Vector potential for a divergence-free field

A vector potential for a field $u$ is a field $A$ whose curl satisfies

u=\nabla\times A.

If $A$ is $C^2$ , then $u$ is divergence-free. Adding a gradient $\nabla\phi$ to $A$ leaves its curl unchanged, so potentials are generally nonunique.

A local construction

On a region star-shaped about zero, a smooth divergence-free field has the potential

A(x)=\int_0^1 t\,u(tx)\times x\,dt.

Indeed, $\nabla\times[t\,u(tx)\times x]=\partial_t[t^2u(tx)]$ when $\operatorname{div}u=0$ ; integration gives $\nabla\times A=u$ . This local formula does not assert that an arbitrary domain admits a global potential with prescribed boundary conditions or compact support.

Localization

If a smooth scalar cutoff $\chi$ equals one in a region, then $\nabla\times(\chi A)=u$ there and stays divergence-free everywhere. The transition region contains the correction $\nabla\chi\times A$ .