$z = (r\cos\theta)^2 - 2(r\cos\theta)(r\sin\theta) + (r\sin\theta)^2$ $z = r^2\cos^2\theta - 2r^2\cos\theta\sin\theta + r^2\sin^2\theta$ $z = r^2(\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)$ $z = r^2(1 - 2\cos\theta\sin\theta)$ $z = r^2(1 - \sin 2\theta)$

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1. 問題の内容

3. $z = x^2 - 2xy + y^2$ かつ $x = r\cos\theta$, $y = r\sin\theta$ のとき、$\frac{\partial z}{\partial r}$ と $\frac{\partial z}{\partial \theta}$ を求めよ。

4. $f(x, y) = \log\sqrt{x^2 + y^2}$ について、$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$ を求めよ。

5. $z = f(x, y)$ かつ $x = r\cos\theta$, $y = r\sin\theta$ のとき、次の等式が成り立つことを証明せよ。

$\qquad \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = \left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2$

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2. 解き方の手順

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3.

1. $z$ を $x$ と $y$ で表された式に $x = r\cos\theta$ と $y = r\sin\theta$ を代入して、$z$ を $r$ と $\theta$ の関数として表す。

$z = (r\cos\theta)^2 - 2(r\cos\theta)(r\sin\theta) + (r\sin\theta)^2$

$z = r^2\cos^2\theta - 2r^2\cos\theta\sin\theta + r^2\sin^2\theta$

$z = r^2(\cos^2\theta - 2\cos\theta\sin\theta + \sin^2\theta)$

$z = r^2(1 - 2\cos\theta\sin\theta)$

$z = r^2(1 - \sin 2\theta)$

2. $\frac{\partial z}{\partial r}$ を計算する。

$\frac{\partial z}{\partial r} = \frac{\partial}{\partial r} [r^2(1 - \sin 2\theta)]$

$\frac{\partial z}{\partial r} = 2r(1 - \sin 2\theta)$

3. $\frac{\partial z}{\partial \theta}$ を計算する。

$\frac{\partial z}{\partial \theta} = \frac{\partial}{\partial \theta} [r^2(1 - \sin 2\theta)]$

$\frac{\partial z}{\partial \theta} = r^2(-2\cos 2\theta)$

$\frac{\partial z}{\partial \theta} = -2r^2\cos 2\theta$

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4.

1. $f(x, y) = \log\sqrt{x^2 + y^2}$ を $f(x, y) = \frac{1}{2}\log(x^2 + y^2)$ と書き換える。

2. $\frac{\partial f}{\partial x}$ を計算する。

$\frac{\partial f}{\partial x} = \frac{1}{2}\cdot\frac{1}{x^2 + y^2}\cdot 2x = \frac{x}{x^2 + y^2}$

3. $\frac{\partial^2 f}{\partial x^2}$ を計算する。

$\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \left[\frac{x}{x^2 + y^2}\right] = \frac{(x^2 + y^2) - x(2x)}{(x^2 + y^2)^2} = \frac{y^2 - x^2}{(x^2 + y^2)^2}$

4. $\frac{\partial f}{\partial y}$ を計算する。

$\frac{\partial f}{\partial y} = \frac{1}{2}\cdot\frac{1}{x^2 + y^2}\cdot 2y = \frac{y}{x^2 + y^2}$

5. $\frac{\partial^2 f}{\partial y^2}$ を計算する。

$\frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} \left[\frac{y}{x^2 + y^2}\right] = \frac{(x^2 + y^2) - y(2y)}{(x^2 + y^2)^2} = \frac{x^2 - y^2}{(x^2 + y^2)^2}$

6. $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$ を計算する。

$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{y^2 - x^2}{(x^2 + y^2)^2} + \frac{x^2 - y^2}{(x^2 + y^2)^2} = 0$

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5.

1. 連鎖律 (chain rule) を用いる。

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

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

2. $\frac{\partial x}{\partial r} = \cos\theta$, $\frac{\partial y}{\partial r} = \sin\theta$, $\frac{\partial x}{\partial \theta} = -r\sin\theta$, $\frac{\partial y}{\partial \theta} = r\cos\theta$ である。

3. 上の結果を代入する。

$\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x}\cos\theta + \frac{\partial z}{\partial y}\sin\theta$

$\frac{\partial z}{\partial \theta} = \frac{\partial z}{\partial x}(-r\sin\theta) + \frac{\partial z}{\partial y}(r\cos\theta)$

4. $\left(\frac{\partial z}{\partial r}\right)^2$ を計算する。

$\left(\frac{\partial z}{\partial r}\right)^2 = \left(\frac{\partial z}{\partial x}\cos\theta + \frac{\partial z}{\partial y}\sin\theta\right)^2 = \left(\frac{\partial z}{\partial x}\right)^2\cos^2\theta + 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\cos\theta\sin\theta + \left(\frac{\partial z}{\partial y}\right)^2\sin^2\theta$

5. $\left(\frac{\partial z}{\partial \theta}\right)^2$ を計算する。

$\left(\frac{\partial z}{\partial \theta}\right)^2 = \left(\frac{\partial z}{\partial x}(-r\sin\theta) + \frac{\partial z}{\partial y}(r\cos\theta)\right)^2 = \left(\frac{\partial z}{\partial x}\right)^2r^2\sin^2\theta - 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}r^2\sin\theta\cos\theta + \left(\frac{\partial z}{\partial y}\right)^2r^2\cos^2\theta$

6. $\left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2$ を計算する。

$\left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2 = \left(\frac{\partial z}{\partial x}\right)^2\cos^2\theta + 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\cos\theta\sin\theta + \left(\frac{\partial z}{\partial y}\right)^2\sin^2\theta + \frac{1}{r^2}\left[\left(\frac{\partial z}{\partial x}\right)^2r^2\sin^2\theta - 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}r^2\sin\theta\cos\theta + \left(\frac{\partial z}{\partial y}\right)^2r^2\cos^2\theta\right]$

$= \left(\frac{\partial z}{\partial x}\right)^2\cos^2\theta + 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\cos\theta\sin\theta + \left(\frac{\partial z}{\partial y}\right)^2\sin^2\theta + \left(\frac{\partial z}{\partial x}\right)^2\sin^2\theta - 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\sin\theta\cos\theta + \left(\frac{\partial z}{\partial y}\right)^2\cos^2\theta$

$= \left(\frac{\partial z}{\partial x}\right)^2(\cos^2\theta + \sin^2\theta) + \left(\frac{\partial z}{\partial y}\right)^2(\sin^2\theta + \cos^2\theta)$

$= \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2$

よって、$\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = \left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2$ が成り立つ。

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3. 最終的な答え

3. $\frac{\partial z}{\partial r} = 2r(1 - \sin 2\theta)$、$\frac{\partial z}{\partial \theta} = -2r^2\cos 2\theta$

4. $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$

5. $\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = \left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2$ が成り立つ。