I Problems with understanding the role of the partition of unity

  • I
  • Thread starter Thread starter Santiago24
  • Start date Start date
  • Tags Tags
    Partition Unity
Santiago24
Messages
32
Reaction score
6
I'm reading "Calculus on manifolds" by Spivak and i can't understand the role that the partition of unity play and why this properties are important , Spivak say:
ml2cb.jpg

What is the purpose of the partition of unity? if someone can give me examples, bibliography or clear my doubt i'll appreciate it.
 
Physics news on Phys.org
We can use them to put together smooth objects, such as functions, that are only defined on parts of 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.
 
  • Like
Likes Santiago24 and fresh_42
andrewkirk said:
We can use them to put together smooth objects, such as functions, that are only defined on parts of 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.
Thanks for the answer and the links.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top