I learned this concept in feedback linearization problems, where it is used for state transformation. For example, are states of a system. Now we have a new set of variables: . The problem is: whether can be a set of states of the system? The answer is: as long as the function is diffeomorphism, it can be used to represent the system.
It should be noted that: from to , DOFs are preserved. But the value region changes. If then . So my question on math stackexchange is not proper. Because may not be an arbitrary vector in . And of course, may not be an arbitrary vector in . But keep in mind that the DOFs are preserved.