立体角/ノート
下記座標のヤコビ行列式の計算
- のとき、
下記座標のヤコビ行列式の計算
- 下図のような座標系を考える。
(ROOT 用ファイル : rec_solid_angle_def.C)
すなわち、半径 r の球面上のある点 (x,y,z) (図中の点 P) を図の θ と φ で指定する。このとき、
- と表せる。このとき、ヤコビ行列式は以下のように計算される。
![\newcommand{\dfrac}{\displaystyle\frac}
\begin{eqnarray*}
\left|\frac{\partial(x,y)}{\partial(\theta,\varphi)}\right|
&=& \left|
\begin{array}{ccc}
\dfrac{\partial x}{\partial\theta} & \dfrac{\partial x}{\partial\varphi} \\[1.7ex]
\dfrac{\partial y}{\partial\theta} & \dfrac{\partial y}{\partial\varphi}
\end{array}
\right|
\\
&=& \left|
\begin{array}{ccc}
\dfrac{c\cos\varphi}{\cos^2\theta} & -c\tan\theta\sin\varphi \\[1.7ex]
\dfrac{c\sin\varphi}{\cos^2\theta} & c\tan\theta\cos\varphi
\end{array}
\right|
\\
&=& c^2 \frac{\tan\theta}{\cos^2\theta}\cos^2\varphi+c^2 \frac{\tan\theta}{\cos^2\theta}\sin^2\varphi\\
&=& c^2\frac{\sin\theta}{\cos^3\theta}
\end{eqnarray*}](http://192.168.224.129/~koba/cgi-bin/moin.cgi/%E7%AB%8B%E4%BD%93%E8%A7%92/%E3%83%8E%E3%83%BC%E3%83%88?action=AttachFile&do=get&target=latex_4893e3df725df348b4941b260a1f74f7444111d0_p1.png "\newcommand{\dfrac}{\displaystyle\frac}
\begin{eqnarray*}
\left|\frac{\partial(x,y)}{\partial(\theta,\varphi)}\right|
&=& \left|
\begin{array}{ccc}
\dfrac{\partial x}{\partial\theta} & \dfrac{\partial x}{\partial\varphi} \\[1.7ex]
\dfrac{\partial y}{\partial\theta} & \dfrac{\partial y}{\partial\varphi}
\end{array}
\right|
\\
&=& \left|
\begin{array}{ccc}
\dfrac{c\cos\varphi}{\cos^2\theta} & -c\tan\theta\sin\varphi \\[1.7ex]
\dfrac{c\sin\varphi}{\cos^2\theta} & c\tan\theta\cos\varphi
\end{array}
\right|
\\
&=& c^2 \frac{\tan\theta}{\cos^2\theta}\cos^2\varphi+c^2 \frac{\tan\theta}{\cos^2\theta}\sin^2\varphi\\
&=& c^2\frac{\sin\theta}{\cos^3\theta}
\end{eqnarray*}")
(ROOT 用ファイル : rec_solid_angle_def.C)
すなわち、半径 r の球面上のある点 (x,y,z) (図中の点 P) を図の θ と φ で指定する。このとき、
![\newcommand{\dfrac}{\displaystyle\frac}
\begin{eqnarray*}
\left|\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}\right|
&=& \left|
\begin{array}{ccc}
\dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial\theta} & \dfrac{\partial x}{\partial\varphi} \\[1.7ex]
\dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial\theta} & \dfrac{\partial y}{\partial\varphi} \\[1.7ex]
\dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial\theta} & \dfrac{\partial z}{\partial\varphi}
\end{array}
\right|
\\
&=& \left|
\begin{array}{ccc}
\dfrac{\tan\theta}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{r(1+\tan^2\varphi)}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{-r\tan\theta\tan\varphi}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}} \\[1.7ex]
\dfrac{\tan\varphi}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{-r\tan\theta\tan\varphi}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{r(1+\tan^2\theta)}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}\\[1.7ex]
\dfrac{1}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{-r\tan\theta}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{-r\tan\varphi}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}
\end{array}
\right|
\\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left|
\begin{array}{ccc}
\tan\theta & 1+\tan^2\varphi & -\tan\theta\tan\varphi \\
\tan\varphi & -\tan\theta\tan\varphi & 1+\tan^2\theta \\
1 & -\tan\theta & -\tan\varphi
\end{array}
\right|
\\
% &=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left[\tan^2\theta\tan^2\varphi+\tan\theta^2\tan^2\varphi+(1+\tan^2\theta)(1+\tan^2\varphi)\right.
