area formula

Let $\mathcal{H}^{m}$ denote the Hausdorff measure. Let $m\leq n$ and consider a Lipschitz function $f\colon\mathbb{R}^{m}\to\mathbb{R}^{n}$. If $A\subset\mathbb{R}^{m}$ is a Lebesgue measurable set, the equality

 $\int_{A}J_{f}(x)\,dx=\int_{\mathbb{R}^{n}}\mathcal{H}^{0}(f^{-1}(\{y\})\cap A)% \,d\mathcal{H}^{m}y$

holds, where

 $J_{f}(x)=\sqrt{\det(Df(x)\cdot Df(x)^{*})}$

is the Jacobian determinant of $f$ in the point $x$ and represent the $m$-volume of the image of the unit cube under the linear map $Df(x)$.

If $u\in L^{1}(\mathbb{R}^{m})$ then one has

 $\int_{\mathbb{R}^{m}}u(x)J_{f}(x)\,dx=\int_{\mathbb{R}^{n}}\sum_{x\in f^{-1}(% \{y\})}u(x)\,d\mathcal{H}^{m}y.$

Notice that this formula is a generalization of the change of variables in integrals on $\mathbb{R}^{n}$.

Title area formula AreaFormula 2013-03-22 14:27:36 2013-03-22 14:27:36 paolini (1187) paolini (1187) 12 paolini (1187) Theorem msc 28A78 ChangeOfVariablesInIntegralOnMathbbRn