等方的ビーム/計算ノート

下記の 4 次元の球面座標系のヤコビアンを求める。
- $\begin{eqnarray*} x_1 &=& r\sin\theta\sin\phi\cos\psi\\ x_2 &=& r\sin\theta\sin\phi\sin\psi\\ x_3 &=& r\sin\theta\cos\phi\\ x_4 &=& r\cos\theta \end{eqnarray*}$
$\newcommand{\dfrac}{\displaystyle\frac} \begin{eqnarray*} \left|\frac{\partial(x_1,x_2,x_3,x_4)}{\partial(r,\theta,\phi,\psi)}\right| &=& \left| \begin{array}{cccc} \dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} & \dfrac{\partial x_1}{\partial \psi} \\[1.7ex] \dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} & \dfrac{\partial x_2}{\partial \psi} \\[1.7ex] \dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} & \dfrac{\partial x_3}{\partial \psi} \\[1.7ex] \dfrac{\partial x_4}{\partial r} & \dfrac{\partial x_4}{\partial \theta} & \dfrac{\partial x_4}{\partial \phi} & \dfrac{\partial x_4}{\partial \psi} \end{array} \right| \\ &=& \left| \begin{array}{cccc} \sin\theta\sin\phi\cos\psi & r\cos\theta\sin\phi\cos\psi & r\sin\theta\cos\phi\cos\psi & -r\sin\theta\sin\phi\sin\psi\\ \sin\theta\sin\phi\sin\psi & r\cos\theta\sin\phi\sin\psi & r\sin\theta\cos\phi\sin\psi & r\sin\theta\sin\phi\cos\psi\\ \sin\theta\cos\phi & r\cos\theta\cos\phi & -r\sin\theta\sin\phi & 0 \\ \cos\theta & -r\sin\theta & 0 & 0 \end{array} \right| \\ \end{eqnarray*}$

おぼえがき