If we have a surface, say $S$, and a smooth map $f: S \to R$ (or $\mathbb{R}^3$), and a parametrization of that surface $ x: U\subseteq \mathbb{R}^2 \to S$, how do we find the directional derivative of $f$ along a tangent vector $v$ at $p \in x(U)$ ?
Almost all the books defines the directional derivative of such a map as the derivative of $f\circ c$ where $c$ is a curve on $S$ passing through $p$ with derivative $v$ at $p$; however, finding such a curve, I think, is not practical, in general.Therefore, I would like to calculate $\nabla_v f$ with the help of the parametrisation $x$.