# Problems with understanding the role of the partition of unity

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• Santiago24
In summary, the partition of unity gives us two smooth functions that add up to 1 everywhere on the unit circle. This theorem can be used to prove that any manifold can have smooth vector fields that are not everywhere zero.
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.

aclaret
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 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.

## 1. What is the partition of unity and why is it important?

The partition of unity is a mathematical concept used in fields such as differential geometry and functional analysis. It refers to a set of non-negative functions that sum up to 1 on a given domain. It is important because it allows us to break down a complex function or space into simpler parts, making it easier to study and analyze.

## 2. How does the partition of unity help in solving problems?

The partition of unity helps in solving problems by providing a way to decompose a problem into smaller, more manageable parts. This allows for a more systematic and organized approach to problem-solving, making it easier to find solutions.

## 3. What are some common challenges in understanding the role of the partition of unity?

Some common challenges in understanding the role of the partition of unity include grasping the concept of a function summing up to 1, understanding its applications in different fields, and visualizing how it breaks down a complex space or function.

## 4. Can the partition of unity be used in any field of science?

Yes, the partition of unity can be used in various fields of science, including physics, engineering, and computer science. It is a fundamental concept in many mathematical theories and has numerous applications in different scientific disciplines.

## 5. Are there any limitations to the use of the partition of unity?

While the partition of unity is a powerful tool in problem-solving, it also has some limitations. For example, it may not be applicable to all types of functions or spaces, and its use may lead to some loss of information. It is important to understand these limitations and use the partition of unity appropriately in scientific research.

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