Definition:Tangent Vector

Definition 1
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

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

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

Then a tangent vector $X_m$ on $M$ at $m$ is a linear mapping $X_m : C^\infty \left({V, \R}\right) \to \R$ satisfying the Leibniz law:
 * $\displaystyle X_m \left({f g}\right) = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)$

Definition 2
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

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

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

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

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

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
 * Definition:Tangent Space
 * Definition:Tangent Bundle