Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 8 / b / Solution

To obtain the volume integral of a curl, test its components with an arbitrary constant vector $c$. By part (a), $\nabla \cdot (G\times c)=c\cdot (\nabla \times G)$. Applying the divergence theorem to the closed boundary, with outward unit normal $n$, givesThe last step is cyclic symmetry of the scalar triple product. Since this holds for every $c$, all vector components agree:The same regularity conditions as the divergence theorem apply: for example a continuously differentiable field on a neighbourhood of a bounded volume with piecewise smooth boundary.