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 $\omega = \varphi \wedge dX$.