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Math Help - Jordan Curve (high level, graduation level)

  1. #1
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    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 \Gamma be a Jordan closed curve. Set \epsilon>0
    and set:

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

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

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

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


    Now, I have to show that for \epsilon small enough,
    we have:
    m(\Gamma_\epsilon)=\int^b_a\left(\int^{\epsilon}_{-\epsilon} <br />
\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
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  2. #2
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    I made it.
    It was to hard for the forum to answer.
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  3. #3
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    Quote Originally Posted by deubelte View Post
    I made it.
    It was to hard for the forum to answer.
    Maybe you could post your solution for eveyone to enjoy.
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  4. #4
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    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
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  5. #5
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    Quote Originally Posted by deubelte View Post
    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
    If you post it I will change the tags from $ to [tex] etc.
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