502: Function over C∞ Manifold with Boundary

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definition of function over $C^{\mathrm{\infty }}$ manifold with boundary

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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: Structured Description
• 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 Euclidean $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of map.

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Target Context

• The reader will have a definition of function over $C^{\mathrm{\infty }}$ manifold with boundary.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\mathbb{R}$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold }$
$\ast f$: $:M\to \mathbb{R}$
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Conditions:
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2: Note

'function over $M$' is nothing but '$0$-form over $M$'.

The term, 'function', has the general meaning that means any relation such that each element of the domain has the unique corresponding element in the range, but just 'function' in the manifold-context usually means the definition by this article, not just a general function.

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References

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