is a video series by explaining the Faraday 2-form and its relations to Maxwell's equations.
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.
or using the index notation with square brackets for the antisymmetric part of the tensor:
The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is
Substituting this into the Euler–Lagrange equation of motion for a field:
The quantity in parentheses above is just the field tensor, so this finally simplifies to