Level-set normalization of a surface delta
= Level-set normalization of a surface delta
{title2=$\delta_S=|\nabla g|\delta(g)$}
If $g$ is smooth and $\nabla g\ne0$ on $S=\{g=0\}$, transverse local coordinates give $\langle\delta(g),\varphi\rangle=\int_S\varphi/|\nabla g|\,dS$. Thus $\delta_S=|\nabla g|\delta(g)$ as <distributions>. In particular $\delta_{S^1}=2\delta(|x|^2-1)$. Omitting this Jacobian confuses the delta of a defining function with a geometric <surface delta distribution>.