Why is $\partial_\nu\partial_\mu F^{\mu\nu}=0$?

Question

I must admit: I've no idea about tensor calculus, I just started to read some basics about contraction and index lowering/pushing. Still, I can't figure out why $\partial_\nu\partial_\mu F^{\mu\nu}=0$ for an antisymmetric tensor $F^{\mu\nu}$. Any hints?

Answer 1

Note that $\partial_\nu\partial_\mu=\partial_\mu\partial_\nu$. This means $$ \partial_\nu\partial_\mu F^{\mu\nu}=\partial_\mu\partial_\nu F^{\mu\nu}=-\partial_\mu\partial_\nu F^{\nu\mu} $$ Now, the only difference between the first and the last expression is a renaming of the indices (which doesn't change anything) and a sign. So it's equal to its own negative, and therefore zero.