$z = f(x, y)$, $x=\varphi(u, v)$, $y=\psi(u, v)$ がすべて $C^2$ 級のとき, \begin{align*} \frac{\partial^2z}{\partial u^2} &= \frac{\partial^2 z}{\partial x^2}\left( \frac{\partial x}{\partial u} \right)^2 + \frac{\partial^2 z}{\partial y^2}\left( \frac{\partial y}{\partial u} \right)^2 + 2\frac{\partial^2 z}{\partial x\partial y}\frac{\partial x}{\partial u}\frac{\partial y}{\partial u} + \frac{\partial z}{\partial x}\frac{\partial^2 x}{\partial u^2} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial u^2}, \\ \frac{\partial^2z}{\partial u\partial v} &= \frac{\partial^2 z}{\partial x^2}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v} + \frac{\partial^2 z}{\partial y^2}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v} + \frac{\partial^2 z}{\partial x\partial y}\left( \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} + \frac{\partial x}{\partial v}\frac{\partial y}{\partial u} \right) + \frac{\partial z}{\partial x}\frac{\partial^2 x}{\partial u\partial v} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial u\partial v} \end{align*} が成り立つことを証明せよ.
解答例 1
合成関数の偏微分の公式 \begin{align*} \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}, \\ \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} \end{align*} の両辺をさらに $u$, $v$ で偏微分すると, \begin{align} \frac{\partial^2z}{\partial u^2} &= \left(\frac{\partial}{\partial u}\frac{\partial z}{\partial x}\right)\frac{\partial x}{\partial u} + \frac{\partial z}{\partial x}\frac{\partial^2x}{\partial u^2} + \left(\frac{\partial}{\partial u}\frac{\partial z}{\partial y}\right)\frac{\partial y}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial^2y}{\partial u^2}, \tag{1} \\ \frac{\partial^2z}{\partial u\partial v} &= \frac{\partial}{\partial u}\frac{\partial z}{\partial v} = \left(\frac{\partial}{\partial u}\frac{\partial z}{\partial x}\right)\frac{\partial x}{\partial v} + \frac{\partial z}{\partial x}\frac{\partial^2x}{\partial u\partial v} + \left(\frac{\partial}{\partial u}\frac{\partial z}{\partial y}\right)\frac{\partial y}{\partial v} + \frac{\partial z}{\partial y}\frac{\partial^2y}{\partial u\partial v}. \tag{2} \end{align} 一方, 合成関数の偏微分の公式より, \begin{align*} \frac{\partial}{\partial u}\frac{\partial z}{\partial x} &= \frac{\partial^2 z}{\partial x^2}\frac{\partial x}{\partial u} + \frac{\partial^2 z}{\partial x\partial y}\frac{\partial y}{\partial u}, \\ \frac{\partial}{\partial u}\frac{\partial z}{\partial y} &= \frac{\partial^2 z}{\partial x\partial y}\frac{\partial x}{\partial u} + \frac{\partial^2 z}{\partial y^2}\frac{\partial y}{\partial u}. \end{align*} これらを (1), (2) に代入すれば, 求める等式が得られる.
最終更新日：2011年11月02日