= Coarea formula
{title2=$\int g|\nabla\Phi|\,dx=\int_{\mathbb R}\int_{\Phi^{-1}(s)}g\,dS\,ds$}
{wiki}
For a scalar smooth map $\Phi:\mathbb R^n\to\mathbb R$, integration can be decomposed into its level surfaces with the <gradient> weight displayed in the title. Near a regular level, using $\Phi$ as the normal coordinate gives the volume element $dx=dS\,ds/|\nabla\Phi|$, which explains the formula by <integration by substitution>. In particular, $\int W(x)\delta(\Phi(x))\,dx=\int_{\Phi=0}W/|\nabla\Phi|\,dS$ when the zero level is regular and the integrals are well-defined. General versions allow <Lipschitz functions> and maps into higher-dimensional spaces.
Back to article page
