57: Germ of Ck Functions at Point

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A definition of germ of $C^k$ functions at point, $C_p^k(M)$

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Topics

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

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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 $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of function over $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of germ of $C^k$ functions at point, $C_p^k(T)$.

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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 $C^{\mathrm{\infty }}$ manifold with (possibly empty) boundary, $M$, and any point, $p\in M$, the equivalence class of $\{U_{p,\alpha },f_{\alpha }\}$ where $U_{p,\alpha }$ is any neighborhood of $p$ and $f_{\alpha }$ is any $C^k$ function, $f_{\alpha }:U_{p,\alpha }\to \mathbb{R}$, such that $(U_{p,{\alpha }_1},f_{{\alpha }_1})\sim (U_{p,{\alpha }_2},f_{{\alpha }_2})$ if and only if there is a neighborhood, $U_{p,{\alpha }_3}$, such that $U_{p,{\alpha }_3}\subseteq U_{p,{\alpha }_1}\cap U_{p,{\alpha }_2}$ and $f_{{\alpha }_1}=f_{{\alpha }_2}$ on $U_{p,{\alpha }_3}$

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2: Note

Strictly speaking, $C^{\mathrm{\infty }}$ manifold with boundary is not required for $k\ne \mathrm{\infty }$, but just topological manifold with boundary does not suffice, because $C^k$-ness is not defined there.

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References

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