Commutator between covariant derivative, field strength

[oliveriandrea]

Hello,
i try to prove that
∂_μF^μ$\nu$ + ig[A_μ, F^μ$\nu$] = [D_μ,F^μ$\nu$]
with the D_μ = ∂_μ + igA_μ

but i have a problem with the term F^μ$\nu$∂_μ ...
i try to demonstrate that is nil, but i don't know if it's right...

F^μ$\nu$∂_μ $\Psi$ = $\int$ (∂_$\nu$F^μ$\nu$) (∂_μ$\Psi$) + $\int$ F^μ$\nu$∂_μ∂_$\nu$ $\Psi$ = (∂_$\nu$F^μ$\nu$) [$\Psi$ ]^∞_∞ - $\int$$\Psi$∂_μ∂_$\nu$F^μ$\nu$ = 0

with $\Psi$ a smooth function, nil at infinity

if it's wrong please do you post the right answers? and why it is wrong...
thank you

 

[vanhees71]

This is right. Of course, if you need it mathematically accurate, you have to think about, how fast you test function has to go to 0 at infinity, but if it's a physics question, what you did should be sufficient.

 

[oliveriandrea]

Thank you (yes it's physics question)