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 $\map {C^\infty} {V, \R}$ be defined as the set of all smooth mappings $f: V \to \R$.

Definition 1

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:

$\ds \map {X_m} {f g} = \map {X_m} f \map g m + \map f m \map {X_m} g$

Definition 2

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

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

defined by:

$\map {X_m} f := \map {\dfrac \d {\d \tau} {\restriction_0} } {\map {f \circ \gamma} \tau}$

for all $f \in \map {C^\infty} {V, \R}$.

Also see

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

• Results about tangent vectors can be found here.

Sources

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