58: Derivation at Point of Ck Functions

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A definition of derivation at point of $C^k$ functions

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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 germ of $C^k$ functions at point.

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Target Context

• The reader will have a definition of derivation at point of $C^k$ functions.

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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, $p\in M$, any $\mathbb{R}$-linear map, $D_v:C_p^k(M)\to \mathbb{R}$, that satisfies the Leibniz rule, $D_v(f_1{f}_2)=D_v(f_1)f_2(p)+f_1(p)D_v(f_2)$

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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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