PlanetMath: partition of unity

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partition of unity

(Definition)

Let be a topological space. A partition of unity is a collection of continuous functions $\{\varepsilon_i \colon X \to [0,1]\}$ such that

$\displaystyle \sum_i \varepsilon_i(x) = 1$ $\displaystyle \mbox{for all $x \in X$}$ $\displaystyle .$

(1)

A partition of unity is locally finite if each in is contained in an open set on which only a finite number of $\varepsilon_i$ are non-zero. That is, if the cover $\{\varepsilon_i^{-1}((0,1])\}$ is locally finite.

A partition of unity is subordinate to an open cover $\{U_i\}$ of if each $\varepsilon_i$ is zero on the complement of .

Example 1 (Circle)
A partition of unity for $\mathbb{S}^1$ is given by $\{ \sin^2(\theta/2), \cos^2(\theta/2) \}$ subordinate to the covering $\{ (0, 2\pi), (-\pi, \pi) \}$ .

Application to integration

Let be an orientable manifold with volume form $\omega$ and a partition of unity $\{\varepsilon_i(x)\}$ . Then, the integral of a function over is given by

$\displaystyle \int_M f(x) \omega = \sum_i \int_{U_i} \varepsilon_i(x) f(x) \omega.$

It is independent of the choice of partition of unity.

"partition of unity" is owned by mhale.

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Also defines:

locally finite partition of unity, subordinate to an open cover

Attachments:: existence of partitions of unity (Theorem) by asteroid

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Cross-references: function, integral, volume form, orientable manifold, application, covering, complement, open cover, locally finite, cover, number, open set, contained, finite, continuous functions, collection, unity, topological space
There are 7 references to this entry.

This is version 7 of partition of unity, born on 2003-02-26, modified 2006-09-08.
Object id is 4063, canonical name is PartitionOfUnity.
Accessed 7786 times total.

Classification:

AMS MSC:	58A05 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differentiable manifolds, foundations)
	54D20 (General topology :: Fairly general properties :: Noncompact covering properties )

Pending Errata and Addenda

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