How does one compute the following integral
$$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}$$
if $g: \mathbb{R}^n \to \mathbb{R}^m$. In this question and the Wikipedia page on the Dirac delta function an answer is given for $m=1$ as
$$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _{g^{-1}(0)}\frac{f(\pmb{r})}{|\nabla g(\pmb{r})|}d\sigma\ .$$
So my question concerns the case wheere $m>1$. Is there also some good reference for citing the corresponding result?