Gauß's law for magnetic flux is satisfied iff the -field is incompressible, which is equivalent to the statement that there exists a vector potential with . Then inspecting Faraday's law of induction,

we find it holds iff is irrotational, i.e. admits a scalar potential such that . Therefore .

Applying the identity it immediately follows that gives the same -field as , however if we want to also be the same we require that

hence .