- #1

- 210

- 0

You should upgrade or use an alternative browser.

In summary, a bump function is used to extend a locally defined function to a smooth function on the whole manifold.

- #1

- 210

- 0

Physics news on Phys.org

- #2

Science Advisor

- 1,089

- 10

- #3

- 210

- 0

i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.

i don't see how such an extention is done.

thank's

Hedi

- #4

Science Advisor

Homework Helper

Gold Member

- 4,807

- 32

hedipaldi said:

i quote from the text:If F is a smooth function on a neighbourhood of x,we can multiply it by a bump function to extend it to M ".here M is a differential manifold so there are local coordinates at each point.F is a real function.

i don't see how such an extention is done.

thank's

Hedi

For instance, say F is defined on a coordinate chart U diffeomorphic to R^n. Let D(1) be the closed subset of U corresponding to the closed unit disk under the coordinate map, and let B(2) be the open subset of U corresponding to the open disk of radius 2 under the coordinate map. Then you know there is a smooth bump function

F'(x):=b(x)F(x) if x is in U

F'(x)=0 otherwise

It is obvious that F' is smooth since it is smooth on U and it is smooth "away from U": the only potentially problematic points are at the boundary of U. But at those points, F' is locally identically 0 by construction so all is well.

Now F' is not an extension of F in the strictest sense since we have modified it outside of D(1). But for many purposes, this is just fine.

- #5

- 210

- 0

It was very helpfull indeed.Thank"s a lot.

Hedi

Hedi

A smooth extension of a locally defined function on a manifold is a way of extending a function defined on a subset of a manifold to a larger domain while maintaining smoothness. This allows the function to be defined and continuous on the entire manifold, rather than just a portion of it.

Smooth extension is important in manifold theory because it allows for the study of functions on the entire manifold, rather than just on a subset. This can provide a more complete understanding of the manifold and its properties, and is often necessary in order to apply other mathematical tools and techniques.

A smooth extension of a locally defined function on a manifold can be achieved by using techniques such as partitions of unity or bump functions. These methods allow for the construction of a smooth function that agrees with the original function on the subset where it is defined, and is also smooth on the rest of the manifold.

In order for a smooth extension to exist, the original function must be continuous and at least C^k (k times continuously differentiable) on the subset of the manifold where it is defined. Additionally, the manifold must be a paracompact space, which ensures that a partition of unity exists.

No, not all locally defined functions on a manifold can be smoothly extended. There are certain functions that are not continuous or differentiable, and therefore cannot be extended in a smooth manner. Additionally, the existence of a smooth extension depends on the specific properties of the manifold and the function in question.

Share:

- Replies
- 21

- Views
- 490

- Replies
- 13

- Views
- 2K

- Replies
- 20

- Views
- 2K

- Replies
- 11

- Views
- 594

- Replies
- 14

- Views
- 676

- Replies
- 3

- Views
- 2K

- Replies
- 10

- Views
- 2K

- Replies
- 16

- Views
- 3K

- Replies
- 4

- Views
- 2K

- Replies
- 4

- Views
- 2K