Definition:Tangent Vector/Definition 2

Definition
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

Let $V$ be an open neighborhood of $m$.

Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$.

Let $I$ be an open real interval with $0 \in I$.

Let $\gamma: I \to M$ be a smooth curve with $\gamma \left({0}\right) = m$.

Then a tangent vector $X_m$ at a point $m \in M$ is a mapping
 * $X_m: C^\infty \left({V, \R}\right) \to \R$

defined by:
 * $X_m \left({f} \right) := \dfrac {\mathrm d} {\mathrm d \tau {\restriction_0} } \, f \circ \gamma \left({\tau}\right)$

for all $f \in C^\infty \left({V, \R}\right)$.

Also see

 * Equivalence of Definitions of Tangent Vector