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Definition: X in R

^{N}is a k-dimensional manifold if it is locally diffeomorphic to R

^{k}.

Suppose U is an open subset of R

^{k}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 R

^{k}, that is easier to work with as oppose to some abstract space X. I not to sure of how to interpret the parametrization f.