试证明: 当流场为无旋, 即 $\rot{\bf u}={\bf 0}$ 时, 理想流体的 Euler 方程可写为如下形式: $$\bex \cfrac{\p {\bf u}}{\p t}+\n \cfrac{u^2}{2}+\cfrac{1}{\rho}\n p={\bf F}. \eex$$

 

证明: 仅须注意到 $$\bex ({\bf u}\cdot\n){\bf u}=\Div({\bf u}\otimes{\bf u})= (\Div{\bf u}){\bf u}+\rot{\bf u}\times {\bf u}+\cfrac{1}{2}\n u^2=\n\cfrac{u^2}{2}. \eex$$