definition of standard basis for tangent vectors space at point on \(C^\infty\) manifold with boundary w.r.t. chart
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vectors space at point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of Euclidean \(C^\infty\) manifold.
- The reader knows a definition of closed upper half Euclidean \(C^\infty\) manifold with boundary.
- The reader knows a definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
Target Context
- The reader will have a definition of standard basis for tangent vectors space at point on \(C^\infty\) manifold with boundary with respect to chart.
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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( (U \subseteq M, \phi)\): \(\in \{\text{ the charts }\}\)
\( m\): \(\in U\)
\( T_mM\): \(= \text{ the tangent vectors space at } m\)
\( \phi (U)\): \(= \text{ the open submanifold with boundary of } \mathbb{R}^d \text{ or } \mathbb{H}^d\)
\( \{\partial_j \vert_{\phi (m)} \vert j \in \{1, ..., d\}\}\): \(= \text{ the standard basis for } T_{\phi (m)}\phi (U)\)
\(*\{\partial / \partial x^j \vert_m := d \phi^{-1} \vert_{\phi (m)} (\partial_j \vert_{\phi (m)}) \vert j \in \{1, ..., d\}\}\), \(\in \{\text{ the bases for } T_mM\}\)
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Conditions:
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2: Note
\(\{\partial_j \vert_{\phi (m)} \vert j \in \{1, ..., d\}\}\) is indeed a basis for \(T_{\phi (m)}\phi (U)\), by the proposition that for any open subset of \(\mathbb{R}^d\) or \(\mathbb{H}^d\), there is the standard basis for the tangent vectors space at each point on the open subset, where \(\partial_j \vert_{\phi (m)}\) is the partial derivative at \(\phi (m)\) by the \(j\)-th component.
By the proposition that for any diffeomorphism from any \(C^\infty\) manifold with boundary onto any neighborhood of any point image on any \(C^\infty\) manifold with boundary, the differential of the diffeomorphism or any its codomain extension at the point is a 'vectors spaces - linear morphisms' isomorphism, \(\{\partial / \partial x^j \vert_m := d \phi^{-1} \vert_{\phi (m)} (\partial_j \vert_{\phi (m)}) \vert j \in \{1, ..., d\}\}\) is a basis for \(T_mM\), because \(\phi^{-1}: \phi (U) \to U\) is a diffeomorphism.
While \(\partial / \partial x^j \vert_m\) deceptively looks like the partial derivative, it is not so because \(\partial / \partial x^j \vert_m \in T_mM\) and it operates on a function, \(f: M \to \mathbb{R}\), not on any function from \(\mathbb{R}^d\) or its open subset.
According to the definition of differential, \(\partial / \partial x^j \vert_m (f) = \partial_j \vert_{\phi (m)} (f \circ \phi^{-1})\) where \(f \circ \phi^{-1}: \phi (U) \to \mathbb{R}\).
