$v(x, y)$ は $R^2$ 上の $C^2$ 級関数とする。$\theta \in [0, 2\pi)$ を定数として、変数変換 $(x, y) \mapsto (X, Y)$ を $X = x \cos\theta - y \sin\theta, Y = x \sin\theta + y \cos\theta$ によって与える。この変換を通じて $v$ を $(X, Y)$ の関数とみなしたとき、 (1) $\frac{\partial v}{\partial X}, \frac{\partial v}{\partial Y}$ を $\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$ を用いて表せ。 (2) $(\frac{\partial v}{\partial X})^2 + (\frac{\partial v}{\partial Y})^2$ および $\frac{\partial^2 v}{\partial X^2} + \frac{\partial^2 v}{\partial Y^2}$ を $\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}, \frac{\partial^2 v}{\partial x^2}, \frac{\partial^2 v}{\partial x\partial y}, \frac{\partial^2 v}{\partial y^2}$ 等によって表せ。

1. 問題の内容

$v(x, y)$ は $R^2$ 上の $C^2$ 級関数とする。$\theta \in [0, 2\pi)$ を定数として、変数変換 $(x, y) \mapsto (X, Y)$ を

$X = x \cos\theta - y \sin\theta, Y = x \sin\theta + y \cos\theta$

によって与える。この変換を通じて $v$ を $(X, Y)$ の関数とみなしたとき、

(1) $\frac{\partial v}{\partial X}, \frac{\partial v}{\partial Y}$ を $\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$ を用いて表せ。

(2) $(\frac{\partial v}{\partial X})^2 + (\frac{\partial v}{\partial Y})^2$ および $\frac{\partial^2 v}{\partial X^2} + \frac{\partial^2 v}{\partial Y^2}$ を $\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}, \frac{\partial^2 v}{\partial x^2}, \frac{\partial^2 v}{\partial x\partial y}, \frac{\partial^2 v}{\partial y^2}$ 等によって表せ。

2. 解き方の手順

(1) 連鎖律を用いて $\frac{\partial v}{\partial X}, \frac{\partial v}{\partial Y}$ を $\frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$ で表す。

$\frac{\partial v}{\partial X} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial X} + \frac{\partial v}{\partial y} \frac{\partial y}{\partial X}$

$\frac{\partial v}{\partial Y} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial Y} + \frac{\partial v}{\partial y} \frac{\partial y}{\partial Y}$

$X = x \cos\theta - y \sin\theta, Y = x \sin\theta + y \cos\theta$ より、

$x = X \cos\theta + Y \sin\theta, y = -X \sin\theta + Y \cos\theta$

よって、

$\frac{\partial x}{\partial X} = \cos\theta, \frac{\partial y}{\partial X} = -\sin\theta, \frac{\partial x}{\partial Y} = \sin\theta, \frac{\partial y}{\partial Y} = \cos\theta$

$\frac{\partial v}{\partial X} = \frac{\partial v}{\partial x} \cos\theta - \frac{\partial v}{\partial y} \sin\theta$

$\frac{\partial v}{\partial Y} = \frac{\partial v}{\partial x} \sin\theta + \frac{\partial v}{\partial y} \cos\theta$

(2) $(\frac{\partial v}{\partial X})^2 + (\frac{\partial v}{\partial Y})^2$ を計算する。

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

$(\frac{\partial v}{\partial Y})^2 = (\frac{\partial v}{\partial x})^2 \sin^2\theta + 2 \frac{\partial v}{\partial x} \frac{\partial v}{\partial y} \sin\theta \cos\theta + (\frac{\partial v}{\partial y})^2 \cos^2\theta$

$(\frac{\partial v}{\partial X})^2 + (\frac{\partial v}{\partial Y})^2 = (\frac{\partial v}{\partial x})^2 (\cos^2\theta + \sin^2\theta) + (\frac{\partial v}{\partial y})^2 (\sin^2\theta + \cos^2\theta) = (\frac{\partial v}{\partial x})^2 + (\frac{\partial v}{\partial y})^2$

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

$\frac{\partial^2 v}{\partial Y^2} = \frac{\partial}{\partial Y} (\frac{\partial v}{\partial Y}) = \frac{\partial}{\partial Y} (\frac{\partial v}{\partial x} \sin\theta + \frac{\partial v}{\partial y} \cos\theta) = (\frac{\partial^2 v}{\partial x^2} \frac{\partial x}{\partial Y} + \frac{\partial^2 v}{\partial x \partial y} \frac{\partial y}{\partial Y}) \sin\theta + (\frac{\partial^2 v}{\partial y \partial x} \frac{\partial x}{\partial Y} + \frac{\partial^2 v}{\partial y^2} \frac{\partial y}{\partial Y}) \cos\theta = (\frac{\partial^2 v}{\partial x^2} \sin\theta + \frac{\partial^2 v}{\partial x \partial y} \cos\theta) \sin\theta + (\frac{\partial^2 v}{\partial y \partial x} \sin\theta + \frac{\partial^2 v}{\partial y^2} \cos\theta) \cos\theta = \frac{\partial^2 v}{\partial x^2} \sin^2\theta + 2\frac{\partial^2 v}{\partial x \partial y} \sin\theta \cos\theta + \frac{\partial^2 v}{\partial y^2} \cos^2\theta$

$\frac{\partial^2 v}{\partial X^2} + \frac{\partial^2 v}{\partial Y^2} = \frac{\partial^2 v}{\partial x^2} (\cos^2\theta + \sin^2\theta) + \frac{\partial^2 v}{\partial y^2} (\sin^2\theta + \cos^2\theta) = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}$

3. 最終的な答え

(1)

$\frac{\partial v}{\partial X} = \frac{\partial v}{\partial x} \cos\theta - \frac{\partial v}{\partial y} \sin\theta$

$\frac{\partial v}{\partial Y} = \frac{\partial v}{\partial x} \sin\theta + \frac{\partial v}{\partial y} \cos\theta$

(2)

$(\frac{\partial v}{\partial X})^2 + (\frac{\partial v}{\partial Y})^2 = (\frac{\partial v}{\partial x})^2 + (\frac{\partial v}{\partial y})^2$

$\frac{\partial^2 v}{\partial X^2} + \frac{\partial^2 v}{\partial Y^2} = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}$