1) Looks correct.
2) First of all, , so this one is zero as well.
As for your parametrization and the limits:
, and the Jacobian is
1) , where C is the circle in an anticlockwise direction.
I got 0. Is that right? (Since .)
2) Now this one I'm sort of stuck on: EDIT: Oops! Corrected sign.
where C is closed curve in an anticlockwise direction.
So that equals:
where D is the region bounded by C (the ellipse with radii 2 and 1).
Now the problem here is computation. I opted to go to polar coords (maybe that was a bad idea?), so I got:
. So far so good?
So then I did:
Now.. do I really have to integrate that? Or is there a simpler way?