description/proof of that for \(C^\infty\) manifold with boundary, on intersection of 2 charts, looks-like-chain-rule-for-partial-derivative-of--composition-of-transitions-of-coordinates rule holds
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 \(C^\infty\) 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.
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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\((U \subseteq M, \phi)\): \(\in \{\text{ the charts }\}\)
\((U' \subseteq M, \phi')\): \(\in \{\text{ the charts }\}\)
\(m\): \(\in U \cap U'\)
\(\{\partial / \partial x^j \vert_m \vert j \in \{1, ..., d\}\}\): \(= \text{ the standard basis for } T_mM \text{ with respect to } U\)
\(\{\partial / \partial x'^j \vert_m \vert j \in \{1, ..., d\}\}\): \(= \text{ the standard basis for } T_mM \text{ with respect to } U'\)
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Statements:
\(\partial x^j / \partial x^l = \partial x^j / \partial x'^m \partial x'^m / \partial x^l = \delta^j_l\)
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2: Note
It deceptively looks like the chain rule of partial derivative of \(x \mapsto x' \mapsto x\), but each of \(\partial / \partial x^l\), \(\partial / \partial x'^m\), and \(\partial / \partial x^l\) is not any partial derivative.
3: Proof
Whole Strategy: Step 1: see that \(\partial x^j / \partial x^l = \partial_l (\phi^j \circ {\phi}^{-1})\); Step 2: apply the chain rule to \(\partial_l (\phi^j \circ {\phi}^{-1}) = \partial_l (\phi^j \circ {\phi'}^{-1} \circ \phi' \circ {\phi}^{-1})\).
Step 1:
\(x^j\) is the function, \(: U \to \mathbb{R}\), \(= \phi^j\).
\(\partial x^j / \partial x^l = \partial / \partial x^l (x^j) = \partial_l (\phi^j \circ {\phi}^{-1})\), by the definition of standard basis for tangent vectors space at point on \(C^\infty\) manifold with boundary with respect to chart.
\(= \delta^j_l\).
Step 2:
On \(\phi (U \cap U')\), \(\phi^j \circ {\phi}^{-1}: \phi (U \cap U') \to \mathbb{R} = \phi^j \circ ({\phi'}^{-1} \circ \phi') \circ {\phi}^{-1} = (\phi^j \circ {\phi'}^{-1}) \circ (\phi' \circ {\phi}^{-1}): \phi (U \cap U') \to \phi' (U \cap U') \to \mathbb{R}\).
Applying the chain rule to it, \(\partial_l (\phi^j \circ {\phi}^{-1}) = \partial_l ((\phi^j \circ {\phi'}^{-1}) \circ (\phi' \circ {\phi}^{-1})) = \partial_m (\phi^j \circ {\phi'}^{-1}) \partial_l (\phi'^m \circ {\phi}^{-1}) = \partial / \partial x'^m (x^j) \partial / \partial x^l (x'^m) = \partial x^j / \partial x'^m \partial x'^m / \partial x^l\), by the definition of standard basis for tangent vectors space at point on \(C^\infty\) manifold with boundary with respect to chart.
So, \(\partial x^j / \partial x^l = \partial x^j / \partial x'^m \partial x'^m / \partial x^l\).
