POINT
【関連記事】
- 曲線座標系におけるデルタ関数の表式．
デルタ関数の座標変換
デルタ関数の座標変換
$\boldsymbol{\xi}=f(\boldsymbol{x})$（注：$f^{-1}(0)$が1点に定まるとする）に対し
\begin{aligned}
&\delta^n(\boldsymbol{\xi})
=J\delta^n(\boldsymbol{x}-f^{-1}(0))\\
&\Biggl(J=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr|=\Biggl| \det{ \biggl(\frac{\partial \xi_i}{\partial x_j}\biggr)} \Biggr|^{-1}\Biggr)
\end{aligned}
（詳細：デルタ関数と公式 - Notes_JP）&\delta^n(\boldsymbol{\xi})
=J\delta^n(\boldsymbol{x}-f^{-1}(0))\\
&\Biggl(J=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr|=\Biggl| \det{ \biggl(\frac{\partial \xi_i}{\partial x_j}\biggr)} \Biggr|^{-1}\Biggr)
\end{aligned}
以下では$\boldsymbol{\xi}$が曲線座標を表し，$\boldsymbol{x}$がデカルト座標系を表すとします．したがって，
\begin{aligned}
&\delta^n(\boldsymbol{x})
=\frac{\delta^n(\boldsymbol{\xi})}{\Biggl| \det{ \biggl(\dfrac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr|}
\end{aligned}
の右辺を計算することが目的となります．&\delta^n(\boldsymbol{x})
=\frac{\delta^n(\boldsymbol{\xi})}{\Biggl| \det{ \biggl(\dfrac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr|}
\end{aligned}
デカルト座標系
3次元デカルト座標系のデルタ関数は\begin{aligned}
\delta^3(\boldsymbol{r})
=\delta(x)\delta(y)\delta(z)
\end{aligned}
で与えられます．\delta^3(\boldsymbol{r})
=\delta(x)\delta(y)\delta(z)
\end{aligned}
円筒座標系
円筒標系のデルタ関数
\begin{aligned}
\delta(x)\delta(y)\delta(z)
&=\frac{\delta(r)\delta(\theta)\delta(z)}{r}
\end{aligned}
\delta(x)\delta(y)\delta(z)
&=\frac{\delta(r)\delta(\theta)\delta(z)}{r}
\end{aligned}
円筒標$\boldsymbol{\xi}=(r,\theta,z)$を考える．円筒座標系とデカルト座標系$\boldsymbol{x}$との関係は
\begin{aligned}
\boldsymbol{x}
=
\begin{pmatrix}
x(r,\theta,z) \\
y(r,\theta,z) \\
z(r,\theta,z)
\end{pmatrix}
=
\begin{pmatrix}
r\cos\theta \\
r\sin\theta \\
z
\end{pmatrix}
\end{aligned}
である．\boldsymbol{x}
=
\begin{pmatrix}
x(r,\theta,z) \\
y(r,\theta,z) \\
z(r,\theta,z)
\end{pmatrix}
=
\begin{pmatrix}
r\cos\theta \\
r\sin\theta \\
z
\end{pmatrix}
\end{aligned}
ヤコビ行列は
\begin{aligned}
&\frac{\partial(x,y,z)}{\partial(r,\theta,z)}
=\biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)_{i,j} \\
&=\begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\
\frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z}
\end{pmatrix}\\
&=\begin{pmatrix}
\cos\theta & -r\sin\theta & 0 \\
\sin\theta & r\cos\theta & 0 \\
0 & 0 & 1
\end{pmatrix}
\end{aligned}
だから，&\frac{\partial(x,y,z)}{\partial(r,\theta,z)}
=\biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)_{i,j} \\
&=\begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\
\frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z}
\end{pmatrix}\\
&=\begin{pmatrix}
\cos\theta & -r\sin\theta & 0 \\
\sin\theta & r\cos\theta & 0 \\
0 & 0 & 1
\end{pmatrix}
\end{aligned}
\begin{aligned}
J&=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr| \\
&=\Bigl|r\cos^2\theta + r\sin^2\theta \Bigr| \\
&=r
\end{aligned}
となる．以上で示された．//J&=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr| \\
&=\Bigl|r\cos^2\theta + r\sin^2\theta \Bigr| \\
&=r
\end{aligned}
極座標系
極座標系のデルタ関数
\begin{aligned}
\delta(x)\delta(y)\delta(z)
