Line integral of vector field whose curl=0

Question

I've got a problem that's asking me to take the line integral of a conservative vector field, given by

V=xi+yj

I'm asked to perform the line integral using stoke's theorem for this field when a) the curve C is the unit circle in the xy plane centered at the origin, and b) when C is the same circle only centered at (1,0).

My thought was, since Stoke's theorem replaces F*dr in the integral with curl(F)*da, that the answer to both would be zero since curl(F)=0.

Does this make sense? I'm not sure since I'm asked to do a line integral of the same function twice.

Thanks!

Answer 1

The vector line integral of a conservative field over any smooth, closed curve is zero. Your justification with Stokes' theorem is fine. It doesn't matter what the smooth, closed curve is. They are likely asking you twice to try and emphasize this key fact about conservative fields.