Is a diffeomorphism if the inverse function

[atomqwerty]

I've read that a function f given by $$f:U\rightarrow V$$ is a diffeomorphism if the inverse function $$f^{-1}$$ exists and is differentiable. I've also read that that function is a local diffeomorphism in a given point $$p\inU$$ if it can be found a range A around p such that the function f verifies f:A -> f(A)
I'm really in troubles with all those definitions. I've to do an exercise in which I've been asked to say if a given function is a diffeomorphism, and my question is: how do I know if a function has the inverse f^{-1}?

thanks a lot!

 

[Office_Shredder]

I'm a bit confused. Are you just asking how to determine if f(x) is invertible? You just need to check that it is both 1-1 and onto. There's nothing topological about that part. Or are you wondering how you prove that the inverse function is differentiable?

It might help if you post some specific functions that you're looking at

 

[atomqwerty]

I'm a bit confused. Are you just asking how to determine if f(x) is invertible? You just need to check that it is both 1-1 and onto. There's nothing topological about that part. Or are you wondering how you prove that the inverse function is differentiable?

It might help if you post some specific functions that you're looking at

The question is mainly how to determine where (in which points) a function f is a local diffeomorphism, that I think leads us to determine when the funtion is a diffeomorfism near of that point p.

Thank you!

 

[Office_Shredder]

The general strategy is to calculate the differential and determine if it's an invertible linear transformation. The points at which it is are where f is locally a diffeomorphism. If you have a specific example you want to look at we can do that

 

[atomqwerty]

The general strategy is to calculate the differential and determine if it's an invertible linear transformation. The points at which it is are where f is locally a diffeomorphism. If you have a specific example you want to look at we can do that

Sure, I quote you an example of spherical coordinates,

Let be the function

f:R⁺ x R x R -> R³
{r,phi,theta} -> {rcos(phi)sin(theta),rsin(phi)sin(theta),rcos(theta)}
Demonstrate that f is differentiable. Calculate de Jacobian matrix and determine in which points is a local diffeomorphism. Calculate the range of R⁺ x R x R in which the function is a global diffeomorphism.

PS. Sorry about not using LaTeX, I had too much troubles with it.

Thank you