Definition:Tangent Vector/Definition 1

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 $f: V \to \R$.

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

which satisfies the Leibniz law:
 * $\displaystyle X_m \left({f g}\right) = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)$

Also see

 * Equivalence of Definitions of Tangent Vector