description/proof of that for \(C^\infty\) map between \(C^\infty\) manifolds with boundary and corresponding charts, components function of differential w.r.t. standard bases is this
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) map between any \(C^\infty\) manifolds with boundary and any corresponding charts, the components function of the differential of the map with respect to the standard bases 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:
\(M_1\): \(\in \{\text{ the } d_1 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d_2 \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ maps }\}\)
\((U_1 \subseteq M_1, \phi_1)\): \(\in \{\text{ the charts for } M_1\}\)
\((U_2 \subseteq M_2, \phi_2)\): \(\in \{\text{ the charts for } M_2\}\), such that \(f (U_1) \subseteq U_2\)
\(m\): \(\in U_1\)
\(d f_m\): \(: T_mM_1 \to T_{f (m)}M_2\), \(= \text{ the differential }\)
\(\{\partial / \partial x^j \vert j \in \{1, ..., d_1\}\}\): \(= \text{ the standard basis for } T_mM_1\) with respect to \((U_1 \subseteq M_1, \phi_1)\)
\(\{\partial / \partial y^j \vert j \in \{1, ..., d_2\}\}\): \(= \text{ the standard basis for } T_{f (m)}M_2\) with respect to \((U_2 \subseteq M_2, \phi_2)\)
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Statements:
\(\forall v = v^j \partial / \partial x^j \in T_mM_1 (d f_m (v) = w^j \partial / \partial y^j \text{ where } w^j = \partial \hat{f}^j / \partial x^l v^l \text{ where } \hat{f} = \phi_2 \circ f \circ {\phi_1}^{-1})\)
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In other words, \(\hat{f}\) is the components function of \(f\) with respect to \((U_1 \subseteq M_1, \phi_1)\) and \((U_2 \subseteq M_2, \phi_2)\).
2: Proof
Whole Strategy: Step 1: for any \(f' \in C^\infty (M_2)\), compute \(d f_m (v) (f')\) as \(v (f' \circ f)\); Step 2: compare the result of Step 1 with \(d f_m (v) (f') = w^j \partial / \partial y^j (f')\).
Step 1:
Let \(f' \in C^\infty (M_2)\) be any.
\(d f_m (v) (f') = v (f' \circ f) = v^j \partial / \partial x^j (f' \circ f) = v^j \partial_j (f' \circ f \circ {\phi_1}^{-1})\), where \(f' \circ f \circ {\phi_1}^{-1}\) is \(: \phi_1 (U_1) \subseteq \mathbb{R}^{d_1} \to \mathbb{R}\), \(= v^j \partial_j ((f' \circ {\phi_2}^{-1}) \circ (\phi_2 \circ f \circ {\phi_1}^{-1}))\), where \((f' \circ {\phi_2}^{-1}) \circ (\phi_2 \circ f \circ {\phi_1}^{-1})\) is \(: \phi_1 (U_1) \subseteq \mathbb{R}^{d_1} \to \phi_2 (U_2) \subseteq \mathbb{R}^{d_2} \to \mathbb{R}\), and applying the chain rule, \(= v^j \partial_l (f' \circ {\phi_2}^{-1}) \partial_j (\phi_2 \circ f \circ {\phi_1}^{-1})^l = v^j \partial_l (f' \circ {\phi_2}^{-1}) \partial_j \hat{f}^l = \partial_j \hat{f}^l v^j \partial / \partial y^l (f') = \partial_l \hat{f}^j v^l \partial / \partial y^j (f')\), which is just renaming the dummy indexes.
Step 2:
\(d f_m (v) (f') = w^j \partial / \partial y^j (f')\).
So, comparing that with the result of Step 1, \(w^j \partial / \partial y^j (f') = \partial_l \hat{f}^j v^l \partial / \partial y^j (f')\).
As \(f'\) is arbitrary and \(\{\partial / \partial y^1, ..., \partial / \partial y^{d_2}\}\) is a basis, \(w^j = \partial_l \hat{f}^j v^l\).
