Calculating the Vector Field from a curl function

Homework Statement

Consider the intersection,R, between two circles : x2+y2=2 and (x-2)2+y2=2
a) Find a 2-Dimensional vector field F=(M(x,y),N(x,y)) such that ∂N/∂x - ∂M/∂y=1

none.

The Attempt at a Solution

There are other parts to the main question but I don't think I will have a problem with them. I know how to calculate the curl of F but I'm unsure of how to go about it so that the vector field relates to the intersection as we use F in an integral later on. What I thought of doing was integrating the curl function with respect to x or y in turn and trying to find M and N. I cant quite see how to involve the equations of the circles in all of this, if someone could point me in the right direction?
Thanks

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HallsofIvy
Homework Helper
What was the full statement of the problem? There doesn't see, to be any statement as to what the "two circles" has to do with the vector you are trying to find.

If the problem were just "Find a two dimensional vector $\underline{F}= (M(x,y),N(x,y))$ such that $\partial N/\partial x- \partial M/\partial y= 1$ then M= y, N=2x so that $\underline{F}= (y, 2x)$ is an obvious solution.

What was the full statement of the problem? There doesn't see, to be any statement as to what the "two circles" has to do with the vector you are trying to find.

If the problem were just "Find a two dimensional vector $\underline{F}= (M(x,y),N(x,y))$ such that $\partial N/\partial x- \partial M/\partial y= 1$ then M= y, N=2x so that $\underline{F}= (y, 2x)$ is an obvious solution.
Part b) of the question asks "Using this F and Greene's theorem, write the area integral ∫∫RdA as a line integral"

Does that help?

How can I solve the
∇×A(r,θ,φ)=B,A(r)=?,A(θ)=?,A(φ)=?
where B is constant.