Level-set representation of the joint density of states
= 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 $\int d^2k\,\delta(D(\mathbf k)-E)=\int_{D=E}dl/|\nabla D|$. Critical points require a limiting or distributional treatment; the apparent small contour at a parabolic edge is compensated by its small <energy> gradient.