解答例 1

\begin{alignat*}{2} \frac{\partial x}{\partial r} &= \cos\theta,\quad &\frac{\partial x}{\partial \theta} &= -r\sin\theta, \\ \frac{\partial y}{\partial r} &= \sin\theta,\quad &\frac{\partial y}{\partial \theta} &= r\cos\theta \end{alignat*} より, \begin{align*} \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 x}\cos\theta + \frac{\partial z}{\partial y}\sin\theta, \\ \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} = \frac{\partial z}{\partial x}(-r\sin\theta) + \frac{\partial z}{\partial y}(r\cos\theta). \end{align*} さらに, \begin{alignat*}{2} \frac{\partial^2 x}{\partial r^2} &= 0,\quad &\frac{\partial^2 x}{\partial \theta^2} &= -r\cos\theta, \\ \frac{\partial^2 y}{\partial r^2} &= 0,\quad &\frac{\partial^2 y}{\partial \theta^2} &= -r\sin\theta \end{alignat*} より, \begin{align*} \frac{\partial^2 z}{\partial r^2} &= \frac{\partial^2 z}{\partial x^2}\left(\frac{\partial x}{\partial r}\right)^2 + \frac{\partial^2 z}{\partial y^2}\left(\frac{\partial y}{\partial r}\right)^2 \\ &\qquad + 2\frac{\partial^2 z}{\partial x\partial y}\frac{\partial x}{\partial r}\frac{\partial y}{\partial r} + \frac{\partial z}{\partial x}\frac{\partial x^2}{\partial r^2} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial r^2} \\ &= \frac{\partial^2 z}{\partial x^2}\cos^2\theta + \frac{\partial^2 z}{\partial y^2}\sin^2\theta + 2\frac{\partial^2 z}{\partial x\partial y}\sin\theta\cos\theta, \\ \frac{\partial^2 z}{\partial \theta^2} &= \frac{\partial^2 z}{\partial x^2}\left(\frac{\partial x}{\partial \theta}\right)^2 + \frac{\partial^2 z}{\partial y^2}\left(\frac{\partial y}{\partial \theta}\right)^2 \\ &\qquad + 2\frac{\partial^2 z}{\partial x\partial y}\frac{\partial x}{\partial \theta}\frac{\partial y}{\partial \theta} + \frac{\partial z}{\partial x}\frac{\partial x^2}{\partial \theta^2} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial \theta^2} \\ &= \frac{\partial^2 z}{\partial x^2}\cdot r^2\sin^2\theta + \frac{\partial^2 z}{\partial y^2}\cdot r^2\cos^2\theta - 2\frac{\partial^2 z}{\partial x\partial y}\cdot r^2\sin\theta\cos\theta \\ &\qquad + \left(\frac{\partial z}{\partial x}(-r\cos\theta) + \frac{\partial z}{\partial y}(-r\sin\theta) \right). \end{align*} よって, \begin{align*} &\frac{\partial^2 z}{\partial r^2} + \frac{1}{r}\frac{\partial z}{\partial r} + \frac{1}{r^2}\frac{\partial^2 z}{\partial \theta^2} \\ & = \frac{\partial^2 z}{\partial x^2}(\cos^2\theta+\sin^2\theta) + \frac{\partial^2 z}{\partial y^2}(\cos^2\theta+\sin^2\theta) \\ & = \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} \end{align*} となる.

最終更新日：2011年11月02日