Suppose \(F\colon M\to N\) is a smooth map, and \((\varphi,U)\) and \((\psi,V)\) are local coordinates of \(M\) and \(N\), respectively. Then \(\hat{F}=\psi\circ F\circ \varphi^{-1}\colon \tilde{U}\to V\) is called the coordinate representation of \(F\) with respect to the given coordinates, where
\[ \tilde{U} = \varphi^{-1}\bigl(\varphi(U)\cap F^{-1}(\psi(V))\bigr). \]