- #1
rayman123
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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