Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 6 / Solution

The field is the gradient of a globally defined polynomial:It is therefore a conservative vector field on all of ${\mathbb{R}}^3$. The fundamental theorem for line integrals givesfor every closed piecewise smooth loop. Constructing the global vector-field potential avoids relying on a zero-curl test without checking the topology of the domain.