$z = f(x, y)$ であり、$x = u + 3v$, $y = 2u + v$ であるとき、以下の問題を解く。 1. $z_u$ と $z_v$ を $z_x$ と $z_y$ を用いて表せ。

1. 問題の内容

$z = f(x, y)$ であり、$x = u + 3v$, $y = 2u + v$ であるとき、以下の問題を解く。

1. $z_u$ と $z_v$ を $z_x$ と $z_y$ を用いて表せ。

2. $z_{uu}$, $z_{uv}$, $z_{vv}$ を $z_{xx}$, $z_{xy}$, $z_{yy}$ を用いて表せ。

2. 解き方の手順

問題1:

連鎖律を用いる。

* $z_u = \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u} = z_x \frac{\partial x}{\partial u} + z_y \frac{\partial y}{\partial u}$

$\frac{\partial x}{\partial u} = 1$, $\frac{\partial y}{\partial u} = 2$ より、

$z_u = z_x + 2z_y$

* $z_v = \frac{\partial z}{\partial v} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial v} = z_x \frac{\partial x}{\partial v} + z_y \frac{\partial y}{\partial v}$

$\frac{\partial x}{\partial v} = 3$, $\frac{\partial y}{\partial v} = 1$ より、

$z_v = 3z_x + z_y$

問題2:

問題1の結果をさらに微分する。

* $z_{uu} = \frac{\partial}{\partial u}(z_u) = \frac{\partial}{\partial u}(z_x + 2z_y)$

$= \frac{\partial z_x}{\partial u} + 2\frac{\partial z_y}{\partial u}$

$= (\frac{\partial z_x}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z_x}{\partial y}\frac{\partial y}{\partial u}) + 2(\frac{\partial z_y}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z_y}{\partial y}\frac{\partial y}{\partial u})$

$= (z_{xx} \cdot 1 + z_{xy} \cdot 2) + 2(z_{yx} \cdot 1 + z_{yy} \cdot 2)$

$= z_{xx} + 2z_{xy} + 2z_{yx} + 4z_{yy}$

$= z_{xx} + 4z_{xy} + 4z_{yy}$ (なぜなら $z_{xy} = z_{yx}$)

* $z_{uv} = \frac{\partial}{\partial v}(z_u) = \frac{\partial}{\partial v}(z_x + 2z_y)$

$= \frac{\partial z_x}{\partial v} + 2\frac{\partial z_y}{\partial v}$

$= (\frac{\partial z_x}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z_x}{\partial y}\frac{\partial y}{\partial v}) + 2(\frac{\partial z_y}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z_y}{\partial y}\frac{\partial y}{\partial v})$

$= (z_{xx} \cdot 3 + z_{xy} \cdot 1) + 2(z_{yx} \cdot 3 + z_{yy} \cdot 1)$

$= 3z_{xx} + z_{xy} + 6z_{yx} + 2z_{yy}$

$= 3z_{xx} + 7z_{xy} + 2z_{yy}$

* $z_{vv} = \frac{\partial}{\partial v}(z_v) = \frac{\partial}{\partial v}(3z_x + z_y)$

$= 3\frac{\partial z_x}{\partial v} + \frac{\partial z_y}{\partial v}$

$= 3(\frac{\partial z_x}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z_x}{\partial y}\frac{\partial y}{\partial v}) + (\frac{\partial z_y}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial z_y}{\partial y}\frac{\partial y}{\partial v})$

$= 3(z_{xx} \cdot 3 + z_{xy} \cdot 1) + (z_{yx} \cdot 3 + z_{yy} \cdot 1)$

$= 9z_{xx} + 3z_{xy} + 3z_{yx} + z_{yy}$

$= 9z_{xx} + 6z_{xy} + z_{yy}$

3. 最終的な答え

問題1:

$z_u = z_x + 2z_y$

$z_v = 3z_x + z_y$

問題2:

$z_{uu} = z_{xx} + 4z_{xy} + 4z_{yy}$

$z_{uv} = 3z_{xx} + 7z_{xy} + 2z_{yy}$

$z_{vv} = 9z_{xx} + 6z_{xy} + z_{yy}$