465: Velocity Vectors Field Along C∞ Curve Is C∞

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

A description/proof of that velocity vectors field along $C^{\mathrm{\infty }}$ curve is $C^{\mathrm{\infty }}$

---

Topics

About: $C^{\mathrm{\infty }}$ manifold

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

---

Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors field along curve.
• The reader knows a definition of velocity of curve.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, any vectors field along the curve is $C^{\mathrm{\infty }}$ if and only if its operation result on any $C^{\mathrm{\infty }}$ function on the manifold is $C^{\mathrm{\infty }}$.

---

Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, the velocity vectors field along the curve is $C^{\mathrm{\infty }}$.

---

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 $C^{\mathrm{\infty }}$ manifold, $M$, any open interval, $I\subseteq \mathbb{R}$, and any $C^{\mathrm{\infty }}$ curve, $c:I\to M$, the velocity vectors field along $c$, $V:I\to c(I)\to TM$, $i\mapsto c^{\prime }(i)$, is $C^{\mathrm{\infty }}$.

---

2: Proof

For any $C^{\mathrm{\infty }}$ function, $f:M\to \mathbb{R}$, $V(i)f=c^{\prime }(i)f=\frac{df(c(i))}{di}$, which is $C^{\mathrm{\infty }}$, because $f(c(i))$ is a composition of $C^{\mathrm{\infty }}$ maps. $\frac{df(c(i))}{di}$ is really $(Vf)\circ c$. So, $V$ is $C^{\mathrm{\infty }}$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold and any $C^{\mathrm{\infty }}$ curve over any open interval, any vectors field along the curve is $C^{\mathrm{\infty }}$ if and only if its operation result on any $C^{\mathrm{\infty }}$ function on the manifold is $C^{\mathrm{\infty }}$.

---

References

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