Inverting Generalized Coordinates In Corben's classical mechanics on pg. 9, it says that given generalized coordinates $q_m = q_m(x_1, ..., x_n,t)$, then if the Jacobian is non-zero everywhere, you may express $x_i = x_i(q_1,...,q_n,t)$.
If the transform $q_m$ is $C^1$, then the inverse function theorem will guarantee that local inverses exist everywhere, but by no means guarantee the existence of a global inverse. What am I missing?
 A: It depends on whether the coordinates are given globally or locally.  In Classical Mechanics, we usually work with a system of coordinates which are global, i.e., they work everywhere.  (Usually, not always.  You have to look at the context to see which is intended.)  Even if they are generalised coordinates.  Now in fact, even if they weren't global, it is automatic if you call them a coordinate system, that the Jacobian never vanishes.  (The motivation for imposing this condition is that the main reason to change coordinates is to see how a partial differential equation transforms, or how various integrals transform, in the hopes the new form will be more enlightening.  If the Jacobian vanished, you couldn't divide by it so the volume integral, for example, would have severe problems.)
Without seeing the text you have in mind, I think you can safely assume the author is just being sloppy, and permissibly so, since context in any practical application will always make the situation perfectly clear.  So, the short answer is, 
No, you are not missing anything, it is simply that the exceptions are not important for anything the text is going to develop later.
