Re: diffeomorphism
- From: The World Wide Wade <waderameyxiii@xxxxxxxxxxxxxxxxxxxx>
- Date: Sun, 27 Nov 2005 22:43:40 -0800
In article
<1133135034.070549.106160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
liting0612@xxxxxxxxx wrote:
> I have been thought about sth like the converse of Inverse Function
> Theorem.
> If f is a diffeomorphism from U to V, f and its inverse are
> continously differentiable of all orders, then, for each p in U, will
> the derivative of f at p is one-to-one or locally invertible?
Yes. Let g denote the inverse. Then f o g = I, the identity. As f
and g are given to be smooth, we can use the chain rule: df(g(x))
o dg(x) = dI(x) = I. It follows that df is invertible at each
point of U.
.
- References:
- diffeomorphism
- From: liting0612
- diffeomorphism
- Prev by Date: Re: Real sine waves
- Next by Date: Math Game.
- Previous by thread: Re: diffeomorphism and derivatives equal to 0
- Next by thread: involution has fixed point
- Index(es):
Relevant Pages
|
