770: Velocity of C∞ Curve at Point on C∞ Manifold with Boundary

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definition of velocity of $C^{\mathrm{\infty }}$ curve at point on $C^{\mathrm{\infty }}$ manifold with boundary

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Topics

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

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of curve on topological space.
• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of velocity of $C^{\mathrm{\infty }}$ curve at point on $C^{\mathrm{\infty }}$ manifold with boundary.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\mathbb{R}$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold }$
$J$: $=(t_1,t_2),[t_1,t_2],(t_1,t_2],\text{ or }[t_1,t_2)\subseteq \mathbb{R}$ such that $t_1<t_2$, as the embedded submanifold with boundary of $\mathbb{R}$
$t_0$: $\in J$
$\gamma$: $:J\to M$, $\in \{\text{ the curves }\}\cap \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$d/dt{|}_{t_0}$: $\in T_{t_0}J$, $:C^{\mathrm{\infty }}(J)\to \mathbb{R},f\mapsto d\tilde{f}/dt{|}_{t_0}$, where $\tilde{f}$ is any extension of $f$ on any open neighborhood of $t_0$, $U_{t_0}\subseteq \mathbb{R}$
$\ast d\gamma (d/dt{|}_{t_0})$: $\in T_{\gamma (t_0)}M$
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Conditions:
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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifold with boundary, $M$, the Euclidean $C^{\mathrm{\infty }}$ manifold, $\mathbb{R}$, any interval on $\mathbb{R}$, $J=(t_1,t_2),[t_1,t_2],(t_1,t_2],\text{ or }[t_1,t_2)$ such that $t_1<t_2$, as the embedded submanifold with boundary of $\mathbb{R}$, any $t_0\in J$, any $C^{\mathrm{\infty }}$ curve, $\gamma :J\to M$, and the tangent vector, $d/dt{|}_{t_0}:C^{\mathrm{\infty }}(J)\to \mathbb{R},f\mapsto d\tilde{f}/dt{|}_{t_0}\in T_{t_0}J$, where $\tilde{f}$ is any extension of $f$ on any open neighborhood of $t_0$, $U_{t_0}\subseteq \mathbb{R}$, $d\gamma (d/dt{|}_{t_0})\in T_{\gamma (t_0)}M$

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3: Note

We need to introduce $\tilde{f}$, because while taking the derivative at $t_0$ requires the function to be defined on an open neighborhood of $t_0$, $U_{t_0}\subseteq \mathbb{R}$, there may not be any $U_{t_0}$ such that $U_{t_0}\subseteq J$ in cases like $J=[t_1,t_2]$ and $t_0=t_1$.

$d/dt{|}_{t_0}$ is well-defined (does not depend on the choice of $\tilde{f}$): when $U_{t_0}$ can be taken to be such that $U_{t_0}\subseteq J$, $\tilde{f}=f{|}_{U_{t_0}}$, and $d\tilde{f}/dt{|}_{t_0}$ depends only on $f$; otherwise, $t_0=t_1$ where $t_1$ is the closed boundary or $t_0=t_2$ where $t_2$ is the closed boundary, and for the former, $U_{t_0}=(t_0-{\delta }_1,t_0+{\delta }_2)$, where ${\delta }_2$ can be taken to be such that $t_0+{\delta }_2<t_2$, and $\tilde{f}{|}_{[t_0,t_0+{\delta }_2)}=f{|}_{[t_0,t_0+{\delta }_2)}$, but $d\tilde{f}/dt{|}_{t_0}=li{m}_{t\mapsto +t_0}(\tilde{f}(t)-\tilde{f}(t_0))/(t-t_0)=li{m}_{t\mapsto +t_0}(f(t)-f(t_0))/(t-t_0)$, which depends only on $f$, and for the latter, $U_{t_0}=(t_0-{\delta }_1,t_0+{\delta }_2)$, where ${\delta }_1$ can be taken to be such that $t_1<t_0-{\delta }_1$, and $\tilde{f}{|}_{(t_0-{\delta }_1,t_0]}=f{|}_{(t_0-{\delta }_1,t_0]}$, but $d\tilde{f}/dt{|}_{t_0}=li{m}_{t\mapsto -t_0}(\tilde{f}(t)-\tilde{f}(t_0))/(t-t_0)=li{m}_{t\mapsto -t_0}(f(t)-f(t_0))/(t-t_0)$, which depends only on $f$.

$d/dt{|}_{t_0}$ is indeed a tangent vector: $d/dt{|}_{t_0}(rf)=d\widetilde{rf}/dt{|}_{t_0}=d(r\tilde{f})/dt{|}_{t_0}=rd\tilde{f}/dt{|}_{t_0}=rd/dt{|}_{t_0}(f)$; $d/dt{|}_{t_0}(f_1{f}_2)=d\widetilde{f_1{f}_2}/dt{|}_{t_0}=d(\widetilde{f_1}\widetilde{f_2})/dt{|}_{t_0}=d\widetilde{f_1}/dt{|}_{t_0}\widetilde{f_2}(t_0)+\widetilde{f_1}(t_0)d\widetilde{f_2}/dt{|}_{t_0}=d/dt{|}_{t_0}(f_1)f_2(t_0)+f_1(t_0)d/dt{|}_{t_0}(f_2)$.

A moral of this definition is that while a velocity is usually taken from a curve from an open interval, we can take a curve from $[t_0,t_2)$ or $(t_1,t_0]$, which may be in fact necessary when $M$ has any nonempty boundary and $\gamma (t_0)$ is on the boundary, and even when $\gamma (t_0)$ is not on the boundary or $M$ has the empty boundary, there is no reason why we cannot take a non-open interval.

There is a related article, what the velocity of any curve at any closed boundary point is.

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References

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