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Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there a way of rewriting the above integral as a surface integral over the level set $S$? Related to this, is there some distributional identity like $$ \delta(g(x)) = \int_S ds dt \frac{\delta^{(3)}(x - X(s,t))}{|g'(X(s,t))|} $$ where $X(s,t)$ is a parameterization of $S$ that would allow one to to this, analogous to the formula $$ \delta(h(x)) = \sum_{x_0\in h^{-1}(0)}\frac{\delta^{(3)}(x-x_0)}{|h'(x_0)|} $$ when the zero set of $h$ is finite?

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I refer you to section 4.5 in the WP page (and references therein): simple layer integral. It turns out you can write $$\int_{\mathbb{R}^3} f(\vec{r})\delta(g(\vec{r}))d\vec{r}=\int_{S}\frac{f(\vec{r})}{|\vec{\nabla}g(\vec{r})|}d\sigma$$

A very similar question to this one is here.

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