How are Maxwell's homogeneous equations related to the following?

Question

ivancsc1996 · Answer

$$F_{\mu \nu , \sigma} + F_{\nu \sigma , \mu} + F_{\sigma \mu , \nu} = 0$$

michele_laino · Answer

by the definition of the electromagnetic tensor $$F ^{\mu \nu} $$
namely:
$$F ^{\mu  \nu}=\partial ^{\mu}A ^{\nu} - \partial ^{\nu} A ^{\mu} $$
where $$A ^{\mu} $$
is the four vector potential

michele_laino · Answer

and $$F ^{\mu \nu} $$
is the electromagnetic tensor

anonymous · Answer

$$F _{\mu \nu,\sigma}+F _{\nu \sigma,\mu}+F _{\sigma \mu,\nu}=0$$
is an expression of Maxwell's equations in 4 dimensional covariant form. Where$$F _{ij,k}=\partial _{k}F _{ij}$$ the F's are as stated by @Michele_Laino .  So if you plug in the F's as stated above you will get Maxwell's equations.