A partition of unity refers to a mathematical concept used in topology, analysis, and differential geometry. Specifically, it is a collection of functions that smoothly "partition" or divide up a given space or manifold.
In more detail, let's consider a manifold, which is a space that locally resembles Euclidean space (such as a curved surface in three dimensions). A partition of unity on this manifold is a collection of continuous functions, typically denoted by φᵢ, where each φᵢ has support (the set of points where the function is non-zero) that lies within an open subset of the manifold. Additionally, these functions are chosen such that their sum at any point in the manifold is equal to one.
The purpose of a partition of unity is to break down a complex space into smaller, more manageable pieces. By selecting functions with appropriate properties, it becomes possible to construct a covering of the manifold using these functions and their supports. This covering can then be used to perform calculations or define functions on the manifold, enabling mathematical analysis.
One of the key properties of a partition of unity is that these functions are typically "smooth" or continuously differentiable. This property ensures that the functions can be used in various mathematical operations and calculations, such as integration. By smoothly dividing up the manifold, a partition of unity enables the application of tools from analysis and differential geometry to study properties of the manifold or solve differential equations.
Overall, a partition of unity provides a powerful mathematical tool for breaking down and analyzing complex spaces or manifolds, allowing for the translation of local properties to global ones, and facilitating techniques such as integration and solving partial differential equations.
