For an uniform flow over a circular cylinder. The flow field may be approximated by the following steady, two$-$dimensional velocity components in the xy $-$ or r $0-$plane.

$u_r=$Vcos$(1-\frac{a^2}{r^2}),u_{}=-$Vsin$(1+\frac{a^2}{r^2})$

$($a$)$ Is this flow field rotational or irrotational? Explain. $(10\%)$

$($b$)$ Find the stream function $(r,)$ for flow over a circular cylinder. $(10\%)$