$u = u(x, y)$ で、$x = r \cos \theta$, $y = r \sin \theta$ とする。 (1) $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ を $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$ で表す。 (2) $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$ を $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ で表す。 (3) $(\frac{\partial u}{\partial x})^2 + (\frac{\partial u}{\partial y})^2$ と $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ を $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ で表す。

1. 問題の内容

$u = u(x, y)$ で、$x = r \cos \theta$, $y = r \sin \theta$ とする。

(1) $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ を $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$ で表す。

(2) $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$ を $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ で表す。

(3) $(\frac{\partial u}{\partial x})^2 + (\frac{\partial u}{\partial y})^2$ と $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$ を $\frac{\partial u}{\partial r}$, $\frac{\partial u}{\partial \theta}$ で表す。

2. 解き方の手順

(1) 連鎖律より、

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

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

(2) (1) より、

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

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

これらを $\frac{\partial u}{\partial x}$ と $\frac{\partial u}{\partial y}$ について解く。

$\cos \theta \frac{\partial u}{\partial x} = \frac{\partial u}{\partial r} - \sin \theta \frac{\partial u}{\partial y}$

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

$\frac{\partial u}{\partial x} = \cos \theta \frac{\partial u}{\partial r} - \frac{\sin \theta}{r} \frac{\partial u}{\partial \theta}$

$\sin \theta \frac{\partial u}{\partial y} = \frac{\partial u}{\partial r} - \cos \theta \frac{\partial u}{\partial x}$

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

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

(3)

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

$= \cos^2 \theta (\frac{\partial u}{\partial r})^2 - 2 \cos \theta \sin \theta \frac{1}{r} \frac{\partial u}{\partial r} \frac{\partial u}{\partial \theta} + \frac{\sin^2 \theta}{r^2} (\frac{\partial u}{\partial \theta})^2 + \sin^2 \theta (\frac{\partial u}{\partial r})^2 + 2 \sin \theta \cos \theta \frac{1}{r} \frac{\partial u}{\partial r} \frac{\partial u}{\partial \theta} + \frac{\cos^2 \theta}{r^2} (\frac{\partial u}{\partial \theta})^2$

$= (\cos^2 \theta + \sin^2 \theta) (\frac{\partial u}{\partial r})^2 + (\frac{\sin^2 \theta}{r^2} + \frac{\cos^2 \theta}{r^2}) (\frac{\partial u}{\partial \theta})^2$

$= (\frac{\partial u}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial u}{\partial \theta})^2$

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}$

ただし、この導出は複雑であり、ここでは省略する。

3. 最終的な答え

(1)

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

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

(2)

$\frac{\partial u}{\partial x} = \cos \theta \frac{\partial u}{\partial r} - \frac{\sin \theta}{r} \frac{\partial u}{\partial \theta}$

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

(3)

$(\frac{\partial u}{\partial x})^2 + (\frac{\partial u}{\partial y})^2 = (\frac{\partial u}{\partial r})^2 + \frac{1}{r^2} (\frac{\partial u}{\partial \theta})^2$

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}$