An Electromagnetic Field Tensor

 From a mathematical perspective electromagnetic theory consists of laws of nature expressed in terms of vector analysis.

But what would we see if we looked at EM theory from a more physical perspective focusing on the forces and changes in momentum instead? We can start with the Lorentz force which involves electric and magnetic fields. The electric force produces changes in the motion of a charged particle in the direction of the electric field. The magnetic force involves changes in position and gives the deflection of the path of motion. So for differential changes in time and position we can associate changes in momentum or impulses acting on the particle.

It turns out that the fields are the coefficients of the differential changes in position and time that give the changes in momentum. We can add the change in energy or work done on the particle to the momentum changes to get a 4-dimension picture of what's happening. And we end up with a field tensor, the result of applying the chain rule to each component of the momentum. So the field tensor can be expressed in terms of 4-dimensional gradients of the components of a momentum flow field describing the paths of identical test particles in neighboring positions. This appears to have been the approach that Maxwell adopted for EM theory.