definition of integral curve of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of integral curve of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary.
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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( V\): \(\in \{\text{ the } C^\infty \text{ vectors fields over } M\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( I\): \(= (t_1, t_2), [t_1, t_2], (t_1, t_2], \text{ or } [t_1, t_2) \subseteq \mathbb{R}\) such that \(t_1 \lt t_2\), as the embedded submanifold with boundary of \(\mathbb{R}\)
\(*\gamma\): \(: I \to M\), \(\in \{\text{ the } C^\infty \text{ curves }\}\)
\( d \gamma / d t\): \(: I \to TM, t \mapsto d \gamma / d t \vert_t\)
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Conditions:
\(\forall t \in I (d \gamma / d t \vert_t = V (\gamma (t)))\)
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