Smoothness of Functions on Manifolds

Definition: Smoothness of Functions on Manifolds

Let $f:M\to {\mathbb{R}}^n$ be a real vector-valued function on a $k$-dimensional smooth manifold.

We say that $f$ is smooth iff for each $p\in M$ there exists a smooth chart $(U,\varphi )$ such that $p\in U$ and the Composition $f\circ {\varphi }^{-1}:\varphi (U)\subseteq {\mathbb{R}}^k\to {\mathbb{R}}^n$ is smooth.