# Definition:Tangent Vector/Definition 2

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## 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 $f: V \to \R$.

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

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

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

## Sources

- 2013: Gerd Rudolph and Matthias Schmidt:
*Differential Geometry and Mathematical Physics*: $\S 1.4$: Tangent Space