If a scalar function
exists for which
, then we'd have
so the scalar function would be given by
where
is an arbitrary constant.
Presumably, this question is being asked in the context of the line integral of
along the given contour
. Since
is conservative - and consequently a scalar potential function
exists - we can simply use the gradient theorem to evaluate the line integral. We would get
(and we get the same answer by parameterizing
and computing the integral as we usually would)