Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12 / a

The components of the curl of a gradient are differences of mixed second derivatives:If $\varphi$ has continuous second partial derivatives in a neighborhood, equality of mixed derivatives makes every component vanish. ThusThe $C^2$ condition is a sufficient regularity hypothesis; no simple-connectivity assumption is needed for this local identity. Topology matters for the converse assertion that every curl-free field has a global potential.