Coordinate Representation of Functions

Definition: Coordinate Representation of a Function

Let $f:M\to {\mathbb{R}}^n$ be a real vector-valued function on a $k$-manifold $M$ and let $(U,\varphi )$ be a chart for $M$.

The coordinate representation of $f$ on $(U,\varphi )$ is the real vector function ${}_{(U,\varphi )}f:\varphi (U)\subseteq {\mathbb{R}}^k\to {\mathbb{R}}^n$ obtained by the compositions of $f$ and the inverse of $\varphi$.

$${}_{(U,\varphi )}f\stackrel{\text{def}}{=}f\circ {\varphi }^{-1}$$

INTUITION

Although all points on a chart can be uniquely represented by their coordinates, the $f$ does not act on coordinates - it acts on points. Hence, we need a way to convert coordinates to the points they represent before $f$ can be used. This is precisely what ${\varphi }^{-1}$ does. If we are given coordinates $(p^1,\dots ,p^k)$, we can pass them to ${\varphi }^{-1}$ to obtain the point $p\in U$ which they represent. After having converted the coordinates $(p^1,\dots ,p^k)$ to the point $p$, we can invoke $f$, i.e. we can obtain the value $f(p)$.

The coordinate representation ${}_{(U,\varphi )}f$ does all of this in one single step. It takes coordinates $(p^1,\dots ,p^k)$, converts them to the point $p$ which they represent and then returns the value of $f$ at $p$.