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Problem:

Let's consider the collection of $C^1$ functions, where $k=1,2,\ldots,(n-1)$: $g_k:\mathbb{R}^k\rightarrow \mathbb{R},$ where: $ g_k=g_k(x_1,x_2,\ldots,x_k)$

Then a new map $f$ is defined as follows: $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that: $f(x_{1},x_{2},\ldots,x_n)=(x_1, g_1(x_1)+2x_2, g_2(x_1,x_2) + 3x_3, \ldots, g_{n-1}(x_1,x_2,\ldots,x_{n-1})+nx_n)$

How can find the volume of $f((0,1)^n)$ where $(0,1)^n$ is an open unit cube $(0,1)^n$?

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    @draks: These problems are for an exam...2012-04-27

1 Answers 1

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By change of variables formula, if $\Omega \subset \mathbb{R^{n}}$ is open and $G: \Omega \mapsto \mathbb{R^{n}}$ is a $C^{1}$ diffeomorphism, we have \begin{equation} \int_{G(\Omega)} f(x) dx = \int_{\Omega)}f \circ G(x)| det D_xG| dx \end{equation} Then $ L^{n}(f((0,1)^{n})) = \int_{f((0,1)^{n})} 1 dx = \int_{(0,1)^{n}} |det D_x f| dx $

But, $ D_xf = \left[ \begin{array}{cccccc} 1 & 0 & 0 & \cdots & 0 & 0 \\ D_1 g_1(x_1) & 2 & 0 & \cdots & 0 & 0\\ D_1 g_2(x_1,x_2) & D_2 g_2(x_1,x_2)& 3 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & 0 & 0\\ D_1 g_{n-1}(x*) & D_2 g_{n-1}(x*) & D_3 g_{n-1}(x*)& \cdots & D_{n-1}g_{n-1}(x*)& n \\ \end{array} \right] $ Where $x* = (x_1, x_2, \cdots , x_{n-1})$

Hence $ L^{n}(f((0,1)^{n}))= n!L^{n}(0,1)^{n}.$