description/proof of that for \(C^\infty\) manifold with boundary and tangent vectors space at point, transition of components of tangent vector w.r.t. standard bases w.r.t. charts is this
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of tangent vector.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and the tangent vectors space at any point, the transition of the standard bases with respect to any charts is this.
- The reader admits the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and the tangent vector at any point, the transition of the components of any tangent vector with respect to the standard bases with respect to any charts is this.
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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(m\): \(\in M\)
\((U_m \subseteq M, \phi_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\((U'_m \subseteq M, \phi'_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\(\{\partial / \partial x^j \vert j \in \{1, ..., d\}\}\): \(= \text{ the standard basis for } T_mM\) by \((U_m \subseteq M, \phi_m)\)
\(\{\partial / \partial x'^j \vert j \in \{1, ..., d\}\}\): \(= \text{ the standard basis for } T_mM\) by \((U'_m \subseteq M, \phi'_m)\)
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Statements:
\(\forall v = v^j \partial / \partial x^j = v'^j \partial / \partial x'^j \in T_mM (v'^j = \partial x'^j / \partial x^l v^l)\)
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\(x'\) as a function of \(x\) is \(\phi'_m \circ {\phi_m}^{-1} \vert_{\phi_m (U_m \cap U'_m)}: \phi_m (U_m \cap U'_m) \to \phi'_m (U_m \cap U'_m)\).
2: Proof
Whole Strategy: Step 1: see that \(\partial / \partial x'^j = \partial x^l / \partial x'^j \partial / \partial x^l\); Step 2: apply the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this.
Step 1:
\(\partial / \partial x'^j = \partial x^l / \partial x'^j \partial / \partial x^l\), by the proposition that for any \(C^\infty\) manifold with boundary and the tangent vectors space at any point, the transition of the standard bases with respect to any charts is this.
Step 2:
By the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this, \(v'^j = \partial x'^j / \partial x^l v^l\), because the inverse of the matrix, \(\begin{pmatrix} \partial x^l / \partial x'^j \end{pmatrix}\), is \(\begin{pmatrix} \partial x'^m / \partial x^n \end{pmatrix}\), because \(\partial x'^m / \partial x^l x^l / \partial x'^j = \partial x'^m / \partial x'^j = \delta^m_j\).
