definition of velocity of \(C^\infty\) curve at point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of \(C^\infty\) 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 \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
Target Context
- The reader will have a definition of velocity of \(C^\infty\) curve at point on \(C^\infty\) manifold with boundary.
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:
\( M\): \(\in \{ \text{ the } C^\infty \text{ manifolds with boundary } \}\)
\( \mathbb{R}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( J\): \(= (t_1, t_2), [t_1, t_2], (t_1, t_2], \text{ or } [t_1, t_2) \subseteq \mathbb{R}\) such that \(t_1 \lt t_2\), as the embedded submanifold with boundary of \(\mathbb{R}\)
\( t_0\): \(\in J\)
\( \gamma\): \(: J \to M\), \(\in \{\text{ the curves }\} \cap \{\text{ the } C^\infty \text{ maps }\}\)
\( d / d t \vert_{t_0}\): \(\in T_{t_0}J\), \(: C^\infty (J) \to \mathbb{R}, f \mapsto d \tilde{f} / d t \vert_{t_0}\), where \(\tilde{f}\) is any extension of \(f\) on any open neighborhood of \(t_0\), \(U_{t_0} \subseteq \mathbb{R}\)
\(*d \gamma (d / d t \vert_{t_0})\): \(\in T_{\gamma (t_0)}M\)
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Conditions:
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2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M\), the Euclidean \(C^\infty\) 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 \(t_1 \lt t_2\), as the embedded submanifold with boundary of \(\mathbb{R}\), any \(t_0 \in J\), any \(C^\infty\) curve, \(\gamma: J \to M\), and the tangent vector, \(d / d t \vert_{t_0}: C^\infty (J) \to \mathbb{R}, f \mapsto d \tilde{f} / d t \vert_{t_0} \in T_{t_0}J \), where \(\tilde{f}\) is any extension of \(f\) on any open neighborhood of \(t_0\), \(U_{t_0} \subseteq \mathbb{R}\), \(d \gamma (d / d t \vert_{t_0}) \in T_{\gamma (t_0)}M\)
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 \(t_0 = t_1\).
\(d / d t \vert_{t_0}\) is well-defined (does not depend on the choice of \(\tilde{f}\)): when \(U_{t_0}\) can be taken to be such that \(U_{t_0} \subseteq J\), \(\tilde{f} = f \vert_{U_{t_0}}\), and \(d \tilde{f} / d t \vert_{t_0}\) 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 \(t_0 + \delta_2 \lt t_2\), and \(\tilde{f} \vert_{[t_0, t_0 + \delta_2)} = f \vert_{[t_0, t_0 + \delta_2)}\), but \(d \tilde{f} / d t \vert_{t_0} = lim_{t \mapsto + t_0} (\tilde{f} (t) - \tilde{f} (t_0)) / (t - t_0) = lim_{t \mapsto + t_0} (f (t) - f (t_0)) / (t - t_0)\), 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 \(t_1 \lt t_0 - \delta_1\), and \(\tilde{f} \vert_{(t_0 - \delta_1, t_0]} = f \vert_{(t_0 - \delta_1, t_0]}\), but \(d \tilde{f} / d t \vert_{t_0} = lim_{t \mapsto - t_0} (\tilde{f} (t) - \tilde{f} (t_0)) / (t - t_0) = lim_{t \mapsto - t_0} (f (t) - f (t_0)) / (t - t_0)\), which depends only on \(f\).
\(d / d t \vert_{t_0}\) is indeed a tangent vector: \(d / d t \vert_{t_0} (r f) = d \widetilde{r f} / d t \vert_{t_0} = d (r \tilde{f}) / d t \vert_{t_0} = r d \tilde{f} / d t \vert_{t_0} = r d / d t \vert_{t_0} (f)\); \(d / d t \vert_{t_0} (f_1 f_2) = d \widetilde{f_1 f_2} / d t \vert_{t_0} = d (\tilde{f_1} \tilde{f_2}) / d t \vert_{t_0} = d \tilde{f_1} / d t \vert_{t_0} \tilde{f_2} (t_0) + \tilde{f_1} (t_0) d \tilde{f_2} / d t \vert_{t_0} = d / d t \vert_{t_0} (f_1) f_2 (t_0) + f_1 (t_0) d / d t \vert_{t_0} (f_2)\).
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.
