definition of rough vectors field over \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vectors bundle over \(C^\infty\) manifold with boundary.
- The reader knows a definition of rough section of continuous surjection.
Target Context
- The reader will have a definition of rough 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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( (TM, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\(*V\): \(: M \to TM\)
//
Conditions:
\(V \in \{\text{ the rough sections of } \pi\}\)
//
2: Note
As \(\pi\) is a continuous surjection (in fact, \(C^\infty\)), the definition is well-defined.
Usually, we need only (non-rough) vectors fields, but we sometimes need to talk about a rough vectors field in order to 1st introduce a may-be-rough vectors field and then prove that it is really a non-rough vectors field.
