Past exam of the mathematical methods II course of the University of Oxford / 2018 / 9 / a

Apply the divergence theorem to $A(r)=(C\cdot r)(C^{\prime }\cdot r)J(r)$. Since $C,C^{\prime }$ are constant and $J$ is a solenoidal vector field, the product rule givesIts boundary flux vanishes when $J$ has compact support, or when its normal component vanishes on the boundary of the integration volume. Integrating therefore provesFor a discontinuous field, the divergence-free hypothesis must include its boundary jump, in the sense of a weak derivative. A field that merely has zero divergence at interior points can still have a nonzero surface source; that weaker hypothesis would not justify discarding the flux. The result is the antisymmetric first moment of a localized solenoidal field.