# Definition A *function on a [[manifold]]* $M$ is a rule or[[map]] that assigns a real number to each point of $M$, $f:M\rightarrow \mathbb{R}$ When a region of $M$ is mapped differentiably to a region of $\mathbb{R}^n$, the function becomes a function on $\mathbb{R}^n$, $f:M\rightarrow \mathbb{R}^n$ ![[Pasted image 20210123193628.png]]