Coordinate Representations#

Definition: Coordinate Representation

Let \(M\) be an \(m\)-manifold, let \(N\) be an \(n\)-manifold and let \(f: M \to N\) be a function. Let \((U_M, \phi_M)\) be a chart on \(M\) and let \((U_N, \phi_N)\) be a chart on \(N\).

The coordinate representation of \(f\) w.r.t. \((U_M, \phi_M)\) and \((U_N, \phi_N)\) is the real vector function

\[\phi_M(U_M \cap f^{-1}(U_N)) \subseteq \mathbb{R}^m \to \phi_N(U_N) \subseteq \mathbb{R}^n\]

defined as the following composition:

\[\phi_N \circ f \circ \phi_M^{-1}\]

The coordinate representation \(\phi_N \circ f \circ \phi_M^{-1}\) serves as a wrapper which allows us to express \(f\) entirely using coordinates. It does this by first taking coordinates \({}_{\phi_M}{p^1}, \dotsc, {}_{\phi_M}{p^m}\) and mapping them to \(p \in M\) via \(\phi_M^{-1}\). It then invokes \(f\) on \(p\) to obtain \(f(p) \in N\). Finally, it uses \(\phi_N\) to map \(f(p)\) to coordinates \({}_{\phi_N}f(p)^{1}, \dotsc, {}_{\phi_N} f(p)^n\).