I am reading Differential Topology by Guillemin and Pollack. Definition: X in RN is a k-dimensional manifold if it is locally diffeomorphic to Rk. Suppose U is an open subset of Rk and V is a neighborhood of a point x in X. A diffeomorphism f:U->V is called a parametrization of the neighborhood V. The inverse mapping f-1:V->U is called a coordinate system. Why is f a parametrization and its inverse a coordinate system? How do these terms fit in the big picture of manifolds? I understand that we are rewriting V in X into a new coordinate system in Rk, that is easier to work with as oppose to some abstract space X. I not to sure of how to interpret the parametrization f.