# Jordan Curve (high level, graduation level)

• Apr 28th 2009, 04:00 AM
deubelte
Jordan Curve (high level, graduation level)
Hi all

I have an exercice which is at a rather high level. I post it here, because I dont know the general level of the forum. But let see if I have some answers.

Let $\displaystyle \Gamma$ be a Jordan closed curve. Set $\displaystyle \epsilon>0$
and set:

$\displaystyle \Gamma_{\epsilon}=\left\{z\in\mathbb{C};d(z,\Gamma )\leq\epsilon\right\}$

1) Let $\displaystyle \gamma:[a,b]\rightarrow\mathbb{C}$ a regular parametrization of $\displaystyle \Gamma$ which is $\displaystyle C^2$.

For $\displaystyle \epsilon>0$, we note $\displaystyle R_{\epsilon}$ the rectangle $\displaystyle [a,b]\times[-\epsilon,\epsilon]$ and we set $\displaystyle \Phi_{\epsilon}\rightarrow\mathbb{C}$ by :
$\displaystyle \Phi_{\epsilon}(t,s)=\gamma(t)+sN(t)$
where $\displaystyle N(t)$ is the normal vector to $\displaystyle \gamma$ at the point $\displaystyle t$.

I have shown that if $\displaystyle \epsilon$ is small enough, then $\displaystyle \Phi_\epsilon$ is a $\displaystyle C^1$ diffeomorphism and that $\displaystyle Im(\Phi_\epsilon)=\Gamma_\epsilon$

Now, I have to show that for \epsilon small enough,
we have:
$\displaystyle m(\Gamma_\epsilon)=\int^b_a\left(\int^{\epsilon}_{-\epsilon} \left|\gamma'(t)\right|+s\varphi(t)ds\right)dt$

I don't know how to do this. I know that I have to use the jacobien, the integral transformation (Multiple integral - Wikipedia, the free encyclopedia)

Thank you
• Apr 28th 2009, 05:39 AM
deubelte
It was to hard for the forum to answer.
• Apr 29th 2009, 01:10 AM
The Second Solution
Quote:

Originally Posted by deubelte
yes, I could, but the pb is that I have written it in latex, using the $symbols. I am to lazzy to transforme it to with MATH][/MATH • Apr 29th 2009, 05:24 AM mr fantastic Quote: Originally Posted by deubelte yes, I could, but the pb is that I have written it in latex, using the$ symbols.