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I am studying partitions of unity in Munkres' $\textit{Analysis on Manifolds}$ book. Are partitions of unity just theoretical tools, i.e. used to prove theorems, or do people actually apply them concretely to compute integrals in practice?

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    $\begingroup$ In my experience, you rarely have an explicit partition of unity. So, in my experience, that is mostly theoretical tool. $\endgroup$ – Giuseppe Negro Mar 31 '16 at 15:37
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Partitions of unity arise naturally in many applications related to affine spaces and affine transformations. In fact, barycentric coordinates are partitions of unity. So, there are lots of applications in geometry.

In other fields, also, there is often the idea of a "weighted average" where the weights necessarily sum to 1. This is common in finance/economics, for example.

The spline techniques mention by @William Krinsman are another example. Many curves in CAGD are so-called "blended curves" of the form $$ \mathbf{P}(t) = \sum_{i=0}^n \phi_i(t) \mathbf{P}_i $$ The blending functions must sum to 1, or else this definition is not well defined (it doesn't make sense to form an arbitrary sum of points).

The Bernstein polynomials are one very common example of such blending functions -- they are used to construct Bézier curves and surfaces. But they have many other uses in approximation theory and probability, too.

Lagrange polynomials also form a partition of unity, and these have many uses.

So the short answer is that partitions of unity occur in many places in applied mathematics, computing, and engineering -- they are not just a theoretical concept.

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I'm not an expert on numerical analysis, but based on the textbooks I have read on the subject, partitions of unity are used when defining basises of splines (see below). I think B-splines or Bezier splines; they are used in computer graphics.

http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/bspline-property.html

http://arxiv.org/pdf/1410.5413.pdf

https://www.cs.unc.edu/~dm/UNC/COMP258/Papers/bsplbasic.pdf

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