Problems with understanding the role of the partition of unity

[Santiago24]

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:

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

 

[andrewkirk]

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.

 

[Santiago24]

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.