A description/proof of that for maps between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, composition is \(C^k\) at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader admits the proposition that for any maps between any topological spaces continuous at corresponding points, the composition is continuous at the point.
Target Context
- The reader will have a description and a proof of the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the 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: Description
For any Euclidean \(C^\infty\) manifolds, \(\mathbb{R}^{d_1}, \mathbb{R}^{d_2}, \mathbb{R}^{d_3}\), any subsets, \(S_1 \subseteq \mathbb{R}^{d_1}, S_2, S'_2 \subseteq \mathbb{R}^{d_2}, S_3 \subseteq \mathbb{R}^{d_3}\), such that \(S_2 \subseteq S'_2\), any point, \(p \in S_1\), any natural number (including 0) or \(\infty\) \(k\), and any maps, \(f_1: S_1 \to S_2, f_2: S'_2 \to S_3\), such that \(f_1\) and \(f_2\) are \(C^k\) at \(p\) and \(f_1 (p)\), \(f_2 \circ f_1: S_1 \to S_3\) is \(C^k\) at \(p\).
2: Proof
Let us suppose that \(k = 0\).
\(f_2 \circ f_1\) is continuous at \(p\), by the proposition that for any maps between any topological spaces continuous at corresponding points, the composition is continuous at the point.
Let us suppose that \(1 \le k\) including \(\infty\).
There are an open neighborhood, \(U'_{f_1 (p)} \subseteq \mathbb{R}^{d_2}\), of \(f_1 (p)\) and a map, \(f'_2: U'_{f_1 (p)} \to \mathbb{R}^{d_3}\), \(C^k\) at \(f_1 (p)\) such that \(f'_2 \vert_{U'_{f_1 (p)} \cap S'_2} = f_2 \vert_{U'_{f_1 (p)} \cap S'_2}\). There are an open neighborhood, \(U'_p \subseteq \mathbb{R}^{d_1}\), of \(p\) and a map, \(f'_1: U'_p \to \mathbb{R}^{d_2}\), \(C^k\) at \(p\) such that \(f'_1 \vert_{U'_p \cap S_1} = f_1 \vert_{U'_p \cap S_1}\). As \(f'_1\) is continuous at \(p\) (see Note of the definition), there is an open neighborhood, \(U''_p \subseteq U'_p\), of \(p\) such that \(f'_1 (U''_p) \subseteq U'_{f_1 (p)}\). \(f'_2 \circ f'_1 \vert_{U''_p}: U''_p \to \mathbb{R}^{d_3}\) is \(C^k\) at \(p\) as a usual composition of maps between open subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points. \(f'_2 \circ f'_1 \vert_{U''_p} \vert_{U''_p \cap S_1} = f'_2 \circ f_1 \vert_{U''_p \cap S_1} = f_2 \circ f_1 \vert_{U''_p \cap S_1}\), because \(f_1 (U''_p \cap S_1) \subseteq U'_{f_1 (p)} \cap S_2 \subseteq U'_{f_1 (p)} \cap S'_2\).
