description/proof of that for \(C^\infty\) manifold with boundary and \((p, q)\)-tensors 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 \((p, q)\)-tensors 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 finite-dimensional vectors space, the transition of the dual bases for the covectors space with respect to any bases for the vectors space is this.
- The reader admits the proposition that for the tensor product of any \(k\) finite-dimensional vectors spaces over any field, the transition of the standard bases with respect to any bases for the vectors spaces 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 \((p, q)\)-tensors 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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(m\): \(\in M\)
\(p\): \(\in \mathbb{N}\)
\(q\): \(\in \mathbb{N}\)
\(T^p_q (T_mM)\): \(= \text{ the } (p, q) \text{ -tensors space at } m\)
\((U_m \subseteq M, \phi_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\((U'_m \subseteq M, \phi'_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\(B\): \(= \text{ the standard basis for } T^p_q (T_mM)\) with respect to \((U_m \subseteq M, \phi_m)\), \(= \{[((\partial / \partial x^{j_1}, ..., \partial / \partial x^{j_p}, d x^{l_1}, ..., d x^{l_q}))]\}\)
\(B'\): \(= \text{ the standard basis for } T^p_q (T_mM)\) with respect to \((U'_m \subseteq M, \phi'_m)\), \(= \{[((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))]\}\)
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Statements:
\([((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))] = \partial x^{m_1} / \partial x'^{j_1} ... \partial x^{m_p} / \partial x'^{j_p} \partial x'^{l_1} / \partial x^{n_1} ... \partial x'^{l_q} / \partial x^{n_q} [((\partial / \partial x^{m_1}, ..., \partial / \partial x^{m_p}, d x^{n_1}, ..., d x^{n_q}))]\)
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\(x\) as a function of \(x'\) is \(\phi_m \circ {\phi'_m}^{-1} \vert_{\phi'_m (U_m \cap U'_m)}: \phi'_m (U_m \cap U'_m) \to \phi_m (U_m \cap U'_m)\); \(x'\) 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, the proposition that for any finite-dimensional vectors space, the transition of the dual bases for the covectors space with respect to any bases for the vectors space is this, and the proposition that for the tensor product of any \(k\) finite-dimensional vectors spaces over any field, the transition of the standard bases with respect to any bases for the vectors spaces 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 finite-dimensional vectors space, the transition of the dual bases for the covectors space with respect to any bases for the vectors space is this, \(d x'^l = \partial x'^l / \partial x^n d x^n\).
By the proposition that for the tensor product of any \(k\) finite-dimensional vectors spaces over any field, the transition of the standard bases with respect to any bases for the vectors spaces is this, \([((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))] = \partial x^{m_1} / \partial x'^{j_1} ... \partial x^{m_p} / \partial x'^{j_p} \partial x'^{l_1} / \partial x^{n_1} ... \partial x'^{l_q} / \partial x^{n_q} [((\partial / \partial x^{m_1}, ..., \partial / \partial x^{m_p}, d x^{n_1}, ..., d x^{n_q}))]\).
