I've got a question on a statement in do Carmo's "Differential Forms and Applications". On pages 46/47 he explains what a representation of a differential form $\omega$ on a manifold is. He defines it like this (p. 47): $$\omega_\alpha(v_1,\dots,v_k)=\omega(df_\alpha(v_1),\dots,df_\alpha(v_k)).$$
I don't really understand why this is a reasonable way to define it, because this means the parametrizations $f_\alpha$ are a differentiable maps (from a subset of $\mathbb{R}^n$ on $M$). But how do I know, that this is always true, because we only have this property in the definiton of regular surfaces (Example 5, p. 43).
Then he claims that the deriviate $d\omega$ defined in the way he explained is unique. But after looking in Spivaks "Integration on manifolds" (p. 117) I figured out that this uses that $df_a$ has to be invertible - how do I know this is true in the abstract setting described by do Carmo?
Thank you for your help!