Consider the two-dimensional velocity field $\mathbf{u}=(u, v)$ with

$$u(x, y)=x^{2}-y^{2}, \quad v(x, y)=-2 x y$$

(i) Show that the flow is incompressible and irrotational.

(ii) Derive the velocity potential, $\phi$, and the streamfunction, $\psi$.

(iii) Plot all streamlines passing through the origin.

(iv) Show that the complex function $w=\phi+i \psi$ (where $i^{2}=-1$ ) can be written solely as a function of the complex coordinate $z=x+i y$ and determine that function.