(i) Show that for a two-dimensional incompressible flow $(u(x, y), v(x, y), 0)$, the vorticity is given by $\boldsymbol{\omega} \equiv \omega_{z} \hat{\mathbf{e}}_{z}=\left(0,0,-\nabla^{2} \psi\right)$ where $\psi$ is the stream function.

(ii) Express the $z$-component of the vorticity equation

$$\frac{\partial \boldsymbol{\omega}}{\partial t}+(\mathbf{u} \cdot \nabla) \boldsymbol{\omega}=(\boldsymbol{\omega} \cdot \nabla) \mathbf{u}$$

in terms of the stream function $\psi$.