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Suppose we have a vectorfield who is defined everywhere in R^2 except at the origin where it has a singularity, and suppose it's curl is zero. We then have that it is conservative in every open, simply connected subset in R^2 not containing the origin.

Now if we have a closed curve around the origin we can't generally say that the curve integral is zero right? That depends on the given vectorfield? If we have calculated the curve integral around a circle with centre in the origin and shown that it is zero, can we then say that every closed curve around the origin has curve integral zero?

Suppose for example we have a square around the origin. Can we calculate the curveintegral for the whole square by adding the curveintegrals along the 4 sides?