Maxwell Equations in Tensor Notation

[TheEtherWind]

²A^$\mu$=-$\mu$_oJ^$\mu$

Griffith's Introduction to Electrodynamics refers to this 4-vector equation as "the most elegant (and the simplest) formulation of Maxwell's equations." But does this encapsulate the homogeneous Maxwell Equations? I see how the temporal components lead to Gauss' Law, and I'm assuming, though I haven't shown it to myself, that the spatial components lead to the Ampere-Maxwell Law. What about Faraday's Law and the divergence of B?

 

[Matterwave]

The other two laws are basically obtained by definition of the E and B fields. For example, by defining B as the curl of a vector potential, it is then divergence-less by definition.

I would say that the "most elegant" way to formulate Maxwell's equations is by using the Faraday tensor ($F\equiv dA$, where d is the exterior derivative) :

$$dF=0$$

$$d*F=4\pi*J$$

But this requires a little bit of differential geometry to understand.