489: Differential of C∞ Map Between C∞ Manifolds with Boundary at Point

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A definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary 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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.
• The reader knows a definition of tangent vector.

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Target Context

• The reader will have a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at 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: Definition

For any $C^{\mathrm{\infty }}$ manifolds with boundary, $M_1,M_2$, any $C^{\mathrm{\infty }}$ map, $f:M_1\to M_2$, and any point, $p\in M_1$, the map, $d{f}_p:T_p{M}_1\to T_{f(p)}{M}_2$, $v\mapsto d{f}_p(v)$, such that for any $f^{\prime }\in C^{\mathrm{\infty }}(M_2)$, $d{f}_p(v)(f^{\prime })=v(f^{\prime }\circ f)$

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2: Note

$d{f}_p(v)$ is indeed a derivation, because for any $r\in \mathbb{R}$, $d{f}_p(v)(r{f}^{\prime })=v(r{f}^{\prime }\circ f)=rv(f^{\prime }\circ f)=rd{f}_p(v)(f^{\prime })$, being $\mathbb{R}$ linear; $d{f}_p(v)(f_1^{\prime }{f}_2^{\prime })=v((f_1^{\prime }{f}_2^{\prime })\circ f)=v((f_1^{\prime }\circ f)(f_2^{\prime }\circ f))=v(f_1^{\prime }\circ f)(f_2^{\prime }\circ f(p))+(f_1^{\prime }\circ f(p))v(f_2^{\prime }\circ f)=d{f}_p(v)(f_1^{\prime })f_2^{\prime }(f(p))+f_1^{\prime }(f(p))d{f}_p(v)(f_2^{\prime })$, satisfying the Leibniz rule.

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References

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