Is the Transition Map Smooth in the Intersecting Set?

In summary, a smooth transition map is a mathematical tool used in topology to describe continuous transformations between spaces. It differs from regular transition maps by being differentiable and having no sharp edges. Creating a smooth transition map can be complex, but it has many applications in mathematics and physics. However, it may not be applicable to highly complex spaces and can be computationally intensive. Alternative methods may be used in these cases.
  • #1
sk1001
7
0
Smooth transition map (easy!?)

Homework Statement


Check the transition map
http://img132.imageshack.us/img132/4341/18142532.png
is smooth in the set for which their images intersect

The Attempt at a Solution


I have thought of two ways to show this.

(1) Show that Φ is a composition of two smooth functions and is therefore smooth.
(2) compute the composite function and then prove that is smooth

Which way do you suggest?
I have attempted method (1) to some extent, but wondering if method (2) is easier.
 
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  • #2


bump please
 

FAQ: Is the Transition Map Smooth in the Intersecting Set?

What is a smooth transition map?

A smooth transition map is a mathematical tool used in the field of topology to describe how one space can be continuously transformed into another space. It is often used in the study of manifolds and surfaces.

How is a smooth transition map different from a regular transition map?

A smooth transition map is a type of transition map that is differentiable, meaning it has no sharp edges or corners. This allows for a smooth and continuous transformation between two spaces. In contrast, a regular transition map may have abrupt changes or discontinuities.

Is it easy to create a smooth transition map?

Creating a smooth transition map can be a complex process that requires a good understanding of topology and differential geometry. It may involve advanced mathematical concepts and calculations. However, for simple cases, such as mapping between two-dimensional surfaces, it can be relatively straightforward.

What are some applications of smooth transition maps?

Smooth transition maps have many applications in mathematics and physics. They are commonly used in the study of dynamical systems, where they help describe the behavior of systems over time. They are also used in computer graphics to create smooth and realistic animations. Additionally, smooth transition maps are used in fluid mechanics to model the flow of fluids.

Are there any limitations to using smooth transition maps?

While smooth transition maps are a powerful tool, they have some limitations. They may not be applicable to highly complex or irregular spaces. Additionally, the creation of smooth transition maps can be computationally intensive, making them impractical for certain applications. In some cases, alternative methods may be used to approximate the desired transformation.

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