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1. I Differential forms and bases

Yes, I get it now. See my reply to @fresh_42. Thank you too!
2. I Differential forms and bases

@fresh_42 I see it now. Thank you so much for your very detailed post! The book I'm reading does define the pullback of maps on manifolds. I got confused because it doesn't give an explicit formula for the pullback of forms. Instead, it says that the pullback can be extended to differential...
3. I Differential forms and bases

It seems to me fresh_42 gave the same exact definition I'm using: $(\phi^* \nu)(p) = \nu(\phi(p)) = (\nu\circ\phi)(p)$. His expression for differential forms is just a property of the $d$ operator, according to my book. In $(f^*(w))(X_p) := w(f_* X_p)$ you do the pullback on $w$ by...

11. I Cordinates on a manifold

Maybe I'm starting to see the problem with my definition: the coordinates would just be a local parametrization of the curve but maybe I'd lack enough structure to do regular calculus over them. The cleanest way is to define $\phi$ from $M$ to $\mathbb{R}^n$ and then the coordinates on...