996: C∞ Vectors Field Along C∞ Curve

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definition of $C^{\mathrm{\infty }}$ vectors field along $C^{\mathrm{\infty }}$ curve

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of curve on topological space.
• The reader knows a definition of tangent vectors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ vectors field along $C^{\mathrm{\infty }}$ curve.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\mathbb{R}$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold }$
$J$: $=(t_1,t_2),[t_1,t_2],(t_1,t_2],\text{ or }[t_1,t_2)\subseteq \mathbb{R}$ such that $t_1<t_2$, as the embedded submanifold with boundary of $\mathbb{R}$
$\gamma$: $:J\to M$, $\in \{\text{ the curves }\}\cap \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$\ast V$: $:J\to TM$
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Conditions:
$\mathrm{\forall }j\in J(V(j)\in T_{\gamma (j)}M)$
$\wedge$
$V\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
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2: Note

This concept is different from section along subset of codomain of continuous surjection, which would be a map from $\gamma (J)$, while this concept is a map from $J$: this concept is not talking about $C^{\mathrm{\infty }}$-ness of $:\gamma (J)\to TM$.

This concept is not about velocity of $C^{\mathrm{\infty }}$ curve at point on $C^{\mathrm{\infty }}$ manifold with boundary: $V(j)$ does not need to be $d\gamma (d/dt{|}_j)$.

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References

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