# partition of unity

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

 $\sum_{i}\varepsilon_{i}(x)=1\quad\mbox{for all x\in X}.$ (1)

A partition of unity is if each $x$ in $X$ 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 $X$ if each $\varepsilon_{i}$ is zero on the complement of $U_{i}$.

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

 $\int_{M}f(x)\omega=\sum_{i}\int_{U_{i}}\varepsilon_{i}(x)f(x)\omega.$

It is of the choice of partition of unity.

Title partition of unity PartitionOfUnity 2013-03-22 13:29:23 2013-03-22 13:29:23 mhale (572) mhale (572) 10 mhale (572) Definition msc 54D20 msc 58A05 locally finite partition of unity subordinate to an open cover