Coarea formula ($\int g|\nabla \Phi |dx={\int }_{\mathbb{R}}{\int }_{{\Phi }^{-1}(s)}gdSds$)

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 $dx=dSds/|\nabla \Phi |$, 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.