definition of \(C^\infty\) vectors field on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of tangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of section of continuous map.
- 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 on \(C^\infty\) 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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( (TM, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\(*V\): \(: M \to TM\)
//
Conditions:
\(V \in \{\text{ the } C^\infty \text{ sections of } \pi\}\)
//
2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M\), and its tangent vectors bundle, \((TM, M, \pi)\), any \(C^\infty\) section of \(\pi\), \(V: M \to TM\)
3: Note
As \(\pi\) is continuous (in fact, \(C^\infty\)), the definition is well-defined.
