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There is this problem, I cant even figure out how to start. Any help here is very welcome.

Let $U,V\subset\mathbb{R}^d$ be open sets and $\Phi:U\rightarrow V$ be a homeomorphism. Suppose $\Phi$ is differentiable in $x_0$ and that $\det D\Phi(x_0)=0$. Let $\{C_n\}$ be a sequence of open(or closed) cubes in $U$ such that $x_0$ is inside the cubes and with its sides going to $0$ when $n\rightarrow\infty$. Denoting the $d$-dimension volume of a set by $\operatorname{Vol}(.)$, show that $$\lim_{n\rightarrow\infty}\frac{\operatorname{Vol}(\Phi(C_n))}{\operatorname{Vol}(C_n)}=0$$

I know that $\Phi$ cant be a diffeomorphism in $x_0$, but a have know idea how to use this, or how to do anything different. Thanks for helping.

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Use the fact that $D\Phi(x_0)$ approximates $\Phi$ near $x_0$ and that $\text{Vol}(D\Phi(x_0)(C_n))=\det(D\Phi(x_0))C_n=0$ –  Alex Youcis Sep 15 '12 at 3:46
    
Do you mean $x_0\in U$, that $\Phi$ is differentiable in $U$ or some neighborhood of $x_0$, and $x_0\in C_n$? –  Alex Becker Sep 15 '12 at 7:31
    
I think i made a mistake, i mean the cubes are open(or closed) in $U$ and $x_0$ is inside the cubes. –  Integral Sep 15 '12 at 20:20

2 Answers 2

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You should try using the change of variables formula for integration on Euclidean space: $$ \mathrm{Vol}(\Phi(C_n)) = \int_{\Phi(C_n)} dx = \int_{C_n} |\det(D\Phi)(x)| dx $$ However, it would seem that you should need some additional condition, in particular that $\Phi$ is continuously differentiable about $x_0$.

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Let $r_n$ be the sidelength of $C_n$. The definition of derivative gives you an upper bound on $|\Phi(x)-\Phi(x_0)|$ which says that $\Phi(C_n)$ is contained in a ball of radius about $r_n$ (same order of magnitude). Of course, this is not enough to show that $\mathrm{vol}(\Phi(C_n))/\mathrm{vol}(C_n)$ is small. The key additional fact is that the range of $D\Phi(x_0)$ is a proper subspace of $\mathbb R^d$. Let $V$ be this subspace. Upon a closer inspection, the definition of derivative will tell you that for every $x\in C_n$ the distance of $\Phi(x)$ to $V$ is much less than $r_n$, i.e., its ratio to $r_n$ tends to zero. This means that $\Phi(C_n)$ is contained in some cylinder-like shape, the volume of which you can estimate geometrically, "base times height".

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