2025-06-22

1173: C 1-Form Operated on C Vectors Field Along C Curve Is C Function over Domain of Curve

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that C 1-Form operated on C vector field along C curve is C function over domain of curve

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 and any C curve on it, any C 1-form operated on any C vector field along the curve is a C function over the domain of the curve.

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 }
I: =(s1,s2)R as the open submanifold of the Euclidean C manifold
λ: :IM, { the C curves }
V: :Iλ(I)TM, { the C vectors fields along λ}
t: :MT10(TM), { the C1 -forms }
//

Statements:
t(V):IR{ the C maps }
//


2: Proof


For each sI, let us take a chart, (Uλ(s)M,ϕλ(s)), and the induced chart, (π1(Uλ(s))TM,ϕλ(s)~).

As λ is continuous, there is an open neighborhood of s, UsI, such that λ(Us)Uλ(s).

Over Us, V(λ(s))=Vj(λ(s))/xj, where Vj(λ(s)) s are C as some functions of s, because V is C.

t(V(λ(s)))=t(Vj(λ(s))/xj)=Vj(λ(s))t(/xj), but as /xj is a C vectors field over Uλ(s), t(/xj) is a C function over Uλ(s), by the proposition that any (0,q)-tensors field over C manifold with boundary is C if and only if the operation result on any C vectors fields is C, and Vj(λ(s))t(/xj)=Vj(λ(s))(t(/xj))(λ(s)), and (t(/xj))(λ(s)) is a C function of s because λ is C, so, t(V(λ(s))) is a C function of s.


References


<The previous article in this series | The table of contents of this series | The next article in this series>