- #1

Santiago24

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What is the purpose of the partition of unity? if someone can give me examples, bibliography or clear my doubt i'll appreciate it.

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- #1

Santiago24

- 32

- 6

What is the purpose of the partition of unity? if someone can give me examples, bibliography or clear my doubt i'll appreciate it.

- #2

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Here 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 this wolfram page 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.

This lists other applications. I find the signal processing filter particularly interesting.

- #3

Santiago24

- 32

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Thanks for the answer and the links.partsof a manifold (patches) to make a smooth global object.

Here 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 this wolfram page 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.

This lists other applications. I find the signal processing filter particularly interesting.

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MHB
Properties of Riemann Integrable Functions with Compact Support .... Proof of D&K Theorem 6.2.8 ....

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