I don't understand what the "clutching function" in this example is actually doing. What does it mean to "read off the coordinates of $v_{-}, w_{-}$ in the $v_+, w_+$ coordinate system"? What is the formal equation for $f$? Also, why is it that when "going around the equator", the "angle of rotation when viewed from above" (which angle is this exactly???) goes from $0$ to $4\pi$?
I'm new to studying vector bundles, so this picture-argument is just really confusing. I think I will understand it better if the explanation is more formal (and less handwavy).
