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 be a Jordan closed curve. Set

and set:

1) Let a regular parametrization of which is .

For , we note the rectangle and we set by :

where is the normal vector to at the point .

I have shown that if is small enough, then is a diffeomorphism and that

Now, I have to show that for \epsilon small enough,

we have:

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