Pullback and exterior derivative

[Silviu]

Homework Statement

Let ##\omega \in \Omega^r(N)## and let ##f:M \to N##. Show that ##d(f^*\omega)=f^*(d\omega)##

Homework Equations

##\Omega^r(N)## is the vector field of r-form at a given point in the manifold N, ##f^*## is the pullback function and ##d## is the exterior derivative

##(f^*\omega)(X_1, . . ., X_r) = \omega(f_* X_1, ..., f_* X_r)##, for r vectors ##X_1## and ##f_*## is the differential map

The Attempt at a Solution

I am not sure how to express the pullback in general, without making it act on some vectors, and I am not sure how to act on vectors when I have the exterior derivative, too. For example in ##d(f^*\omega)## do I first act on r vectors with ##f^*\omega## and then apply ##d##, or I apply ##d## and act with everything on r+1 vectors? Any help would be appreciated. Thank you!

 

[fresh_42]

Formally you can proceed by induction on ##r## and write ##\omega = \varphi \wedge dX##.