第4章 問題1 解答

渦度と速度の関係は式(1.11)から \begin{equation} \xi =\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z},\quad \eta =\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x},\quad \zeta=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \end{equation} 与式に代入すれば， \begin{eqnarray} &&\frac{\partial \xi}{\partial x}+\frac{\partial \eta}{\partial y} +\frac{\partial \zeta}{\partial z}\nonumber\\ &=&\frac{\partial }{\partial x} \left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)+ \frac{\partial }{\partial y} \left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right)+ \frac{\partial }{\partial z} \left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right) \label{eq:solenoid} \end{eqnarray} $\, u\, $，$\, v\, $，$\, w\, $が微分可能であれば，微分の順序が可能であるから \begin{equation} \frac{\partial^2 w}{\partial x\partial y} =\frac{\partial^2 w}{\partial y\partial x},\quad \frac{\partial^2 u}{\partial y\partial z} =\frac{\partial^2 u}{\partial z\partial y},\quad \frac{\partial^2 v}{\partial z\partial x} =\frac{\partial^2 v}{\partial x\partial z}\label{eq:cross} \end{equation} それゆえ，式\eqref{eq:solenoid}，\eqref{eq:cross}より， \begin{equation} \frac{\partial \xi}{\partial x}+\frac{\partial \eta}{\partial y} +\frac{\partial \zeta}{\partial z}=0 \end{equation}