A definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
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 \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
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: Definition
For any Euclidean \(C^\infty\) manifolds, \(\mathbb{R}^{d_1}, \mathbb{R}^{d_2}\), any subsets, \(S_1 \subseteq \mathbb{R}^{d_1}, S_2 \subseteq \mathbb{R}^{d_2}\), any point, \(p \in S\), and any natural number (excluding 0) or \(\infty\) \(k\), any map, \(f: S_1 \to S_2\), such that there are an open neighborhood, \(U'_p \subseteq \mathbb{R}^{d_1}\), of \(p\) and a map, \(f': U'_p \to \mathbb{R}^{d_2}\), such that \(f' \vert_{U'_p \cap S_1} = f \vert_{U'_p \cap S_1}\) and \(f'\) is \(C^k\) at \(p\) by the definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
When this definition is satisfied, there is the open subset, \(U_p\), cited in the definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), on the whole of which \(f'\) is \(C^k\) and \(f' \vert_{U_p \cap S_1} = f \vert_{U_p \cap S_1}\), because \(U_p \subseteq U'_p\).
2: Note
\(k = 0\) is excluded because that case has been already defined as map continuous at point.
But when \(f\) is \(C^k\) at \(p\) where \(1 \le k\), \(f\) is \(C^0\) at \(p\): for any open neighborhood, \(U_{f (p)} \subseteq S_2\), of \(f (p)\), \(U_{f (p)} = U'_{f (p)} \cap S_2\) where \(U'_{f (p)} \subseteq \mathbb{R}^{d_2}\) is open, and as \(f': U'_p \to \mathbb{R}^{d_2}\) is \(C^k\) at \(p\), \(f'\) is continuous at \(p\), and there is an open neighborhood, \(U''_p \subseteq U'_p\), such that \(f' (U''_p) \subseteq U'_{f (p)}\), but \(U''_p \cap S_1 \subseteq S_1\) is an open neighborhood of \(p\) and \(f (U''_p \cap S_1) = f' (U''_p \cap S_1) \subseteq U'_{f (p)} \cap S_2 = U_{f (p)}\).
As the derivatives of \(f\) cannot be necessarily taken on \(S\), we need to introduce \(U'_p\).
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 \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\), because in that case, if \(f\) satisfies the latter definition, \(U_p \subseteq S_1\) for the latter definition exists, and \(U'_p\) and \(f'\) can be taken to be \(U_p\) and \(f \vert_{U_p}\) 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 \(\mathbb{R}^{d_1}\), \(U'_p \cap S_1\) and \(f' \vert_{U'_p \cap S_1}\) can be taken instead, and \(U_p\) for the former definition satisfies \(U_p \subseteq U'_p \cap S_1 \subseteq S_1\), and so, \(U_p\) can be taken for the latter definition.
Although called "\(C^k\) 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 \(C^k\) at \(0\) for any \(k\), because \(f': \mathbb{R} \to \mathbb{R}\), \(r \mapsto a r\), satisfies \(f' (0) = f (0)\) and is \(C^k\) for any \(a \in \mathbb{R}\), but the 1st derivative, \(a\), depends on the choice of \(f'\). Still, the definition of \(C^k\)-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), \(C^k\)-ness of \(f\) is talked about in such a case.
