I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it.
I have gotten to page 29. I am confused about proposition 3.2 and the way a differential form is defined on page 21. They define it as a collection of forms $\omega_U$ which produce the same pullback on intersection of charts. Essentially they identify the differential forms on open charts with $R^n$ which can be made compatible nothing more. If you read the text you see on page 29, proposition 3.2, they show the pull back of the standard $n$ form $R^n$. I don't see why the existence of positive global form from their definition implies there is a positive $f_\alpha$ and also how the same thing would work in opposite direction for $\Phi_\alpha^{-1}$ ? Explaining that paragraph generally would be helpful. thanks.