% \\ && \left. -\tan^2\theta\tan^2\varphi+\tan^2\theta(1+\tan^2\theta)+\tan^2\varphi(1+\tan^2\varphi)\right]
% \\ &=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left[1+2\tan^2\theta+2\tan^2\varphi+2\tan^2\theta\tan^2\varphi+\tan^4\theta+\tan^4\varphi\right] \\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} (1+\tan\theta^2+\tan^2\varphi)^2
\\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}
\\
&=& \frac{r^2\cos\theta\cos\varphi}{(\cos^2\theta\cos^2\varphi+\sin\theta^2\cos^2\varphi+\sin^2\varphi\cos^2\theta)^{3/2}}
\\
&=& \frac{r^2\cos\theta\cos\varphi}{(1-\sin^2\theta\sin^2\varphi)^{3/2}}
\end{eqnarray*}](http://192.168.224.129/~koba/cgi-bin/moin.cgi/%E7%AB%8B%E4%BD%93%E8%A7%92/%E3%83%8E%E3%83%BC%E3%83%88?action=AttachFile&do=get&target=latex_27c7fb2c080d1ee09644b75f63847bd404d172f4_p1.png "\newcommand{\dfrac}{\displaystyle\frac}
\begin{eqnarray*}
\left|\frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}\right|
&=& \left|
\begin{array}{ccc}
\dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial\theta} & \dfrac{\partial x}{\partial\varphi} \\[1.7ex]
\dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial\theta} & \dfrac{\partial y}{\partial\varphi} \\[1.7ex]
\dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial\theta} & \dfrac{\partial z}{\partial\varphi}
\end{array}
\right|
\\
&=& \left|
\begin{array}{ccc}
\dfrac{\tan\theta}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{r(1+\tan^2\varphi)}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{-r\tan\theta\tan\varphi}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}} \\[1.7ex]
\dfrac{\tan\varphi}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{-r\tan\theta\tan\varphi}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{r(1+\tan^2\theta)}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}\\[1.7ex]
\dfrac{1}{\sqrt{1+\tan\theta^2+\tan^2\varphi}} & \dfrac{-r\tan\theta}{\cos^2\theta(1+\tan\theta^2+\tan^2\varphi)^{3/2}} & \dfrac{-r\tan\varphi}{\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}
\end{array}
\right|
\\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left|
\begin{array}{ccc}
\tan\theta & 1+\tan^2\varphi & -\tan\theta\tan\varphi \\
\tan\varphi & -\tan\theta\tan\varphi & 1+\tan^2\theta \\
1 & -\tan\theta & -\tan\varphi
\end{array}
\right|
\\
% &=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left[\tan^2\theta\tan^2\varphi+\tan\theta^2\tan^2\varphi+(1+\tan^2\theta)(1+\tan^2\varphi)\right.
% \\ && \left. -\tan^2\theta\tan^2\varphi+\tan^2\theta(1+\tan^2\theta)+\tan^2\varphi(1+\tan^2\varphi)\right]
% \\ &=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} \left[1+2\tan^2\theta+2\tan^2\varphi+2\tan^2\theta\tan^2\varphi+\tan^4\theta+\tan^4\varphi\right] \\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{7/2}} (1+\tan\theta^2+\tan^2\varphi)^2
\\
&=& \frac{r^2}{\cos^2\theta\cos^2\varphi(1+\tan\theta^2+\tan^2\varphi)^{3/2}}
\\
&=& \frac{r^2\cos\theta\cos\varphi}{(\cos^2\theta\cos^2\varphi+\sin\theta^2\cos^2\varphi+\sin^2\varphi\cos^2\theta)^{3/2}}
\\
&=& \frac{r^2\cos\theta\cos\varphi}{(1-\sin^2\theta\sin^2\varphi)^{3/2}}
\end{eqnarray*}")