Differential forms and their meaning

Question

I have been reading the book "An Introduction to Manifolds" and i have a question. So we know that for a $C^\infty$ function we can get a 1-form by doing its differential. My question is, when we write $dx$ for example in $\mathbb{R}^3$ what is the function that we are doing the differential ?Is it the function $w=x$? is it the projection onto the first coordinate? Thanks.

Answer 1

Yes, it's the projection $(x,y,z)\mapsto x$.

In a general manifold, any chart $\varphi:U\to\Bbb R^n$ determines a local coordinate system on $U$, and the coordinate maps are $x_i=\pi_i\circ\varphi$.