# partition of unity

Let $X$ be a topological space^{}.
A partition of unity^{} is a collection^{} of continuous functions^{} $\{{\epsilon}_{i}:X\to [0,1]\}$ such that

$$\sum _{i}{\epsilon}_{i}(x)=1\mathit{\hspace{1em}}\text{for all}x\in X.$$ | (1) |

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

A partition of unity is subordinate to an open cover $\{{U}_{i}\}$ of $X$ if each ${\epsilon}_{i}$ is zero on the complement of ${U}_{i}$.

###### Example 1 (Circle)

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

Application to integration

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

$${\int}_{M}f(x)\omega =\sum _{i}{\int}_{{U}_{i}}{\epsilon}_{i}(x)f(x)\omega .$$ |

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 |