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