How can curl of 4-vector or 6-vector be writen?

[The Count]

How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A⁴=(a₁,a₂,a₃,a₄)
how can we write in details the ∇×A⁴

Can we follow the same procedure for 6-vector?

 

[Orodruin]

The normal curl is an operation which is restricted to three-dimensions. If you look at the 4-vector as a one-form in space-time you can find the exterior derivative, which will be a two-form. Essentially, this will be the thing corresponding to the three-dimensional curl and the set of two-forms is 6-dimensional in a 4-dimensional space-time.

This construction appears, eg, when considering the electromagnetic field tensor as the exterior derivative of the 4-potential $F = dA$. As you may be aware, the field tensor $F$ has six independent components, three for the electric field and three for the magnetic.