All the understanding should be elevated from 3D to spacetime 4D, hence the Lagrangian of point particles q(t) should move to fields φ(Xµ), hence

Note the vector field/spin-1, force mediator particles in EM field, there is the strength tensor, This tensor encapsulates 16 values in a compact form.

Note the derivation of contravariant can be derived step by step, then it’s just the work to convert contravariant form of tensor to covariant form of tensor: