Commutator between covariant derivative, field strength

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

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