description/proof of that for \(C^\infty\) manifold with boundary and \((p, q)\)-tensors space at point, transition of components of tensor 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 (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 \((p, q)\)-tensors space at any point, the transition of the standard bases with respect to any charts is this.
- The reader admits the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses.
- 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 \((p, q)\)-tensors space at any point, the transition of the components of any tensor 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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(m\): \(\in M\)
\(p\): \(\in \mathbb{N}\)
\(q\): \(\in \mathbb{N}\)
\(T^p_q (T_mM)\): \(= \text{ the } (p, q) \text{ -tensors space at } m\)
\((U_m \subseteq M, \phi_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\((U'_m \subseteq M, \phi'_m)\): \(\in \{\text{ the charts for } M \text{ around } m\}\)
\(B\): \(= \text{ the standard basis for } T^p_q (T_mM)\) 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 \((U'_m \subseteq M, \phi'_m)\), \(= \{[((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))]\}\)
//
Statements:
\(\forall t = t^{m_1, ..., m_p}_{n_1, ..., n_q} [((\partial / \partial x^{m_1}, ..., \partial / \partial x^{m_p}, d x^{n_1}, ..., d x^{n_q}))] = t'^{j_1, ..., j_p}_{l_1, ..., l_q} [((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))]\)
(
\(t'^{j_1, ..., j_p}_{l_1, ..., l_q} = \partial x'^{j_1} / \partial x^{m_1} ... \partial x'^{j_p} / \partial x^{m_p} \partial x^{n_1} / \partial x'^{l_1} ... \partial x^{n_q} / \partial x'^{l_q} t^{m_1, ..., m_p}_{n_1, ..., n_q}\)
)
//
\(x\) 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 \((p, q)\)-tensors space at any point, the transition of the standard bases with respect to any charts is this, the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses, and 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:
By 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, \([((\partial / \partial x'^{j_1}, ..., \partial / \partial x'^{j_p}, d x'^{l_1}, ..., d x'^{l_q}))] = \partial x^{m_1} / \partial x'^{j_1} ... \partial x^{m_p} / \partial x'^{j_p} \partial x'^{l_1} / \partial x^{n_1} ... \partial x'^{l_q} / \partial x^{n_q} [((\partial / \partial x^{m_1}, ..., \partial / \partial x^{m_p}, d x^{n_1}, ..., d x^{n_q}))]\).
By the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses, that is a transition by a square matrix and the inverse matrix is \(\partial x'^{j_1} / \partial x^{m_1} ... \partial x'^{j_p} / \partial x^{m_p} \partial x^{n_1} / \partial x'^{l_1} ... \partial x^{n_q} / \partial x'^{l_q}\).
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, \(t'^{j_1, ..., j_p}_{l_1, ..., l_q} = \partial x'^{j_1} / \partial x^{m_1} ... \partial x'^{j_p} / \partial x^{m_p} \partial x^{n_1} / \partial x'^{l_1} ... \partial x^{n_q} / \partial x'^{l_q} t^{m_1, ..., m_p}_{n_1, ..., n_q}\).
