Definition:Tangent Vector

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 $f: V \to \R$.

Also see

 * Equivalence of Definitions of Tangent Vector
 * Definition:Tangent Space
 * Definition:Tangent Bundle