A description of what velocity of curve at closed boundary point is
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) curve.
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition.
Target Context
- The reader will have a description of what the velocity of any curve at any closed boundary point is.
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: Note
The conclusion is in fact what will be rather prevalently guessed intuitively, but this is about more rigorously confirming that that guess is correct.
2: Description
For any \(C^\infty\) manifold with (possibly empty) boundary, \(M\), any (possibly half) closed interval, \(J \subseteq \mathbb{R}\), with any closed boundary point, \(t_j\), and any \(C^\infty\) curve, \(\gamma: J \to M\), what is the velocity, \(d \gamma / d t \vert_{t_j}\), of \(\gamma\) at \(t_j\)?
This argument is profusely based on a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
Let us suppose that \(J = [t_1, t_2)\) or \([t_1, t_2]\) where \(t_1 \lt t_2\) (the \([t_1, t_1]\) case will not be worth considering, because the velocity would not be determined), and we will talk about the velocity at \(t_1\).
\(J\) is regarded to be the canonical \(C^\infty\) manifold with boundary.
There is a chart, \(([t_1, t_3) \subseteq J, id)\), where what exactly \(t_3\) is does not matter if \(t_1 \lt t_3 \le t_2\).
On the chart, \(d / d t \vert_{t_1}\) is defined to be the vector on \(T_{t_1}J\) such that for any \(C^\infty\) function, \(f: J \to \mathbb{R}\), \(d / d t \vert_{t_1} (f) = d f' / d t \vert_{t_1}\), where \(f'\) is any \(C^\infty\) map (called extension of \(f\)), \(f': U'_{t_1} \to \mathbb{R}\), where \(U'_{t_1}\) is an open neighborhood, \(U'_{t_1} \subseteq \mathbb{R}\), of \(t_1\) on \(\mathbb{R}\), such that \(f' \vert_{U'_{t_1} \cap J} = f \vert_{U'_{t_1} \cap J}\). The result does not really depend on the choice of \(f'\), because it has to equal the one-sided derivative of \(f\). That is indeed a derivation, because while \(d / d t \vert_{t_1} (f g) := d (f g)' / d t \vert_{t_1}\), \((f g)'\) can be taken to be \(f' g'\), because while there are some extensions, \(f': U'_{f, t_1} \to \mathbb{R}\) of \(f\) and \(g': U'_{g, t_1} \to \mathbb{R}\) of \(g\), \(f' \vert_{U'_{f, t_1} \cap U'_{g, t_1}} g' \vert_{U'_{f, t_1} \cap U'_{g, t_1}} \) is an extension of \(f g\), and so, \(= d (f' g') / d t \vert_{t_1} = (d f' / d t g') \vert_{t_1} + (f' d g' / d t) \vert_{t_1} = d f' / d t \vert_{t_1} g' (t_1) + f' (t_1) d g' / d t \vert_{t_1} = d / d t \vert_{t_1} (f) g (t_1) + f (t_1) d / d t \vert_{t_1} (g)\).
\(d \gamma / d t \vert_{t_1}\) is defined to be \(d \gamma \vert_{t_1} (d / d t \vert_{t_1})\), where \(d \gamma \vert_{t_j}\) is the differential of \(\gamma\) at \(t_j\).
For any \(C^\infty\) function, \(f: M \to \mathbb{R}\), \(d \gamma / d t \vert_{t_1} (f) = (d \gamma \vert_{t_1} (d / d t \vert_{t_1})) (f) = d / d t \vert_{t_1} (f \circ \gamma) = d (f \circ \gamma)' / d t \vert_{t_1}\) where \((f \circ \gamma)'\) is a \(C^\infty\) extension of \(f \circ \gamma\), \((f \circ \gamma)': U_{t_1} \to \mathbb{R}\).
There is a chart, \((U_{\gamma (t_1)} \subseteq M, \phi_{\gamma (t_1)})\), and \(id \circ f \circ {\phi_{\gamma (t_1)}}^{-1} = f \circ {\phi_{\gamma (t_1)}}^{-1}: \phi_{\gamma (t_1)} (U_{\gamma (t_1)}) \to \mathbb{R}\) is \(C^\infty\) at \(\phi_{\gamma (t_1)} (\gamma (t_1))\), which implies that there are an open neighborhood, \(U'_{\phi_{\gamma (t_1)} (\gamma (t_1))} \subseteq \mathbb{R}^d\), of \(\phi_{\gamma (t_1)} (\gamma (t_1))\) and a \(C^\infty\) extension of \(f \circ {\phi_{\gamma (t_1)}}^{-1}\), \(f': U'_{\phi_{\gamma (t_1)} (\gamma (t_1))} \to \mathbb{R}\).
