- #1

JonnyG

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In summary, a P.O.U. will allow us to break up a map into smaller maps in order to concentrate the map just into a small open set.

- #1

JonnyG

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Physics news on Phys.org

- #2

ToBePhysicist

- 31

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few Words.

Hope I have provided the right answer in my answer...[pssst: I don't know what is this!I just know how to ultra google and tera- research!]

- #3

JonnyG

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I read that already, but it didn't give me the clearest picture.

- #4

Bacle2

Science Advisor

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- #5

ToBePhysicist

- 31

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I already said that.Bacle2 said:

Where is @JonnyG anyways? (Huh, oops...That will show a new alert on his page )

- #6

JonnyG

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- #7

mathwonk

Science Advisor

Homework Helper

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- #8

ToBePhysicist

- 31

- 5

Noticing you have not taken advantage of the link I provided...All I can say is that you didn't read the whole Spivak's Calculus.JonnyG said:

- #9

JonnyG

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Partition of unity is a mathematical concept used in fields such as differential geometry and topology. It involves breaking up a space into smaller subsets, each with its own function that sums up to 1. This technique is useful for constructing global functions from local ones, which can be especially valuable in solving differential equations.

Partition of unity allows for the construction of smooth functions that are globally defined but have different behavior on different subsets of a space. This is useful in solving differential equations, as it allows for the use of local solutions to find a global solution. It also helps in the application of boundary conditions, as the local functions can be adjusted to satisfy the conditions.

Yes, partition of unity has applications in various fields such as physics, computer graphics, and data analysis. It is also used in finite element methods, where it helps in approximating solutions to partial differential equations.

Partition of unity offers a more flexible and efficient approach in solving differential equations and other mathematical problems. It allows for the use of local information to construct global solutions, which can be more accurate and easier to work with. It also helps in dealing with complex boundary conditions and irregular domains.

Partition of unity can be challenging to implement in certain cases, as it requires careful selection of the local functions and their supports. It also introduces additional parameters that need to be determined, which can complicate the solution process. Additionally, partition of unity may not always be applicable, such as in non-smooth or discontinuous problems.

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