I've done (i).
First of all Poincare's Lemma says that if the domain of is simply connected then:
is irrotational is conservative.
So for (ii)(a), does being simply connected (is it or not?) mean Poincare's Lemma implies there is a potential function for (since it would mean is conservative and hence is a gradient)
DESPITE not being simply connected - it has a hole in the middle at (0,0)?
... and the ones that won't