In the Wikipedia page for the Dirac Delta function this formula appears under "Properties in $n$ dimension". $$ \int f(x) \delta(g(x)) dx = \int_{g^{-1}(0)} \frac{f(x)}{|\nabla g(x)|} d\sigma(x) $$ It is said that this is a consequence of the Co-Area formula but no proof is given and the only reference ("Hörmander (1983), The analysis of linear partial differential operators I") doesn't seem to have this formula in it.
I have a few questions, in order of importance.
- What is a proof of this statement?
- What other references are there about this statement and its generalizations to a function $g:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m > 1$?
- In the above the author uses $\delta (g(x)) dx$ as if $\delta$ was a function, where in fact it is a Schwartz distribution or a measure. What did they mean? Especially because now it is concatenated with another function.
Definition of Dirac Distribution
It's a linear functional that maps test functions $\varphi$ to $$ \delta_x[\varphi] = \int \varphi(y) \delta_x^{\text{measure}}(dy) = \varphi(y) $$ where $\delta_x^{\text{measure}}$ is the Dirac Measure which for any measurable set $A$ is defined as $$ \delta_x^{\text{measure}}(A) = \begin{cases} 1 & x\in A \\ 0 & x\notin A \end{cases} $$
Co-Area Formula for Lipschitz Functions
If $g:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m$ then $$ \int_{\mathbb{R}^n} f(x) dx = \int_{\mathbb{R}^m} \left[\int_{g^{-1}(y)} f(x) |J_g(x) J_g(x)^\top|^{-1/2} \mathcal{H}^{n-m}(dx) \right]dy $$ where $J_g(x)$ is the Jacobian matrix of $g$.
They mean $$\int_{\Bbb{R}^N} f(x)\delta(g(x))dx=\lim_{n\to \infty} \int_{\Bbb{R}^N} f(x)\frac{n}2 1_{|g(x)|< 1/n}dx$$ where $f\in C^\infty_c(\Bbb{R}^N)$ and $g\in C^\infty(\Bbb{R}^N)$ with $\|\nabla g\| \ne 0$. What is $d\sigma(x)$? It is just the measure/distribution defined by $$\int_{Z(g)} f(x)d\sigma(x) = \lim_{n\to \infty} \frac{n}2\int_{dist(Z(g),x)<1/n} f(x)dx$$ where $Z(g)$ is the vanishing set of $g$ and $$dist(Z(g),x)=\inf_{a\in Z(g)} \|x-a\|$$
For $x$ close to $a\in Z(g)$ we have $g(x)\approx \nabla g(a) \cdot (x-a)$ from which $$dist(Z(g),x) \approx \frac{|g(x)|}{\|\nabla g(a)\|}$$
Whence for $f_a\in C^\infty_c(\Bbb{R}^N)$ supported on a small ball around $a$ we have $$\lim_{n\to \infty} \int_{\Bbb{R}^N} f_a(x)\frac{n}2 1_{|g(x)|< 1/n}dx\approx \int_{Z(g)} \frac{f_a(x)}{\|\nabla g(a)\|}d\sigma(x)$$ Writing $f$ as a sum of smooth functions supported on very small balls we get the formula $$\int_{\Bbb{R}^N} f(x)\delta (g(x))dx=\int_{Z(g)} \frac{f(x)}{\|\nabla g(x)\|}d\sigma(x)$$