Suppose we have an n-dimensional manifold Mn and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝn. So far I'm clear on this. However, where I become confused is when some books say that φ-1 is called a parameterization of U and basically leave it at that. I want to understand this. What exactly does a parameterization mean and how is it manifested? The only place that I've come across parameterizations is when using parametric equations. Could someone give a simple example showing φ and φ-1 explicitly? Thanks.