Commutator between covariant derivative, field strength

[oliveriandrea]

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

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

F^{μ[itex]\nu[/itex]}∂_μ [itex]\Psi[/itex] = [itex]\int[/itex] (∂_{[itex]\nu[/itex]}F^{μ[itex]\nu[/itex]}) (∂_μ[itex]\Psi[/itex]) + [itex]\int[/itex] F^{μ[itex]\nu[/itex]}∂_μ∂_{[itex]\nu[/itex]} [itex]\Psi[/itex] = (∂_{[itex]\nu[/itex]}F^{μ[itex]\nu[/itex]}) [[itex]\Psi[/itex] ]^∞_∞ - [itex]\int[/itex][itex]\Psi[/itex]∂_μ∂_{[itex]\nu[/itex]}F^{μ[itex]\nu[/itex]} = 0

with [itex]\Psi[/itex] 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)