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 $\map {C^\infty} {V, \R}$ 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
 * $X_m: \map {C^\infty} {V, \R} \to \R$

defined by:
 * $\map {X_m} f := \map {\dfrac \d {\d \tau} {\restriction_0} } {\map {f \circ \gamma} \tau}$

for all $f \in \map {C^\infty} {V, \R}$.

Also see

 * Equivalence of Definitions of Tangent Vector