Homeomorphism and diffeomorphism

In summary, the conversation discusses the concepts of homeomorphism and diffeomorphism in mathematics, focusing on the example of the function f(x) = x^3. It is explained that f is a homeomorphism, but not a diffeomorphism because its inverse function, f^-1, is not differentiable at 0. The concept of a function having derivatives of all orders is also discussed and clarified.
  • #1
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Homework Statement



I have some problems with understanding these two things.

Homeomoprhism is a function f [tex] f: M\rightarrow N[/tex] is a homeomorphism if if is bijective and invertible and if both [tex] f, f^{-1}[/tex] are continuous.
Here comes an example, let's take function
[tex] f(x) = x^{3}[/tex] it is clear that f(x) is continuous and bijective and at the same [tex] f^{-1}(x)\rightarrow x^{\frac{1}{3}}[/tex] is also continuous which indicates that f is a homeomorphism. No strange things here

Diffeomorphism- given two manifolds M, N and a bijective map f is called a diffeomorphism if both
[tex] f: M\Rightarrow N[/tex]
[tex] f^{-1}: N\Rightarrow M[/tex] are of class [tex] C^{\infty}[/tex] (if these functions have derivatives of all orders) f is called diffeomorphism


Going back to my example.

f is clearly a homeomorphism but :

why is [tex] f(x)=x^{3}[/tex] of a class [tex] C^{\infty}[/tex]?? what do they mean by 'all orders'
[tex] \frac{df}{dx}=3x^{2}[/tex]
[tex] \frac{d^2f}{dx^2}=6x [/tex]
[tex]\frac{d^3f}{dx^3}=6[/tex]
[tex]\frac{d^4f}{dx^4}=0...[/tex]


and its inversion
[tex] f^{-1}=x^{\frac{1}{3}}[/tex] is not of a [tex] C^{\infty}[/tex]...because
[tex] \frac{df^{-1}}{dx}=\frac{1}{3}x^{\frac{-2}{3}}[/tex]
it is not definied at x=0
 
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  • #2
So f is a homeomorphism but not a diffeomorphism. What's wrong with that?
 
  • #3
nothing is wrong with that, you did not read my message carefully and overlooked what I was asking for. The problem was that I did not exactly understand the concept of a function which has derivatives of all orders, what do they mean by that?
Apparently f has 4 derivatives until we get 0 but why its inversion is not a smooth function (because it is not definied at x=0?)

We could differentiate the inversion function as many times as we wanted to...
 
  • #4
rayman123 said:
nothing is wrong with that, you did not read my message carefully. The problem was that I did not exactly understand the concept of a function which has derivatives of all orders, what do they mean by that?
Apparently f has 4 derivatives until we get 0 but why its inversion is not a smooth function (because it is not definied at x=0?)

We could differentiate the inversion function as many times as we wanted to...

f is C^infinity. It has derivatives of all orders. f^(-1) isn't. It isn't even differentiable at 0. On the other hand if you define them on a domain that doesn't include 0, like [1,infinity) then they are both C^infinity and it's also a diffeomorphism.
 

1. What is a homeomorphism?

A homeomorphism is a type of mathematical function that preserves the topological properties of a space. In simpler terms, it is a continuous and invertible mapping between two topological spaces, meaning that the shape of the space does not change under the mapping.

2. How is a homeomorphism different from an isomorphism?

While both homeomorphisms and isomorphisms are types of mathematical functions, they differ in the type of properties they preserve. A homeomorphism preserves topological properties, while an isomorphism preserves algebraic structures, such as group operations or vector space operations.

3. What is a diffeomorphism?

A diffeomorphism is a type of smooth and invertible mapping between differentiable manifolds. In other words, it is a function that preserves differentiability between two spaces. This is useful in fields such as physics and engineering, where smoothness and differentiability are important concepts in understanding the behavior of systems.

4. How are homeomorphisms and diffeomorphisms related?

Both homeomorphisms and diffeomorphisms are types of mappings that preserve certain properties between two spaces. However, they differ in the types of properties they preserve. While a homeomorphism preserves topological properties, a diffeomorphism preserves differentiability.

5. What are some real-world applications of homeomorphisms and diffeomorphisms?

Homeomorphisms have several applications in fields such as computer graphics, image processing, and topology. For example, they can be used to create seamless transitions between different images or to map complex shapes onto simpler ones. Diffeomorphisms have applications in fields such as fluid dynamics, where they can be used to model the flow of fluids around objects.

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