Let $M = \overline{\mathbb{C}}$ with coordinates: \begin{align*} \varphi_1: \mathbb{C}\to \mathbb{C} \ &,\ \varphi_1(z)=z\\ \varphi_2: \overline{\mathbb{C}} \setminus \lbrace 0\rbrace\to \mathbb{C}\ &,\ \varphi_2(w) = \frac{1}{w} \end{align*} So the transition function is $z=\varphi_1\circ \varphi_2^{-1}(w) = \frac{1}{w}$
We can take a differential $\eta = \lbrace (dz,\mathbb{C})\ ,\ (-\frac{dw}{w^2}\ ,\ \overline{\mathbb{C}}\setminus\lbrace0\rbrace)\rbrace$, and [my lecture notes] conclude that $M$ has no constant differentials (I assume this mean every function is constant). But does this depend on the coordinates we used? I mean, there are so many coordinates we could use, how do we know we can't find some that give us a constant differential?
Also, is it true that if we introduce some 3rd chart $\varphi_3$ such that its domain overlaps with $\varphi_1$ and $\varphi_2$ that the new $\eta'$ with this third chart added is "different" from $\eta$? i.e. is $\eta$ only well-defined w.r.t to a specific collection of charts? I worry because if this is so, then in my initial question you would basically have to check every collection of charts and show that none of the differentials associated with those charts is constant.