partition of unity

Let X be a topological spaceMathworldPlanetmath. A partition of unityMathworldPlanetmath is a collectionMathworldPlanetmath of continuous functionsMathworldPlanetmathPlanetmath {εi:X[0,1]} such that

iεi(x)=1for all xX. (1)

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

A partition of unity is subordinate to an open cover {Ui} of X if each εi is zero on the complement of Ui.

Example 1 (Circle)

A partition of unity for S1 is given by {sin2(θ/2),cos2(θ/2)} subordinate to the covering {(0,2π),(-π,π)}.

Application to integration

Let M be an orientable manifoldMathworldPlanetmath with volume form ω and a partition of unity {εi(x)}. Then, the integral of a function f(x) over M is given by


It is of the choice of partition of unity.

Title partition of unity
Canonical name PartitionOfUnity
Date of creation 2013-03-22 13:29:23
Last modified on 2013-03-22 13:29:23
Owner mhale (572)
Last modified by mhale (572)
Numerical id 10
Author mhale (572)
Entry type Definition
Classification msc 54D20
Classification msc 58A05
Defines locally finite partition of unity
Defines subordinate to an open cover