1202: Rough Vectors Field over C∞ Manifold with Boundary

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definition of rough vectors field over $C^{\mathrm{\infty }}$ manifold with boundary

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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 tangent vectors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of rough section of continuous surjection.

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Target Context

• The reader will have a definition of rough vectors field over $C^{\mathrm{\infty }}$ manifold with boundary.

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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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$(TM,M,\pi )$: $=\text{ the tangent vectors bundle over }M$
$\ast V$: $:M\to TM$
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Conditions:
$V\in \{\text{ the rough sections of }\pi \}$
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2: Note

As $\pi$ is a continuous surjection (in fact, $C^{\mathrm{\infty }}$), 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.

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References

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