There is a chart, \((U'_{t_1} \subseteq J, id)\), around \(t_1\) such that \(\gamma (U'_{t_1}) \subseteq U_{\gamma (t_1)}\) and \(\phi_{\gamma (t_1)} \circ \gamma \circ {id}^{-1} = \phi_{\gamma (t_1)} \circ \gamma: U'_{t_1} \to \phi_{\gamma (t_1)} (U_{\gamma (t_1)})\) is \(C^\infty\) at \(t_1\), by the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), any pair of domain chart around the point and codomain chart around the corresponding point such that the intersection of the domain chart and the domain is mapped into the codomain chart satisfies the condition of the definition, which implies that there are an open neighborhood, \(U''_{t_1} \subseteq \mathbb{R}\), of \(t_1\) and a \(C^\infty\) extension of \(\phi_{\gamma (t_1)} \circ \gamma\), \(\gamma': U''_{t_1} \to \mathbb{R}^d\).
As \(\gamma'\) is continuous at \(t_1\), there is an open neighborhood, \(U'''_{t_1} \subseteq U''_{t_1}\), of \(t_1\) such that \(\gamma' (U'''_{t_1}) \subseteq U'_{\phi_{\gamma (t_1)} (\gamma (t_1))}\), and let us take \(\gamma'\) as from \(U'''_{t_1}\). Then, \(f' \circ \gamma': U'''_{t_1} \to \mathbb{R}\) is a \(C^\infty\) extension of \(f \circ \gamma\), because \(f' \circ \gamma' \vert_{U'''_{t_1} \cap J} = f' \circ \phi_{\gamma (t_1)} \circ \gamma \vert_{U'''_{t_1} \cap J} = f \circ {\phi_{\gamma (t_1)}}^{-1} \circ \phi_{\gamma (t_1)} \circ \gamma \vert_{U'''_{t_1} \cap J} = f \circ \gamma \vert_{U'''_{t_1} \cap J}\) while it is \(C^\infty\) as a composition of \(C^\infty\) maps. So, \((f \circ \gamma)'\) can be taken to be \(f' \circ \gamma'\).
So, \(d \gamma / d t \vert_{t_1} (f) = d (f' \circ \gamma') / d t \vert_{t_1} = \partial f' / \partial x^k \vert_{\gamma' (t_1)} d \gamma'^k / d t \vert_{t_1}\), where \(k\) denotes the \(k\)-th component on \(U'_{\phi_{\gamma (t_1)} (\gamma (t_1))}\).
Although the result expression contains the extensions, \(f'\) and \(\gamma'\), it really does not depend on the choices of the extensions, because \(d \gamma'^k / d t \vert_{t_1}\) has to equal the one-sided derivative, \(d (\phi_{\gamma (t_1)} \circ \gamma)^k / d t \vert_{t_1}\), and \(\partial f' / \partial x^k \vert_{\gamma' (t_1)}\) has to equal the one-sided derivative (when \(\gamma (t_1)\) is on the boundary of \(M\) and \(k = d\)) or the normal derivative (otherwise), \(\partial (f \circ {\phi_{\gamma (t_1)}}^{-1}) / \partial x^k \vert_{\phi_{\gamma (t_1)} (\gamma (t_1))}\).
So, after all, \(d \gamma / d t \vert_{t_1} (f) = \partial (f \circ {\phi_{\gamma (t_1)}}^{-1}) / \partial x^k \vert_{\phi_{\gamma (t_1)} (\gamma (t_1))} d (\phi_{\gamma (t_1)} \circ \gamma)^k / d t \vert_{t_1}\) with the necessary one-sided or full derivatives.
Likewise, let us suppose that \(J = (t_1, t_2]\) or \([t_1, t_2]\) where \(t_1 \lt t_2\) (the \([t_2, t_2]\) case will not be worth considering), and we will talk about the velocity at \(t_2\).
\(J\) is regarded to be the canonical \(C^\infty\) manifold with boundary.
There is the chart, \(((t_3, t_2] \subseteq J, id)\).
Omitting the parallel descriptions (replace \(t_1\) with \(t_2\)), \(d \gamma / d t \vert_{t_2} (f) = d (f' \circ \gamma') / d t \vert_{t_2} = \partial f' / \partial x^k \vert_{\gamma' (t_2)} d \gamma'^k / d t \vert_{t_2} = \partial (f \circ {\phi_{\gamma (t_2)}}^{-1}) / \partial x^k \vert_{\phi_{\gamma (t_2)} (\gamma (t_2))} d (\phi_{\gamma (t_2)} \circ \gamma)^k / d t \vert_{t_2} \), where \(k\) denotes the \(k\)-th component on \(U'_{\phi_{\gamma (t_2)} (\gamma (t_2))}\) with the necessary one-sided or full derivatives.
Furthermore, as \(\partial (f \circ {\phi_{\gamma (t_j)}}^{-1}) / \partial x^k \vert_{\phi_{\gamma (t_j)} (\gamma (t_j))} = \partial / \partial x^k \vert_{\gamma (t_j)} (f)\), \(d (\phi_{\gamma (t_j)} \circ \gamma)^k / d t \vert_{t_j}\) with the necessary one-sided or full derivative is the \(k\)-th component of \(d \gamma / d t \vert_{t_j}\) with respect to the tangent vectors space basis induced by the chart, \((U_{\gamma (t_1)} \subseteq M, \phi_{\gamma (t_1)})\).
