area formula

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

$${\int }_A{J}_f(x)𝑑x={\int }_{{ℝ}^n}{\mathscr{H}}^0(f^{-1}(\{y\})\cap A)𝑑{\mathscr{H}}^{m}y$$

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 $u\in L^1({ℝ}^m)$ then one has

$${\int }_{{ℝ}^m}u(x)J_f(x)𝑑x={\int }_{{ℝ}^n}\sum _{x\in f^{-1}(\{y\})}u(x)d{\mathscr{H}}^{m}y.$$

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