Use the Euler equation for the motion of an inviscid fluid to derive the vorticity equation in the form

$$D \boldsymbol{\omega} / D t=\boldsymbol{\omega} \cdot \nabla \mathbf{u} .$$

Give a physical interpretation of the terms in this equation and deduce that irrotational flows remain irrotational.

In a plane flow the vorticity at time $t=0$ has the uniform value $\boldsymbol{\omega}_{0} \neq \mathbf{0}$. Find the vorticity everywhere at times $t>0$.