Expansion of the commutator of two vector fields

[tut_einstein]

Hi,

I don't understand a particular coordinate expansion of the commutator of 2 vector fields:

[X, Y ]f = X(Y f) − Y (Xf) = X_be_b(Y _ae_af) − Y _be_b(X_ae_af)
= (X_b(e_bY_ a) − Y _b(e_bX_a))e_af + X_aY _b[e_a, e_b]f


X,Y = Vector fields
f = function

X_i = Components of X and same for Y
e_i = coordinates of the vector space

I don't understand how to get the third term in the 2nd line. I can tell that it's probably a product rule but I don't see how to get it.

Thanks!

 

[Eynstone]

If we write X= (sum)X_i d/dxi &c., we only have to notice that the commutator is bilinear & the terms follow.