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Topology and Analysis
Problems with understanding the role of the partition of unity
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[QUOTE="andrewkirk, post: 6456538, member: 265790"] We can use them to put together smooth objects, such as functions, that are only defined on [I]parts[/I] of a manifold (patches) to make a smooth global object. [URL='https://en.wikipedia.org/wiki/Partition_of_unity#Example']Here[/URL] is an example of creating a partition of unity that gives us two nonzero, smooth functions defined on the unit circle ##S^1##, that add to 1 everywhere. The second para of [URL='https://mathworld.wolfram.com/PartitionofUnity.html']this wolfram page[/URL] gives an example of how we can use the general partition of unity theorem (of which the theorem you quote above is a special case, using the manifold ##\mathbb R^n##) to prove that any manifold can have smooth vector fields on it that are not everywhere zero. [URL='https://en.wikipedia.org/wiki/Partition_of_unity#Applications']This[/URL] lists other applications. I find the signal processing filter particularly interesting. [/QUOTE]
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Problems with understanding the role of the partition of unity
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