partial \theta^2} +\frac{\sin^2\theta}{r}\frac{\partial}{\partial r} + \frac{\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} \end{align*}}$

${ \varphi }$ 微分の項

　　 ${ \displaystyle \frac{\partial}{\partial \varphi} \frac{\partial}{\partial z} = 0 }$

より

　　 ${ \displaystyle \frac{\cos\varphi}{r\sin\theta} \frac{\partial}{\partial \varphi} \frac{\partial}{\partial z} = 0 }$

${ z }$ の2回微分

以上をまとめると

　　 ${ \displaystyle\begin{align*} \frac{\partial^2}{\partial z^2} &= \cos^2\theta\frac{\partial^2}{\partial r^2} + \frac{\sin^2\theta}{r^2}\frac{\partial^2}{\partial \theta^2} \\[2mm] &\qquad - \frac{2\sin\theta\cos\theta}{r}\frac{\partial^2}{\partial r \partial\theta} \\[2mm] &\qquad\qquad +\frac{\sin^2\theta}{r}\frac{\partial}{\partial r} \\ &\qquad\qquad + \frac{2\sin\theta\cos\theta}{r^2}\frac{\partial}{\partial \theta} \end{align*} }$

ふぅ、やっと準備完了。

3次元ラプラシアンの極座標表示

さて、以上の結果をまとめましょう。

2階微分の項

2階微分の項は、同変数の2階微分以外の項は打ち消しあって

　　 ${ \displaystyle \frac{\partial^2}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} +\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} }$

1階微分の項

1階微分の項は残る項や消える項がそれぞれあって

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

3次元のラプラシアンの極座標表示

以上をまとめると

　　 ${ \displaystyle\begin{align*} \triangle &= \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial \theta^2} + \frac{\cos\theta}{r^2 \sin\theta}\frac{\partial}{\partial \theta} +\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \\[2mm] &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \varphi^2} \end{align*}}$