471: For Maps Between Arbitrary Subsets of Euclidean C∞ Manifolds Ck at Corresponding Points, Composition Is Ck at Point

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A description/proof of that for maps between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at corresponding points, composition is $C^k$ at point

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.
• The reader admits the proposition that for any maps between any topological spaces continuous at corresponding points, the composition is continuous at the point.

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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 $C^{\mathrm{\infty }}$ manifolds $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

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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.

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Main Body

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1: Description

For any Euclidean $C^{\mathrm{\infty }}$ manifolds, ${\mathbb{R}}^{d_1},{\mathbb{R}}^{d_2},{\mathbb{R}}^{d_3}$, any subsets, $S_1\subseteq {\mathbb{R}}^{d_1},S_2,S_2^{\prime }\subseteq {\mathbb{R}}^{d_2},S_3\subseteq {\mathbb{R}}^{d_3}$, such that $S_2\subseteq S_2^{\prime }$, any point, $p\in S_1$, any natural number (including 0) or $\mathrm{\infty }$ $k$, and any maps, $f_1:S_1\to S_2,f_2:S_2^{\prime }\to S_3$, 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 $p$.

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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 $\mathrm{\infty }$.

There are an open neighborhood, $U_{f_1(p)}^{\prime }\subseteq {\mathbb{R}}^{d_2}$, of $f_1(p)$ and a map, $f_2^{\prime }:U_{f_1(p)}^{\prime }\to {\mathbb{R}}^{d_3}$, $C^k$ at $f_1(p)$ such that $f_2^{\prime }{|}_{U_{f_1(p)}^{\prime }\cap S_2^{\prime }}=f_2{|}_{U_{f_1(p)}^{\prime }\cap S_2^{\prime }}$. There are an open neighborhood, $U_p^{\prime }\subseteq {\mathbb{R}}^{d_1}$, of $p$ and a map, $f_1^{\prime }:U_p^{\prime }\to {\mathbb{R}}^{d_2}$, $C^k$ at $p$ such that $f_1^{\prime }{|}_{U_p^{\prime }\cap S_1}=f_1{|}_{U_p^{\prime }\cap S_1}$. As $f_1^{\prime }$ is continuous at $p$ (see Note of the definition), there is an open neighborhood, $U_p^{″}\subseteq U_p^{\prime }$, of $p$ such that $f_1^{\prime }(U_p^{″})\subseteq U_{f_1(p)}^{\prime }$. $f_2^{\prime }\circ f_1^{\prime }{|}_{U_p^{″}}:U_p^{″}\to {\mathbb{R}}^{d_3}$ is $C^k$ at $p$ as a usual composition of maps between open subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at corresponding points. $f_2^{\prime }\circ f_1^{\prime }{|}_{U_p^{″}}{|}_{U_p^{″}\cap S_1}=f_2^{\prime }\circ f_1{|}_{U_p^{″}\cap S_1}=f_2\circ f_1{|}_{U_p^{″}\cap S_1}$, because $f_1(U_p^{″}\cap S_1)\subseteq U_{f_1(p)}^{\prime }\cap S_2\subseteq U_{f_1(p)}^{\prime }\cap S_2^{\prime }$.

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References

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