Why does this rotating vector field have zero curl but non-zero line integral?

[peterpang1994]

Homework Statement

Given a vector field
F=-y/(x^2+y^2) i +x/(x^2 +y^2)
Calculate the curl of it the line integral of it in a unit circle centered at O

Homework Equations

The Attempt at a Solution

I calculated that the curl is 0 but the line integral is 2π. I don't think this field should have a zero curl as you can imagine that the field is rotating.

 

[vela]

Show your work. Just telling us your incorrect answers doesn't let us see where your mistakes are (if there are indeed any).

You might want to look more carefully at the requirements for applying Stoke's theorem.