A description/proof of that velocity vectors field along \(C^\infty\) curve is \(C^\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors field along curve.
- The reader knows a definition of velocity of curve.
- The reader admits the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold is \(C^\infty\).
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, the velocity vectors field along the curve is \(C^\infty\).
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: Description
For any \(C^\infty\) manifold, \(M\), any open interval, \(I \subseteq \mathbb{R}\), and any \(C^\infty\) curve, \(c: I \to M\), the velocity vectors field along \(c\), \(V: I \to c (I) \to TM\), \(i \mapsto c' (i)\), is \(C^\infty\).
2: Proof
For any \(C^\infty\) function, \(f: M \to \mathbb{R}\), \(V (i) f = c' (i) f = \frac{d f (c (i))}{d i}\), which is \(C^\infty\), because \(f (c (i))\) is a composition of \(C^\infty\) maps. \(\frac{d f (c (i))}{d i}\) is really \((V f) \circ c\). So, \(V\) is \(C^\infty\), by the proposition that for any \(C^\infty\) manifold and any \(C^\infty\) curve over any open interval, any vectors field along the curve is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function on the manifold is \(C^\infty\).
