Need help to compute the integral using Green's Theorem

Question

calculate integral F dot dr where F(x,y) = ((2xy)i + (y^2-x^2)j) / (x^2+y^2)^2

and C is any positively oriented simple closed curve that encloses the origin.

Answer 1

Did you mean for the i component to also be divided by the last (x^2+y^2)^2 too?

If so, Greens Theorem is just a 2D Stoke's Theorem so instead you should just memorize Stoke's which says the line integral over any closed path is equal to the curl of F dot dA where dA is area of region enclosed by the path.

If you wanted the i component divided by that as well, the curl of F is zero so the line integral would also be zero. If not, however, the curl is not zero and it would depend on the area of the curve.