The virtue of the Hansen solutions is that they ``automatically'' work
to decompose field components into parts that are mutual curls (as
required by Faraday/Ampere's laws for the fields) or divergences (as
required by Gauss's laws for the fields):

(13.5) | |||

(13.6) | |||

(13.7) |

Hence and are divergenceless, while the divergence of is a scalar solution to the HHE! is related to the scalar field and the gauge invariance of the theory in an interesting way we will develop. Also:

(13.8) | |||

(13.9) | |||

(13.10) |

which shows how and are now ideally suited to form the components of electric and magnetic multipole fields mutually linked by Ampere's and Faraday's law.