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 \(\infty\)
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of Euclidean \(C^\infty\) manifold.
- The reader knows a definition of map.
Target Context
- The reader will have a 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 \(\infty\).
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:
\( \mathbb{R}^{d_1}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( \mathbb{R}^{d_2}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( U\): \(\in \{\text{ the open subsets of } \mathbb{R}^{d_1}\}\)
\( S\): \(\in \{\text{ the subsets of } \mathbb{R}^{d_2}\}\)
\( p\): \(\in U\)
\( k\): \(\in (\mathbb{N} \setminus \{0\}) \cup \{\infty\}\)
\(*f\): \(: U \to S\)
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Conditions:
\(\exists U_p \in \{\text{ the open neighborhoods of } p \text{ on } \mathbb{R}^{d_1}\} \text{ such that } U_p \subseteq U (f \text{ 's all the up-to-} k \text{-th partial derivatives (when } k = \infty \text{, that means all the partial derivatives of all the orders) exist on } U_p \land \text{ the derivative maps as the maps from } U_p \text { into } \mathbb{R}^{d_2} \text {are continuous on } U_p)\)
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2: Natural Language Description
For any Euclidean \(C^\infty\) manifolds, \(\mathbb{R}^{d_1}, \mathbb{R}^{d_2}\), any open subset, \(U \subseteq \mathbb{R}^{d_1}\), any subset, \(S \subseteq \mathbb{R}^{d_2}\), any point, \(p \in U\), and any natural number (excluding 0) or \(\infty\) \(k\), any map, \(f: U \to S\), such that there is an open neighborhood, \(U_p \subseteq \mathbb{R}^{d_1}\), of \(p\) such that \(U_p \subseteq U\) and \(f\)'s all the up-to-\(k\)-th partial derivatives (when \(k = \infty\), that means all the partial derivatives of all the orders) exist on \(U_p\) and the derivative maps as the maps from \(U_p\) into \(\mathbb{R}^{d_2}\) are continuous on \(U_p\)
3: 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, \(U_{f (p)} \subseteq S\), of \(f (p)\), \(U_{f (p)} = U'_{f (p)} \cap S\) where \(U'_{f (p)} \subseteq \mathbb{R}^{d_2}\) is open, and as \(f \vert_{U_p}: U_p \to \mathbb{R}^{d_2}\) is \(C^k\), \(f \vert_{U_p}\) is continuous at \(p\), and there is an open neighborhood, \(U'_p \subseteq U_p \subseteq U\), such that \(f \vert_{U_p} (U'_p) \subseteq U'_{f (p)}\), but \(f (U'_p) \subseteq U'_{f (p)} \cap S = U_{f (p)}\).
\(f\) is required to be from an open subset of \(\mathbb{R}^n\) in order for the derivatives of \(f\) to exist at \(p\).
There is a definition for the case that the domain is an arbitrary subset of \(\mathbb{R}^n\).
Being called "\(C^k\) at \(p\)", the derivatives are required to exist on the whole \(U_p\), because in addition to that the up-to-\(k - 1\)-th partial derivatives have to exist on a neighborhood of \(p\) in order for the \(k\)-th partial derivatives to exist, judging the continuousness of the derivative maps as the maps from \(\{p\}\) is meaningless as being vacuously continuous as the constant maps.
There can be a weaker definition that requires the continuousnesses only at \(p\), which we have not adopted, because we do not particularly need that "continuous only at \(p\)" case.
\(S\) does not need to be open on \(\mathbb{R}^{d_2}\), because taking the derivatives or judging the continuousnesses does not require \(S\) to be open.
For the \(k = \infty\) case, the definition requires that a common \(U_p\) exists for all the \(k\)-th partial derivatives, while there can be a weaker definition that requires that a \(U_{p, k}\) exists for each \(k\), which is at least directly different from our definition, because \(\cap_k U_{p, k}\) may not be open unless proved otherwise.
When \(f\) is \(C^k\) at each \(p \in U\), \(U_p\) can be taken to be \(U\): while \(U = \cup_{p \in U} U_p\), for each \(p \in U_{p'} \cap U_{p''}\), each \(j\)-th partial derivative at \(p\) on \(U_{p'}\) equals that on \(U_{p''}\), and so, there is the derivative map from \(U\) into \(\mathbb{R}^{d_2}\) as the compound of the derivative maps from \(U_p\) s (means that each domain restriction of the derivative map from \(U\) is the derivative map from \(U_p\)), which (the derivative map from \(U\)) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.
