Level-set representation of the joint density of states

For a smooth two-dimensional transition energy $D(\mathbf{k})$ and a regular level $D=E$, the coarea identity gives $\int d^{2}k\delta (D(\mathbf{k})-E)={\int }_{D=E}dl/|\nabla D|$. Critical points require a limiting or distributional treatment; the apparent small contour at a parabolic edge is compensated by its small energy gradient.