# diffeomorphism

• Feb 16th 2011, 10:28 PM
Zennie
diffeomorphism
Looking for help with a textbook problem.

Give an example to demonstrate that a one-to-one and onto mapping need not be a diffeomorphism. (Hint: Take $m=n=1$)

I'm assuming that the hint means let $F: R^1 \rightarrow R^1$

I'm very stuck, however and having trouble thinking this through. Help and/or a couple of examples would be greatly appreciated.
• Feb 16th 2011, 10:51 PM
InvisibleMan
Take F(x)= x/(x+1) for x in [0,1] and F(x)=x for x>1.
(reflect this to the negative side).
I believe it's injective and onto, but F its derivative at x=1 isn't defined.
( $F'^-(1)=1/2 -1/4=1/4$ and $F'^+(1)=1$).
• Feb 16th 2011, 10:54 PM
FernandoRevilla
Quote:

Originally Posted by Zennie
Give an example to demonstrate that a one-to-one and onto mapping need not be a diffeomorphism. (Hint: Take $m=n=1$)

For example, $f(x)=x^3$

Fernando Revilla