Smooth map

A smooth map is a morphism in the category of smooth manifolds. It is a map between smooth manifolds that preserves smooth structure.

Definition

A map between smooth manifolds is called a smooth if for all charts of and of such that the map is smooth.

Note, this makes sense, since we don’t know how to take derivatives on by itself. So if we use the charts to get back to then we can use the Jacobian and do calculus the way we know and love. Picture (without necessary restrictions to open sets):