Starting from Euler's equation for the motion of an inviscid fluid, derive the vorticity equation in the form

$$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \boldsymbol{u}$$

Deduce that an initially irrotational flow remains irrotational.

Consider a plane flow that at time $t=0$ is described by the streamfunction

$$\psi=x^{2}+y^{2} .$$

Calculate the vorticity everywhere at times $t>0$.