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ラプラシアンの極座標表示 : 3次元

ラプラシアン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極座標表示。 ラプラシアン

  { \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:

  1. { x, y, z } 微分の極座標表示
  2. { x } の2階微分の極座標表示
  3. { y } の2階微分の極座標表示
  4. { z } の2階微分の極座標表示
  5. ラプラシアンの極座標表示

直交座標の微分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 }{ x }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 }{ x } 微分。 { \tan^{-1} x }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 }{ x } 微分{ \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階微分極座標表示

上記で得られた { x } 微分極座標表示

  { \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} }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階微分極座標表示

{ x } の場合と同様にして { y } 微分極座標表示

  { \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} }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階微分極座標表示

最後は { z } 微分極座標表示

  { \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} }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*}}

・・・。 4次元以上のラプラシアンの計算するための手始めにするつもりだったけど、この方法で高次元にいくのは現実的ではないね・・・

解析入門 (1)

解析入門 (1)

解析入門 (2)     基礎数学 3

解析入門 (2) 基礎数学 3