&=\frac{\delta(r)\delta(\theta)\delta(\varphi)}{r^2\sin\theta}
\end{aligned}
\delta(x)\delta(y)\delta(z)
&=\frac{\delta(r)\delta(\theta)\delta(\varphi)}{r^2\sin\theta}
\end{aligned}
極座標$\boldsymbol{\xi}=(r,\theta,\varphi)$を考える．極座標系とデカルト座標系$\boldsymbol{x}$との関係は
\begin{aligned}
\boldsymbol{x}
=
\begin{pmatrix}
x(r,\theta,\varphi) \\
y(r,\theta,\varphi)\\
z(r,\theta,\varphi)
\end{pmatrix}
=
\begin{pmatrix}
r\sin\theta \cos\varphi \\
r\sin\theta \sin\varphi \\
r\cos\theta
\end{pmatrix}
\end{aligned}
である．\boldsymbol{x}
=
\begin{pmatrix}
x(r,\theta,\varphi) \\
y(r,\theta,\varphi)\\
z(r,\theta,\varphi)
\end{pmatrix}
=
\begin{pmatrix}
r\sin\theta \cos\varphi \\
r\sin\theta \sin\varphi \\
r\cos\theta
\end{pmatrix}
\end{aligned}
ヤコビ行列は
\begin{aligned}
&\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}
=\biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)_{i,j} \\
&=\begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \varphi} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \varphi} \\
\frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \varphi}
\end{pmatrix}\\
&=\begin{pmatrix}
\sin\theta\cos\varphi & r\cos\theta\cos\varphi & -r \sin\theta\sin\varphi\\
\sin\theta\sin\varphi & r\cos\theta\sin\varphi & r \sin\theta\cos\varphi \\
\cos\theta & -r\sin\theta & 0
\end{pmatrix}
\end{aligned}
だから，&\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}
=\biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)_{i,j} \\
&=\begin{pmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \varphi} \\
\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \varphi} \\
\frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \varphi}
\end{pmatrix}\\
&=\begin{pmatrix}
\sin\theta\cos\varphi & r\cos\theta\cos\varphi & -r \sin\theta\sin\varphi\\
\sin\theta\sin\varphi & r\cos\theta\sin\varphi & r \sin\theta\cos\varphi \\
\cos\theta & -r\sin\theta & 0
\end{pmatrix}
\end{aligned}
\begin{aligned}
J&=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr| \\
&=\Biggl| (0+\textcolor{red}{r^2\sin\theta\cos^2\theta\cos^2\varphi}
+\textcolor{blue}{r^2\sin^3\theta\sin^2\varphi}) \\
&\qquad -(-\textcolor{blue}{r^2\sin^3\theta\cos^2\varphi} + 0
-\textcolor{red}{r^2\sin\theta\cos^2\theta\sin^2\varphi}) \Biggr| \\
&=\Bigl|r^2\sin\theta (\textcolor{red}{\cos^2\theta} + \textcolor{blue}{\sin^2\theta}) \Bigr| \\
&=r^2\sin\theta
\end{aligned}
となる（最後の等号で，極座標系では$r>0,\, 0 \leq \theta < \pi$であることに注意）．以上で示された．//J&=\Biggl| \det{ \biggl(\frac{\partial x_i}{\partial \xi_j}\biggr)} \Biggr| \\
&=\Biggl| (0+\textcolor{red}{r^2\sin\theta\cos^2\theta\cos^2\varphi}
+\textcolor{blue}{r^2\sin^3\theta\sin^2\varphi}) \\
&\qquad -(-\textcolor{blue}{r^2\sin^3\theta\cos^2\varphi} + 0
-\textcolor{red}{r^2\sin\theta\cos^2\theta\sin^2\varphi}) \Biggr| \\
&=\Bigl|r^2\sin\theta (\textcolor{red}{\cos^2\theta} + \textcolor{blue}{\sin^2\theta}) \Bigr| \\
&=r^2\sin\theta
\end{aligned}
参考文献
Green関数の問題でよく現れます（球面境界条件であれば極座標系のデルタ関数が現れるなど）．- 物理とグリーン関数 (物理数学シリーズ 4)：補遺 [B] $\delta$関数
