$z = f(x, y)$は、偏導関数がすべて存在し、かつ偏導関数がすべて連続とする。このとき、$x, y$を座標$(\xi, \eta)$に座標変換すると、以下の式が成り立つことを示せ。 $\frac{\partial z}{\partial x} = \frac{\frac{\partial y}{\partial \eta} \frac{\partial z}{\partial \xi} - \frac{\partial y}{\partial \xi} \frac{\partial z}{\partial \eta}}{J}$ $\frac{\partial z}{\partial y} = \frac{\frac{\partial x}{\partial \xi} \frac{\partial z}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial z}{\partial \xi}}{J}$ ここで、$J = \frac{\partial x}{\partial \xi} \frac{\partial y}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial y}{\partial \xi}$である。

1. 問題の内容

$z = f(x, y)$は、偏導関数がすべて存在し、かつ偏導関数がすべて連続とする。このとき、$x, y$を座標$(\xi, \eta)$に座標変換すると、以下の式が成り立つことを示せ。

$\frac{\partial z}{\partial x} = \frac{\frac{\partial y}{\partial \eta} \frac{\partial z}{\partial \xi} - \frac{\partial y}{\partial \xi} \frac{\partial z}{\partial \eta}}{J}$

$\frac{\partial z}{\partial y} = \frac{\frac{\partial x}{\partial \xi} \frac{\partial z}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial z}{\partial \xi}}{J}$

ここで、$J = \frac{\partial x}{\partial \xi} \frac{\partial y}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial y}{\partial \xi}$である。

2. 解き方の手順

まず、$z$を$\xi$と$\eta$で偏微分することを考える。連鎖律より、

$\frac{\partial z}{\partial \xi} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \xi} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \xi}$

$\frac{\partial z}{\partial \eta} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \eta} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \eta}$

この連立方程式を$\frac{\partial z}{\partial x}$と$\frac{\partial z}{\partial y}$について解くことを目指す。

1つ目の式に$\frac{\partial y}{\partial \eta}$をかけ、2つ目の式に$\frac{\partial y}{\partial \xi}$をかけると、

$\frac{\partial z}{\partial \xi}\frac{\partial y}{\partial \eta} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \xi}\frac{\partial y}{\partial \eta} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \xi}\frac{\partial y}{\partial \eta}$

$\frac{\partial z}{\partial \eta}\frac{\partial y}{\partial \xi} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \eta}\frac{\partial y}{\partial \xi} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \eta}\frac{\partial y}{\partial \xi}$

上の式から下の式を引くと、

$\frac{\partial z}{\partial \xi}\frac{\partial y}{\partial \eta} - \frac{\partial z}{\partial \eta}\frac{\partial y}{\Partial\xi} = \frac{\partial z}{\partial x} (\frac{\partial x}{\partial \xi}\frac{\partial y}{\partial \eta} - \frac{\partial x}{\partial \eta}\frac{\partial y}{\partial \xi})$

よって、

$\frac{\partial z}{\partial x} = \frac{\frac{\partial y}{\partial \eta} \frac{\partial z}{\partial \xi} - \frac{\partial y}{\partial \xi} \frac{\partial z}{\partial \eta}}{\frac{\partial x}{\partial \xi} \frac{\partial y}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial y}{\partial \xi}} = \frac{\frac{\partial y}{\partial \eta} \frac{\partial z}{\partial \xi} - \frac{\partial y}{\partial \xi} \frac{\partial z}{\partial \eta}}{J}$

同様に、1つ目の式に$\frac{\partial x}{\partial \eta}$をかけ、2つ目の式に$\frac{\partial x}{\partial \xi}$をかけると、

$\frac{\partial z}{\partial \xi}\frac{\partial x}{\partial \eta} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \xi}\frac{\partial x}{\partial \eta} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \xi}\frac{\partial x}{\partial \eta}$

$\frac{\partial z}{\partial \eta}\frac{\partial x}{\partial \xi} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial \eta}\frac{\partial x}{\partial \xi} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial \eta}\frac{\partial x}{\partial \xi}$

下の式から上の式を引くと、

$\frac{\partial z}{\partial \eta}\frac{\partial x}{\partial \xi} - \frac{\partial z}{\partial \xi}\frac{\partial x}{\partial \eta} = \frac{\partial z}{\partial y} (\frac{\partial y}{\partial \eta}\frac{\partial x}{\partial \xi} - \frac{\partial y}{\partial \xi}\frac{\partial x}{\partial \eta})$

よって、

$\frac{\partial z}{\partial y} = \frac{\frac{\partial x}{\partial \xi} \frac{\partial z}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial z}{\partial \xi}}{\frac{\partial x}{\partial \xi} \frac{\partial y}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial y}{\partial \xi}} = \frac{\frac{\partial x}{\partial \xi} \frac{\partial z}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial z}{\partial \xi}}{J}$

3. 最終的な答え

$\frac{\partial z}{\partial x} = \frac{\frac{\partial y}{\partial \eta} \frac{\partial z}{\partial \xi} - \frac{\partial y}{\partial \xi} \frac{\partial z}{\partial \eta}}{J}$

$\frac{\partial z}{\partial y} = \frac{\frac{\partial x}{\partial \xi} \frac{\partial z}{\partial \eta} - \frac{\partial x}{\partial \eta} \frac{\partial z}{\partial \xi}}{J}$