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## Main Question or Discussion Point

Suppose we have an n-dimensional manifold M

However, where I become confused is when some books say that φ

The only place that I've come across parameterizations is when using parametric equations.

Could someone give a simple example showing φ and φ

Thanks.

^{n}and take a coordinate neighborhood U with associated coordinate map φ: U → V where V is an open subset of ℝ^{n}. 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.