Relating Dirac mass with surface integral

Question

I know that for a smooth function $g:\mathbb{R}^n\to\mathbb{R}$ with $\nabla g(x)\neq 0$, there holds the formula $$\int_{\mathbb{R}^n}\delta(g(x))f(x)\,dx=\int_{g=0}\frac{1}{|\nabla g(x)|}f(x)\,d\sigma(x)$$ where $\,d\sigma$ is the surface measure induced by $[g=0]$ and $\delta$ the one dimensional Dirac mass. I wonder, what is the equivalent expression for $g:\mathbb{R}^n\to\mathbb{R}^m$ for $m<n$, when $Dg(x)\neq 0$? In other words what is the scaling factor in the surface integral to compensate for the $m$-dimensional Dirac mass? Is it something like $|Dg(x)|^m$?