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

The tangent space at $m$, denoted by $T_m M$, is the set of all tangent vectors at $m$.

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 $\gamma\colon I \to M$ 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
 * $\displaystyle X_m : C^\infty\left({V, \R} \right) \to \R$

defined by
 * $\displaystyle X_m \left({f} \right) := \frac{\mathrm{d}{}}{\mathrm{d}{\tau}}_{{\restriction}_0} \, f \circ \gamma \left({\tau} \right)$

for all $f \in C^\infty\left({V, \R} \right)$.

The tangent space at $m$, again denoted by $T_m M$, is the set of all tangent vectors at $m$.

Also see

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