a. We're looking for a scalar function
such that
. This means


Integrate both sides of the first PDE with respect to
:

Differentating both sides with respect to
gives

so that
. A potential function exists, so the fundamental theorem does apply.
Green's theorem also applies because
is a simple and smooth curve.
b. Now with (and I'm guessing as to what
is supposed to be)

we want to find
such that


Same procedure as in (a): integrating the first PDE wrt
gives

Differentiating wrt
gives

so that
, which is undefined whenever
, and the fundamental theorem applies, and Green's theorem also applies for the same reason as in (a).
c. Same as (b) with slight changes. Again, I'm assuming the same format for
as I did for
, i.e.

Now



which is undefined at the point (0, 0). Again, both the fundamental theorem and Green's theorem apply.