Why do the extra terms cancel in the derivation of the EM field strength tensor?

[dyn]

Hi
I am trying to follow the derivation in some notes I have for the field strength tensor using covariant derivatives defined by D_u = ∂_u - iqA_u . The field strength is the defined by [ D_u , D_v ] = -iqF_uv
The given answer is F_uv = ∂_uA_v - ∂_vA_u .When I expand the commutator I get this answer but I have 2 terms left over which I presume should cancel but I don't understand why . I have the extra terms iq(A_v∂_u - A_u∂_v ) . Do these terms cancel and if so , why ?
Thanks

 

[Orodruin]

You are forgetting that this commutator is an operator that should act on something and it should be seen as the operator commutator. The $\partial A$ terms represent first multiplying by A then differentiating. Therefore $\partial A f = (\partial A) f + A \partial f$. Add the corresponding indices.

 

[dyn]

Thanks for your reply. It all works out now.