468: Map Between Arbitrary Subsets of Euclidean C∞ Manifolds Ck at Point, Where k Excludes 0 and Includes ∞

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A definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

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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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of Euclidean $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of map from open subset of Euclidean $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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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: Definition

For any Euclidean $C^{\mathrm{\infty }}$ 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 $\mathrm{\infty }$ $k$, any map, $f:S_1\to S_2$, such that there are an open neighborhood, $U_p^{\prime }\subseteq {\mathbb{R}}^{d_1}$, of $p$ and a map, $f^{\prime }:U_p^{\prime }\to {\mathbb{R}}^{d_2}$, such that $f^{\prime }{|}_{U_p^{\prime }\cap S_1}=f{|}_{U_p^{\prime }\cap S_1}$ and $f^{\prime }$ is $C^k$ at $p$ by the definition of map from open subset of Euclidean $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

When this definition is satisfied, there is the open subset, $U_p$, cited in the definition of map from open subset of Euclidean $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$, on the whole of which $f^{\prime }$ is $C^k$ and $f^{\prime }{|}_{U_p\cap S_1}=f{|}_{U_p\cap S_1}$, because $U_p\subseteq U_p^{\prime }$.

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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 $f(p)$, $U_{f(p)}=U_{f(p)}^{\prime }\cap S_2$ where $U_{f(p)}^{\prime }\subseteq {\mathbb{R}}^{d_2}$ is open, and as $f^{\prime }:U_p^{\prime }\to {\mathbb{R}}^{d_2}$ is $C^k$ at $p$, $f^{\prime }$ is continuous at $p$, and there is an open neighborhood, $U_p^{″}\subseteq U_p^{\prime }$, such that $f^{\prime }(U_p^{″})\subseteq U_{f(p)}^{\prime }$, but $U_p^{″}\cap S_1\subseteq S_1$ is an open neighborhood of $p$ and $f(U_p^{″}\cap S_1)=f^{\prime }(U_p^{″}\cap S_1)\subseteq U_{f(p)}^{\prime }\cap S_2=U_{f(p)}$.

As the derivatives of $f$ cannot be necessarily taken on $S$, we need to introduce $U_p^{\prime }$.

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 $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$, because in that case, if $f$ satisfies the latter definition, $U_p\subseteq S_1$ for the latter definition exists, and $U_p^{\prime }$ and $f^{\prime }$ can be taken to be $U_p$ and $f{|}_{U_p}$ for the former definition; if $f$ satisfies the former definition, $U_p^{\prime }$ and $f^{\prime }$ for the former definition exist, but as $U_p^{\prime }\cap S_1$ is an open neighborhood of $p$ on ${\mathbb{R}}^{d_1}$, $U_p^{\prime }\cap S_1$ and $f^{\prime }{|}_{U_p^{\prime }\cap S_1}$ can be taken instead, and $U_p$ for the former definition satisfies $U_p\subseteq U_p^{\prime }\cap S_1\subseteq S_1$, 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 $f^{\prime }$: 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 $f^{\prime }:\mathbb{R}\to \mathbb{R}$, $r\mapsto ar$, satisfies $f^{\prime }(0)=f(0)$ and is $C^k$ for any $a\in \mathbb{R}$, but the 1st derivative, $a$, depends on the choice of $f^{\prime }$. 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 $C^{\mathrm{\infty }}$ manifold with boundary, and in that case, the derivatives are determined independent of the choice of $f^{\prime }$, because the derivatives are really determined by $f$: for example, ${\partial }_{d_1}{f}^{\prime }{|}_0=li{m}_{\delta \to +0}(f(0,...,0,\delta )-f(0,...,0,0))/\delta$. Usually (although not necessarily), $C^k$-ness of $f$ is talked about in such a case.

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References

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