2025-06-22

1172: For C Manifold with Boundary, Wedge Product of C Forms Is C Form

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description/proof of that for C manifold with boundary, wedge product of C forms is C form

Topics


About: C manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any C manifold with boundary, the wedge product of any C forms is a C form.

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: { the d -dimensional C manifolds with boundary }
f1: :MΛq1(TM), { the Cq1 -forms }
f2: :MΛq2(TM), { the Cq2 -forms }
//

Statements:
f1f2{ the C(q1+q2) -forms }
//


2: Proof


Whole Strategy: Step 1: apply the proposition that any q-form over any C manifold with boundary is C if and only if the operation result on any C vectors fields is C.

Step 1:

No matter whether f1f2 is regarded to be :MTq10(TM) or :MΛq1(TM), the proposition that any q-form over any C manifold with boundary is C if and only if the operation result on any C vectors fields is C can be applied.

Let V1:MTM,...,Vq1+q2:MTM be any C vectors fields.

f1f2(V1,...,Vq1+q2)=(q1+q2)!/(q1!q2!)Asym(f1f2)(V1,...,Vq1+q2)=(q1+q2)!/(q1!q2!)1/(q1+q2)!σ(f1f2)(Vσ1,...,Vσq1+q2)=1/(q1!q2!)σ(f1(Vσ1,...,Vσq1)f2(Vσq1+1,...,Vσq1+q2), but each f1(Vσ1,...,Vσq1) and f2(Vσq1+1,...,Vσq1+q2) are C, by the proposition that any q-form over any C manifold with boundary is C if and only if the operation result on any C vectors fields is C.

So, f1f2(V1,...,Vq1+q2) is C.


References


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