description/proof of that for \(C^\infty\) manifold with boundary and \(q\)-covectors space at point, transition of 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 \(q\)-covectors space at point on \(C^\infty\) manifold with boundary.
- 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 \(q\)-covectors space, the transition of the standard bases with respect to any bases for the vectors space is this.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, on the intersection of any 2 charts, the looks-like-chain-rule-for-partial-derivative-of-composition-of-transitions-of-coordinates rule holds.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary and the \(q\)-covectors space at any point, the transition of 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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(m\): \(\in M\)
\(q\): \(\in \mathbb{N}\)
\(\Lambda_q (T_mM)\): \(= \text{ the } q \text{ -covectors space at } m\)
\((U_m \subseteq M, \phi_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\((U'_m \subseteq M, \phi'_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\(B\): \(= \text{ the standard basis for } \Lambda_q (T_mM)\) with respect to \((U_m \subseteq M, \phi_m)\), \(= \{d x^{l_1} \wedge ... \wedge d x^{l_q} \vert \forall l \in \{1, ..., q\} (1 \le j_l \le d) \land j_1 \lt ... \lt j_q\}\)
\(B'\): \(= \text{ the standard basis for } \Lambda_q (T_mM)\) with respect to \((U'_m \subseteq M, \phi'_m)\), \(= \{d x'^{l_1} \wedge ... \wedge d x'^{l_q} \vert \forall l \in \{1, ..., q\} (1 \le j_l \le d) \land j_1 \lt ... \lt j_q\}\)
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Statements:
\(d x'^{j_1} \wedge ... \wedge d x'^{j_q} = \sum_{(l_1, ..., l_q)} \sum_{\sigma} sgn \sigma \partial x'^{j_1} / \partial x^{l_{\sigma_1}} ... \partial x'^{j_q} / \partial x^{l_{\sigma_q}} d x^{l_1} \wedge ... \wedge d x^{l_q}\)
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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: apply 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 and the proposition that for any \(q\)-covectors space, the transition of the standard bases with respect to any bases for the vectors space is this.
Step 1:
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, \(\partial / \partial x'^j = \partial x^m / \partial x'^j \partial / \partial x^m\).
By the proposition that for any \(q\)-covectors space, the transition of the standard bases with respect to any bases for the vectors space is this, \(d x'^{j_1} \wedge ... \wedge d x'^{j_q} = \sum_{(l_1, ..., l_q)} \sum_{\sigma} sgn \sigma \partial x'^{j_1} / \partial x^{l_{\sigma_1}} ... \partial x'^{j_q} / \partial x^{l_{\sigma_q}} d x^{l_1} \wedge ... \wedge d x^{l_q}\): the inverse of the matrix, \(\begin{pmatrix} \partial x^m / \partial x'^j \end{pmatrix}\), is \(\begin{pmatrix} \partial x'^n / \partial x^l \end{pmatrix}\), by the proposition that for any \(C^\infty\) manifold with boundary, on the intersection of any 2 charts, the looks-like-chain-rule-for-partial-derivative-of-composition-of-transitions-of-coordinates rule holds.
