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 $F:M\to N$ between smooth manifolds is called a smooth if for all charts $(U,\varphi )$ of $M$ and $(V,\psi )$ of $N$ such that $F(U)\subset V$ the map $\psi \circ F\circ {\varphi }^{-1}\varphi (U)⟶\psi (V)$ is smooth.

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