和達三樹《物理用數學》習題1.4.3，極座標下的Laplace算子(Laplacian)

==問題==

$x=\rho \cos \varphi, y=\rho \sin \varphi$，試證
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 u}{\partial \phi^2}.$$

==解答==

$u = u(x, y) = u(\rho \cos \varphi, \rho \sin \varphi)$。

\begin{align*}
\frac{\partial u}{\partial \rho} &= \frac{\partial u}{\partial x} \frac{\partial (\rho \cos \varphi)}{\partial \rho} + \frac{\partial u}{\partial y} \frac{\partial (\rho \sin \varphi)}{\partial \rho} \\
&= \frac{\partial u}{\partial x} \cos \varphi + \frac{\partial u}{\partial y} \sin \varphi.
\end{align*}
\begin{align*}
\frac{\partial^2 u}{\partial \rho^2} &= \frac{\partial}{\partial \rho} \left( \frac{\partial u}{\partial \rho} \right) \\
&= \frac{\partial}{\partial \rho} \left( \frac{\partial u}{\partial x} \cos \varphi + \frac{\partial u}{\partial y} \sin \varphi \right) \\
&= \frac{\partial}{\partial \rho} \left( \frac{\partial u}{\partial x} \cos \varphi  \right) + \frac{\partial}{\partial \rho} \left( \frac{\partial u}{\partial y} \sin \varphi \right) \\
&= \cos \varphi \frac{\partial}{\partial \rho} \frac{\partial u}{\partial x} + \sin \varphi \frac{\partial}{\partial \rho} \frac{\partial u}{\partial y} \\
&= \cos \varphi \left[ \frac{\partial}{\partial x} \frac{\partial u}{\partial x} \frac{\partial (\rho \cos \varphi)}{\partial \rho} \right] + \sin \varphi \left[ \frac{\partial}{\partial y} \frac{\partial u}{\partial x} \frac{\partial (\rho \sin \varphi)}{\partial \rho} \right] \\
&= \cos \varphi \left[ \frac{\partial^2 u}{\partial x^2} \cos \phi + \frac{\partial^2 u}{\partial y \partial x} \sin \varphi \right] + \sin \varphi \left[ \frac{\partial^2 u}{\partial x \partial y} \cos \varphi + \frac{\partial^2 u}{\partial y^2} \sin \varphi \right] \\
&= \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + \frac{\partial^2 u}{\partial y \partial x} \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial x \partial y} \sin \varphi \cos \varphi + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi \\
&= \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + \frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi \\
&= \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + 2\frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi  + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi.
\end{align*}
\begin{align*}
\frac{\partial u}{\partial \varphi} &= \frac{\partial u}{\partial x} \frac{\partial (\rho \cos \varphi)}{\partial \varphi} + \frac{\partial u}{\partial y} \frac{\partial (\rho \sin \varphi)}{\partial \varphi} \\
&= \frac{\partial u}{\partial x} (- \rho \sin \varphi) + \frac{\partial u}{\partial y} \rho \cos \varphi \\
&= - \rho \frac{\partial u}{\partial x} \sin \varphi + \rho \frac{\partial u}{\partial y} \cos \varphi.
\end{align*}
\begin{align*} \frac{\partial^2 u}{\partial \varphi^2} &= \frac{\partial}{\partial \varphi} \left( \frac{\partial u}{\partial \varphi} \right)  \\ &= \frac{\partial}{\partial \varphi} \left( - \rho \frac{\partial u}{\partial x} \sin \varphi + \rho \frac{\partial u}{\partial y} \cos \varphi \right)  \\ &= \frac{\partial}{\partial \varphi} \left( - \rho \frac{\partial u}{\partial x} \sin \varphi \right) + \frac{\partial}{\partial \varphi} \left( \rho \frac{\partial u}{\partial y} \cos \varphi \right)  \\ &= -\rho \frac{\partial}{\partial \varphi} \left( \frac{\partial u}{\partial x} \sin \varphi \right) + \rho \frac{\partial}{\partial \varphi} \left( \frac{\partial u}{\partial y} \cos \varphi \right)  \\ &= -\rho \left[ \left( \frac{\partial}{\partial \varphi} \frac{\partial u}{\partial x} \right) \sin \varphi + \frac{\partial u}{\partial x} \left( \frac{\partial}{\partial \varphi} \sin \varphi \right) \right] + \rho \left[ \left( \frac{\partial}{\partial \varphi} \frac{\partial u}{\partial y} \right) \cos \varphi + \frac{\partial u}{\partial y} \left( \frac{\partial}{\partial \varphi} \cos \varphi \right) \right] \\ &= -\rho \left\{ \left[ \frac{\partial}{\partial x} \frac{\partial u}{\partial x} \frac{\partial (\rho \cos \varphi)}{\partial \varphi} + \frac{\partial}{\partial y} \frac{\partial u}{\partial x} \frac{\partial (\rho \sin \varphi)}{\partial \varphi} \right] \sin \varphi + \frac{\partial u}{\partial x} \frac{\partial (\sin \varphi)}{\partial \varphi} \right\} \\ & \quad +\rho \left\{ \left[ \frac{\partial}{\partial x} \frac{\partial u}{\partial y} \frac{\partial (\rho \cos \varphi)}{\partial \varphi} + \frac{\partial}{\partial y} \frac{\partial u}{\partial y} \frac{\partial (\rho \sin \varphi)}{\partial \varphi} \right] \cos \varphi + \frac{\partial u}{\partial y} \frac{\partial (\cos \varphi)}{\partial \varphi} \right\} \\ &= -\rho \left\{ \left[ \frac{\partial^2 u}{\partial x^2} (- \rho \sin \varphi) + \frac{\partial^2 u}{\partial y \partial x} \rho \cos \varphi \right] \sin \varphi + \frac{\partial u}{\partial x} \cos \varphi \right\} \\ & \quad + \rho \left\{ \left[ \frac{\partial^2 u}{\partial x \partial y} (- \rho \sin \varphi) + \frac{\partial^2 u}{\partial y^2} \rho \cos \varphi \right] \cos \varphi + \frac{\partial u}{\partial y} (- \sin \varphi) \right\} \\ &= \frac{\partial^2 u}{\partial x^2} \rho^2 \sin^2 \varphi - \frac{\partial^2 u}{\partial y \partial x} \rho^2 \cos \varphi \sin \varphi - \frac{\partial u}{\partial x} \rho \cos \varphi - \frac{\partial^2 u}{\partial x \partial y} \rho^2 \sin \varphi \cos \varphi + \frac{\partial^2 u}{\partial y^2} \rho^2 \cos^2 \varphi - \frac{\partial u}{\partial y} \rho \sin \varphi \\ &= \frac{\partial^2 u}{\partial x^2} \rho^2 \sin^2 \varphi - \frac{\partial^2 u}{\partial x \partial y} \rho^2 \cos \varphi \sin \varphi - \frac{\partial u}{\partial x} \rho \cos \varphi - \frac{\partial^2 u}{\partial x \partial y} \rho^2 \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial y^2} \rho^2 \cos^2 \varphi - \frac{\partial u}{\partial y} \rho \sin \varphi \\ &= \frac{\partial^2 u}{\partial x^2} \rho^2 \sin^2 \varphi - 2 \frac{\partial^2 u}{\partial x \partial y} \rho^2 \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial y^2} \rho^2 \cos^2 \varphi - \frac{\partial u}{\partial x} \rho \cos \varphi - \frac{\partial u}{\partial y} \rho \sin \varphi.
\end{align*}
\begin{align*} \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} &= \left( \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + 2\frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi  + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi \right) \\ & \quad +\frac{1}{\rho} \left( \frac{\partial u}{\partial x} \cos \varphi + \frac{\partial u}{\partial y} \sin \varphi \right) \\ & \qquad +\frac{1}{\rho^2} \left( \frac{\partial^2 u}{\partial x^2} \rho^2 \sin^2 \varphi - 2 \frac{\partial^2 u}{\partial x \partial y} \rho^2 \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial y^2} \rho^2 \cos^2 \varphi \right. \\ &\qquad \qquad \qquad \left. - \frac{\partial u}{\partial x} \rho \cos \varphi - \frac{\partial u}{\partial y} \rho \sin \varphi \right) \\ &= \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + 2\frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi  + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi + \frac{\partial u}{\partial x} \frac{1}{\rho} \cos \varphi + \frac{\partial u}{\partial y} \frac{1}{\rho} \sin \varphi + \frac{\partial^2 u}{\partial x^2}  \sin^2 \varphi \\ & \quad - 2 \frac{\partial^2 u}{\partial x \partial y} \cos \varphi \sin \varphi + \frac{\partial^2 u}{\partial y^2} \cos^2 \varphi - \frac{\partial u}{\partial x} \frac{1}{\rho} \cos \varphi - \frac{\partial u}{\partial y} \frac{1}{\rho} \sin \varphi \\ &= \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi + \frac{\partial^2 u}{\partial x^2}  \sin^2 \varphi + \frac{\partial^2 u}{\partial y^2} \cos^2 \varphi \\
&= \left( \frac{\partial^2 u}{\partial x^2} \cos^2 \varphi + \frac{\partial^2 u}{\partial x^2}  \sin^2 \varphi \right) + \left( \frac{\partial^2 u}{\partial y^2} \sin^2 \varphi + \frac{\partial^2 u}{\partial y^2} \cos^2 \varphi \right)  \\
&= \frac{\partial^2 u}{\partial x^2} \left( \cos^2 \varphi + \sin^2 \varphi \right) + \frac{\partial^2 u}{\partial y^2} \left( \sin^2 \varphi + \cos^2 \varphi \right)  \\
&= \frac{\partial^2 u}{\partial x^2} \cdot 1 + \frac{\partial^2 u}{\partial y^2} \cdot 1 \\ &= \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}.
\end{align*}

(解答結束)