TITLE: Exterior derivative on surfaces
QUESTION [1 upvotes]: I am studying exterior derivatives and a professor wrote the $r+1$-form:

$d(\varphi^*\omega)$, where $\omega$ is an $r$-form in $S$ and $\varphi:U\subset\mathbb{R}^n\to \varphi(U)\subset S$ a parameterization. 

I understand a little about what $d(\varphi^*\omega)$ is: $\varphi^*\omega$ is a $r$-form in $U$ and $d(\varphi^*\omega)$ its exterior differential, an $r+1$-form at $U$. But $d(\varphi^*\omega)$ takes values at $\wedge_{r+1}(\mathbb{R}^n)$? Could you explain a little bit more about all of these things?
In fact, forms are a little confused to me.
Many thanks in advance!

REPLY [1 votes]: $\omega$ is an $r$-form on $S$.
$\varphi^*\omega$ is an $r$-form on $U$ and is $\omega$ expressed in the coordinates given by $(U, \varphi)$.
$d(\varphi^*\omega)$ is the exterior derivative of $\varphi^*\omega$ and is therefore a $(r+1)$-form on $U$. If $\varphi^*\omega = w_{i_1\cdots i_r} dx^{i_1} \wedge \cdots \wedge dx^{i_r}$ then 
$$
d(\varphi^*\omega) = \partial_{i_{r+1}}w_{i_1\cdots i_r} dx^{i_{r+1}} \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_r}.
$$
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