description/proof of that \(C^\infty\) 1-Form operated on \(C^\infty\) vector field along \(C^\infty\) curve is \(C^\infty\) function over domain of curve
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(C^\infty\) vectors field along \(C^\infty\) curve.
- The reader knows a definition of \(q\)-form over \(C^\infty\) manifold with boundary.
- The reader admits the proposition that any \((0, q)\)-tensors field over \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and any \(C^\infty\) curve on it, any \(C^\infty\) 1-form operated on any \(C^\infty\) vector field along the curve is a \(C^\infty\) 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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(I\): \(= (s_1, s_2) \subseteq \mathbb{R}\) as the open submanifold of the Euclidean \(C^\infty\) manifold
\(\lambda\): \(: I \to M\), \(\in \{\text{ the } C^\infty \text{ curves }\}\)
\(V\): \(:I \to \lambda (I) \to TM\), \(\in \{ \text{ the } C^\infty \text{ vectors fields along } \lambda\}\)
\(t\): \(: M \to T^0_1 (TM)\), \(\in \{\text{ the } C^\infty 1 \text{ -forms }\}\)
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Statements:
\(t (V): I \to \mathbb{R} \in \{\text{ the } C^\infty \text{ maps }\}\)
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2: Proof
For each \(s \in I\), let us take a chart, \((U_{\lambda (s)} \subseteq M, \phi_{\lambda (s)})\), and the induced chart, \((\pi^{-1} (U_{\lambda (s)}) \subseteq TM, \widetilde{\phi_{\lambda (s)}})\).
As \(\lambda\) is continuous, there is an open neighborhood of \(s\), \(U_s \subseteq I\), such that \(\lambda (U_s) \subseteq U_{\lambda (s)}\).
Over \(U_s\), \(V (\lambda (s)) = V^j (\lambda (s)) \partial / \partial x^j\), where \(V^j (\lambda (s))\) s are \(C^\infty\) as some functions of \(s\), because \(V\) is \(C^\infty\).
\(t (V (\lambda (s))) = t (V^j (\lambda (s)) \partial / \partial x^j) = V^j (\lambda (s)) t (\partial / \partial x^j)\), but as \(\partial / \partial x^j\) is a \(C^\infty\) vectors field over \(U_{\lambda (s)}\), \(t (\partial / \partial x^j)\) is a \(C^\infty\) function over \(U_{\lambda (s)}\), by the proposition that any \((0, q)\)-tensors field over \(C^\infty\) manifold with boundary is \(C^\infty\) if and only if the operation result on any \(C^\infty\) vectors fields is \(C^\infty\), and \(V^j (\lambda (s)) t (\partial / \partial x^j) = V^j (\lambda (s)) (t (\partial / \partial x^j)) (\lambda (s))\), and \((t (\partial / \partial x^j)) (\lambda (s))\) is a \(C^\infty\) function of \(s\) because \(\lambda\) is \(C^\infty\), so, \(t (V (\lambda (s)))\) is a \(C^\infty\) function of \(s\).
