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[q201109050700] $U$ を $\mathbb{R}^2$ の開集合とし, $f(x, y)$ を $U$ 上で $C^1$ 級の実数値関数とする. また, $(a, b)$, $(a+s, b+t)\in U$ とし, その $2$ 点を結ぶ線分が $U$ に含まれているとする. このとき, ある実数 $\theta$ ($0<\theta<1$) が存在して, $$ f(a+s, b+t) = f(a, b) + sf_x(a+\theta s, b+\theta t) + tf_y(a+\theta s, b+\theta t) $$ が成り立つことを証明せよ.
Keywords: 平均値の定理
[q201109050800] $z = f(x, y)$, $x=\varphi(t)$, $y=\psi(t)$ がすべて $C^2$ 級のとき, $$ \frac{d^2z}{dt^2} = \frac{\partial^2 z}{\partial x^2}\left( \frac{dx}{dt} \right)^2 + \frac{\partial^2 z}{\partial y^2}\left( \frac{dy}{dt} \right)^2 + 2\frac{\partial^2 z}{\partial x\partial y}\frac{dx}{dt}\frac{dy}{dt} + \frac{\partial z}{\partial x}\frac{d^2x}{dt^2} + \frac{\partial z}{\partial y}\frac{d^2y}{dt^2} $$ が成り立つことを証明せよ.
[q201109050900] $z = f(x, y)$, $x=\varphi(u, v)$, $y=\psi(u, v)$ 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.
[q201109051000] $z=f(x, y)$, $x=r\cos\theta$, $y=r\sin\theta$ のとき, $$ \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2 = \left( \frac{\partial z}{\partial r} \right)^2 + \frac{1}{r^2}\left( \frac{\partial z}{\partial \theta} \right)^2 $$ が成り立つことを証明せよ. ただし, $f(x, y)$ は全微分可能な関数とする.
[q201109051100] $z=f(x, y)$, $x=r\cos\theta$, $y=r\sin\theta$ のとき, $$ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = \frac{\partial^2 z}{\partial r^2} + \frac{1}{r}\frac{\partial z}{\partial r} + \frac{1}{r^2}\frac{\partial^2 z}{\partial \theta^2} $$ が成り立つことを証明せよ. ただし, $f(x, y)$ は $C^2$ 級関数とする.