$x = \rho \cos \phi$, $y = \rho \sin \phi$ のとき、 $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \phi^2}$ を示せ。

1. 問題の内容

$x = \rho \cos \phi$, $y = \rho \sin \phi$ のとき、

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \phi^2}$

を示せ。

2. 解き方の手順

まず、偏微分を計算するために必要な関係式を求める。

$\frac{\partial \rho}{\partial x} = \frac{x}{\rho} = \cos \phi$

$\frac{\partial \rho}{\partial y} = \frac{y}{\rho} = \sin \phi$

$\frac{\partial \phi}{\partial x} = -\frac{y}{\rho^2} = -\frac{\sin \phi}{\rho}$

$\frac{\partial \phi}{\partial y} = \frac{x}{\rho^2} = \frac{\cos \phi}{\rho}$

連鎖律を用いて、$\frac{\partial u}{\partial x}$ と $\frac{\partial u}{\partial y}$ を計算する。

$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial \rho} \frac{\partial \rho}{\partial x} + \frac{\partial u}{\partial \phi} \frac{\partial \phi}{\partial x} = \frac{\partial u}{\partial \rho} \cos \phi - \frac{\partial u}{\partial \phi} \frac{\sin \phi}{\rho}$

$\frac{\partial u}{\partial y} = \frac{\partial u}{\partial \rho} \frac{\partial \rho}{\partial y} + \frac{\partial u}{\partial \phi} \frac{\partial \phi}{\partial y} = \frac{\partial u}{\partial \rho} \sin \phi + \frac{\partial u}{\partial \phi} \frac{\cos \phi}{\rho}$

次に、$\frac{\partial^2 u}{\partial x^2}$ と $\frac{\partial^2 u}{\partial y^2}$ を計算する。

$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} (\frac{\partial u}{\partial x}) = \frac{\partial}{\partial x} (\frac{\partial u}{\partial \rho} \cos \phi - \frac{\partial u}{\partial \phi} \frac{\sin \phi}{\rho})$

$= (\frac{\partial^2 u}{\partial \rho^2} \frac{\partial \rho}{\partial x} + \frac{\partial^2 u}{\partial \phi \partial \rho} \frac{\partial \phi}{\partial x}) \cos \phi + \frac{\partial u}{\partial \rho} (-\sin \phi) \frac{\partial \phi}{\partial x} - (\frac{\partial^2 u}{\partial \rho \partial \phi} \frac{\partial \rho}{\partial x} + \frac{\partial^2 u}{\partial \phi^2} \frac{\partial \phi}{\partial x}) \frac{\sin \phi}{\rho} - \frac{\partial u}{\partial \phi} (-\sin \phi) (-\frac{1}{\rho^2}) \frac{\partial \rho}{\partial x} - \frac{\partial u}{\partial \phi} \frac{\partial}{\partial x} (\frac{\sin \phi}{\rho})$

ここで、$\frac{\partial}{\partial x} (\frac{\sin \phi}{\rho}) = \frac{\cos \phi}{\rho} \frac{\partial \phi}{\partial x} - \frac{\sin \phi}{\rho^2} \frac{\partial \rho}{\partial x} = \frac{\cos \phi}{\rho} (-\frac{\sin \phi}{\rho}) - \frac{\sin \phi}{\rho^2} \cos \phi = -\frac{2\sin\phi\cos\phi}{\rho^2}$.

$\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial \rho^2} \cos^2 \phi - \frac{\partial^2 u}{\partial \phi \partial \rho} \frac{\sin \phi \cos \phi}{\rho} - \frac{\partial u}{\partial \rho} \frac{\sin^2 \phi}{\rho} - \frac{\partial^2 u}{\partial \rho \partial \phi} \frac{\sin \phi \cos \phi}{\rho} + \frac{\partial^2 u}{\partial \phi^2} \frac{\sin^2 \phi}{\rho^2} - \frac{\partial u}{\partial \phi} \frac{\sin \phi \cos \phi}{\rho^2} + \frac{\partial u}{\partial \phi} (\frac{2 \sin\phi\cos\phi}{\rho^2})$

同様に、

$\frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} \sin^2 \phi + \frac{\partial^2 u}{\partial \phi \partial \rho} \frac{\sin \phi \cos \phi}{\rho} + \frac{\partial u}{\partial \rho} \frac{\cos^2 \phi}{\rho} + \frac{\partial^2 u}{\partial \rho \partial \phi} \frac{\sin \phi \cos \phi}{\rho} + \frac{\partial^2 u}{\partial \phi^2} \frac{\cos^2 \phi}{\rho^2} - \frac{\partial u}{\partial \phi} \frac{\sin \phi \cos \phi}{\rho^2} - \frac{\partial u}{\partial \phi} (\frac{2 \sin\phi\cos\phi}{\rho^2})$

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} (\cos^2 \phi + \sin^2 \phi) + \frac{\partial u}{\partial \rho} \frac{\cos^2 \phi + \sin^2 \phi}{\rho} + \frac{\partial^2 u}{\partial \phi^2} \frac{\sin^2 \phi + \cos^2 \phi}{\rho^2} = \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \phi^2}$

3. 最終的な答え

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \phi^2}$