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

which satisfies the Leibniz law:
 * $\ds \map {X_m} {f g} = \map {X_m} f \map g m + \map f m \map {X_m} g$

Also see

 * Equivalence of Definitions of Tangent Vector