Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 2 / c / Solution

The curl isTo establish that $\mathbf{F}$ is a conservative vector field, construct a potential of a conservative vector field directly:This gradient calculation is valid on all of ${\mathbb{R}}^3$, and hence also on the cone. The explicit potential of a conservative vector field supplies the required global justification.