Which of the functions are diffeomorphisms

[brainslush]

Homework Statement

Decide which ones of the following maps f: are diffeomorphisms.
f(x) = 2x, x^2, x3, e^x, e^x + x.

Homework Equations

The Attempt at a Solution

I think 2x, x^3 are diffeomorphisms. They are bijective and their inverses are differentiable

x^2 and e^x are not bijective. => no diffeomorphisms

Well I'm not sure about e^x+x. I guess that's it is an diffeomorphism but I haven't found a way to proove it, since I don't know how to derive the inverse.

 

[micromass]

Are you certain that x^3 is a diffeo? It is certainly bijective and differentiable, but is it's inverse invertible?

I was also under the impression that diffeomorphism were infinitely differentiable (=smooth). You might have defined it another way, but if my impression is correct then you're not done with showing differentiability...

 

[brainslush]

You are right.
According to my textbook and the lecturenotes, a diffeomorphism is a bijecctive map f:M->M' between smooth manifolds s.t. f and f^-1 are both smooth.

Well, this would lead to the conclusion that only e^x + x is a diffeomorphism.
In this case I need a hint how to derive the inverse of e^x+x.

 

[micromass]

You are right.
According to my textbook and the lecturenotes, a diffeomorphism is a bijecctive map f:M->M' between smooth manifolds s.t. f and f^-1 are both smooth.

Well, this would lead to the conclusion that only e^x + x is a diffeomorphism

2x will still be a diffeo too...
e^x+x will probably be a diffeomorphism but did you prove it yet?

 

[brainslush]

And again you are right. I forgot that also a constant can still be diverentiated.

To be honest, I've no idea how to proove that e^x + x is a difeomorphism, without finding its inverse. I read something about the Lambert W-Function but I've no clue how this fits into the image

 

[micromass]

And again you are right. I forgot that also a constant can still be diverentiated.

To be honest, I've no idea how to proove that e^x + x is a difeomorphism, without finding its inverse. I read something about the Lambert W-Function but I've no clue how this fits into the image

I have no idea how to find the inverse of e^x+x, but all is not lost however. You could apply the inverse function theorem. It would be very easy to show that e^x+x is a local diffeomorphism with that.
The only thing which rests you is showing that e^x+x is bijective. You could prove this by showing that e^x+x is a continuous, increasing function whose limits satisfy
$$\lim_{x\rightarrow \pm \infty}{f(x)}=\pm \infty$$.

 

[brainslush]

Thanks. I never used the inverse function theorem before but it looks quite simple.