definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds differentiable at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds differentiable at point.
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:
\( \mathbb{R}^{d_1}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( \mathbb{R}^{d_2}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( S_1\): \(\subseteq \mathbb{R}^{d_1}\)
\( S_2\): \(\subseteq \mathbb{R}^{d_2}\)
\( p\): \(\in S_1\)
\(*f\): \(: S_1 \to S_2\)
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Conditions:
\(\exists U'_p \in \{\text{ the open neighborhoods of } p \text{ on } \mathbb{R}^{d_1}\}, \exists f': U'_p \to \mathbb{R}^{d_2} (f' \vert_{U'_p \cap S_1} = f \vert_{U'_p \cap S_1} \land f' \in \{\text{ the maps differential at } p\})\)
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2: Note
As the derivatives of \(f\) cannot be necessarily taken on \(S_1\), we need to introduce \(U'_p\).
\(f' \in \{\text{ the maps differential at } p\}\) is by the definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold differentiable at point.
When \(S_1\) happens to be open on \(\mathbb{R}^{d_1}\), this definition coincides with the definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold differentiable at point, because in that case, if \(f\) satisfies the latter definition, \(U'_p\) and \(f'\) can be taken to be \(S_1\) and \(f\) for the former definition; if \(f\) satisfies the former definition, \(U'_p\) and \(f'\) for the former definition exist, but as \(U'_p \cap S_1\) is an open neighborhood of \(p\) on \(\mathbb{R}^{d_1}\), \(f' \vert_{U'_p \cap S_1}\) is inevitably differentiable at \(p\), and as \(f' \vert_{U'_p \cap S_1} = f \vert_{U'_p \cap S_1}\), \(f \vert_{U'_p \cap S_1}\) is differentiable at \(p\), which implies that \(f\) is differentiable at \(p\).
Although called "differential at \(p\)", the derivatives of \(f\) at \(p\) are not necessarily determined in general, because they may depend on the choice of \(f'\): for example, for \(S_1 = \{0\} \subseteq \mathbb{R}\) and \(f: S_1 \to \mathbb{R}, 0 \mapsto 0\), \(f\) is differentiable at \(0\), because \(f': \mathbb{R} \to \mathbb{R}, r \mapsto a r\), satisfies \(f' (0) = f (0)\) and is differentiable for any \(a \in \mathbb{R}\), but the 1st derivative, \(a\), depends on the choice of \(f'\). Still, the definition of differentiable-ness is well-defined, because it has no intention of claiming the existences of the derivatives in the 1st place.
But typically, \(S_1\) is like \(S_1 = \mathbb{H}^{d_1} \subseteq \mathbb{R}^{d_1}\) as for a \(C^\infty\) manifold with boundary, and in that case, the derivatives are determined independent of the choice of \(f'\), because the derivatives are really determined by \(f\): for example, \(\partial_{d_1} f' \vert_0 = lim_{\delta \to +0} (f (0, ..., 0, \delta) - f (0, ..., 0, 0)) / \delta\). Usually (although not necessarily), differentiable-ness of \(f\) is talked about in such cases.
