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 $\map {C^\infty} {V, \R}$ 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

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

defined by:

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

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



This article, or a section of it, needs explaining.
In particular: Needs a link to a page which explains what the derivative of a mapping in this context is -- at the moment, calculus has been defined only on the real number line, partly on the complex plane, and to a very rudimentary level on vector spaces.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
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This article, or a section of it, needs explaining.
In particular: The dependency of $X_m$ on $\gamma$ is not mentioned! I think, $X_m$ here is representing the covariance in the direction $\map {\gamma'} 0 \in T_m M$. Or it should look like $\map {X_m} {f, \map {\gamma'} 0 }$
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Also known as

A tangent vector is also known as a derivation.


Also see

  • Results about tangent vectors can be found here.


Sources

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