Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 11 / e / Solution

The only contributions to the Laplacian come from the pure quadratic terms. HenceThe scalar potential is therefore a harmonic function. Since $\mathbf{F}=\nabla \varphi$,Thus this field is both irrotational and divergence-free. By the divergence theorem, its net flux through every sufficiently regular closed surface is zero.