definition of \(C^\infty\) vectors field along \(C^\infty\) curve
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of curve on topological space.
- The reader knows a definition of tangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
Target Context
- The reader will have a definition of \(C^\infty\) vectors field along \(C^\infty\) curve.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( J\): \(= (t_1, t_2), [t_1, t_2], (t_1, t_2], \text{ or } [t_1, t_2) \subseteq \mathbb{R}\) such that \(t_1 \lt t_2\), as the embedded submanifold with boundary of \(\mathbb{R}\)
\( \gamma\): \(: J \to M\), \(\in \{\text{ the curves }\} \cap \{\text{ the } C^\infty \text{ maps }\}\)
\(*V\): \(: J \to T M\)
//
Conditions:
\(\forall j \in J (V (j) \in T_{\gamma (j)}M)\)
\(\land\)
\(V \in \{\text{ the } C^\infty \text{ maps }\}\)
//
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 \(C^\infty\)-ness of \(: \gamma (J) \to T M\).
This concept is not about velocity of \(C^\infty\) curve at point on \(C^\infty\) manifold with boundary: \(V (j)\) does not need to be \(d \gamma (d / d t \vert_{j})\).
