Definition:Tangent Vector
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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 $\map {C^\infty} {V, \R}$ 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
- 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