In K&K's Intro to Mechanics, they kick off the topic of rotation by trying to turn rotations into vector quantities in analogy with position vectors. It's quickly shown, however, that rotations do not commute, making them rather poor vectors. They then show, however, that infinitesimal rotations do commute. The mathematical reasons for this make sense to me, but I'm not sure why finite rotations shouldn't commute. After all, we can construct a finite rotation by summing an infinite number of infinitesimal rotations, all of which commute, so why do finite rotations lose this property? Does this have any relationship with exact and inexact differentials?