Starting from the Euler equation, derive the vorticity equation for the motion of an inviscid incompressible fluid under a conservative body force, and give a physical interpretation of each term in the equation. Deduce that in a flow field of the form $\mathbf{u}=(u(x, y, t), v(x, y, t), 0)$ the vorticity of a material particle is conserved.

Find the vorticity for such a flow in terms of the stream function $\psi$. Deduce that if the flow is steady, there must be a function $f$ such that

$$\nabla^{2} \psi=f(\psi)$$