$z = f(x, y)$, $x = u + v$, $y = uv$ とする。 1. $z_u = z_x + [1] z_y$, $z_v = z_x + [2] z_y$. ただし、$[1]$, $[2]$ は次の選択肢から選ぶ。 (1) $x$ (2) $y$ (3) $u$ (4) $v$. 括弧()はつけずに数字のみ解答。

1. 問題の内容

$z = f(x, y)$, $x = u + v$, $y = uv$ とする。

1. $z_u = z_x + [1] z_y$, $z_v = z_x + [2] z_y$. ただし、$[1]$, $[2]$ は次の選択肢から選ぶ。

(1) $x$ (2) $y$ (3) $u$ (4) $v$. 括弧()はつけずに数字のみ解答。

2. $z_{uu} - 2z_{uv} + z_{vv} = (x^2 - [3]) z_{yy} - [4] z_y$.

2. 解き方の手順

1. まず、$z_u$ と $z_v$ を求める。

$z_u = \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u} = z_x \cdot 1 + z_y \cdot v = z_x + vz_y$

$z_v = \frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v} = z_x \cdot 1 + z_y \cdot u = z_x + uz_y$

よって、$[1] = v$ と $[2] = u$ であるから、答えはそれぞれ (4) と (3) 。

2. 次に、$z_{uu}$, $z_{uv}$, $z_{vv}$ を求める。

$z_u = z_x + vz_y$ より、

$z_{uu} = \frac{\partial}{\partial u} (z_x + vz_y) = \frac{\partial z_x}{\partial u} + v \frac{\partial z_y}{\partial u} = \frac{\partial z_x}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z_x}{\partial y} \frac{\partial y}{\partial u} + v (\frac{\partial z_y}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z_y}{\partial y} \frac{\partial y}{\partial u}) = z_{xx} + vz_{xy} + v(z_{yx} + vz_{yy}) = z_{xx} + 2vz_{xy} + v^2 z_{yy}$

$z_v = z_x + uz_y$ より、

$z_{vv} = \frac{\partial}{\partial v} (z_x + uz_y) = \frac{\partial z_x}{\partial v} + u \frac{\partial z_y}{\partial v} = \frac{\partial z_x}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z_x}{\partial y} \frac{\partial y}{\partial v} + u (\frac{\partial z_y}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z_y}{\partial y} \frac{\partial y}{\partial v}) = z_{xx} + uz_{xy} + u(z_{yx} + uz_{yy}) = z_{xx} + 2uz_{xy} + u^2 z_{yy}$

$z_{uv} = \frac{\partial}{\partial v} (z_x + vz_y) = \frac{\partial z_x}{\partial v} + z_y + v \frac{\partial z_y}{\partial v} = z_{xx} + u z_{xy} + z_y + v(z_{xy} + uz_{yy}) = z_{xx} + (u+v)z_{xy} + z_y + uv z_{yy} = z_{xx} + (u+v)z_{xy} + z_y + yz_{yy}$

したがって、

$z_{uu} - 2z_{uv} + z_{vv} = (z_{xx} + 2vz_{xy} + v^2 z_{yy}) - 2(z_{xx} + (u+v)z_{xy} + z_y + yz_{yy}) + (z_{xx} + 2uz_{xy} + u^2 z_{yy}) = (v^2 - 2y + u^2) z_{yy} - 2 z_y = (u^2 + v^2 - 2uv) z_{yy} - 2 z_y = ((u+v)^2 - 4uv) z_{yy} - 2 z_y = (x^2 - 4y) z_{yy} - 2 z_y$

よって、$[3] = 4$ と $[4] = 2$。

3. 最終的な答え

1. $[1] = 4$, $[2] = 3$

2. $[3] = 4$, $[4] = 2$