Covariant Formulation of E&M Can anybody explain me what does mean the "covariant formulation of electrodynamics"? What does the covariant here mean? 
Invariance of Maxwell equations under Lorentz Transformations? In what way? Invariance under mathematical base change? 
Does it relate to co- and contravariant derivatives from differential geometry? There  the words covariant and contravariant refer to how objects transform under general coordinate transformations. 
 A: It means the theory is expressed/discussed in the language of tensor fields, where the tensors are quantities that transform between mutually moving inertial frames according to the Lorentz transformation. All differential equations are expressed as relations between tensor fields and their derivatives. For example, the equation 
$$
\frac{dp_k}{dt} = q E_k + q \epsilon_{kij} v_i B_j
$$
which uses the notation of 3-vectors, is not covariant, because, although it does have the same form in all frames, the quantities involved are not tensors that would transform between moving frames according to the Lorentz transformation. They only transform as cartesian tensors between frames that are at rest with respect to each other.
The covariant formulation of the same law is
$$
\frac{d p_\mu}{d\tau} = q F_{\mu\nu} u^\nu,
$$
using components of 4-vectors $p,u$ and 4-tensor $F$. This is because in other frame the equation has the same form and the quantities involved - $p,u,F$ - transform as tensors according to the Lorentz transformation.
