homework and exercises - Question about tensor form of Maxwell equation

By variating the Maxwell Lagrangian we get the equation of motion. The remaining two Maxwell equations can be written as $$\epsilon_{\mu\nu\rho\sigma}\partial^{\rho} F^{\mu\nu} = 0.$$ I have also seen it written as the Bianchi identity: $$\partial_{[\lambda}F_{\mu\nu]} = 0.$$ Why are these two forms equivalent?

Answer

It's basically just a duality relation analogous to the cross product in three dimensions. But if you want to do some work to show the equivalence, then:

Going from the second equation to the first is easy, just hit it with $\epsilon_{\mu\nu\rho\sigma}$.

Going from the first to the second equation, is a little trickier and relies on knowing how to evaluate the products of Levi-Civita symbols. The basic idea is that you should contact the first equation with $\epsilon^{\mu'\nu'\lambda'\sigma}$ and compare the resulting antisymmetric combination of $\delta$s with the antisymmetrization of the indices in the second equation.