I am trying to follow the proof in this paper but I am having a hard time figuring out/visualizing how the following expression came about:
Let $T(\theta), \, \sigma(\theta), \, f(\theta)$ be smooth functions of $\theta \in \mathbb{R}^d$. The paper then claims that:
$ \displaystyle \int_{T(\theta) \in (s - \sigma(\theta),s+ \sigma(\theta))} \,f(\theta) \,d \theta = \int_{T(\theta) = s} \, (h^-(\theta) - h^+(\theta))\,f(\theta) \, dS$
where:
$dS$ is the surface element defined by $T(\theta) = s$,
$h^+,h^-$ satisfy $T(\theta + h^\pm(\theta) \mathbf{n}) \pm \sigma(\theta + h^\pm(\theta) \mathbf{n}) = s$ with
$\mathbf{n} = \nabla T/\|\nabla T\|$ being the unit normal tot he surface $T(\theta) = s$.
To me, this doesn't resemble anything like the use of the coarea formula. Can anyone help me visualize or understand the transition from the integral on $\mathbb{R}^d$ to the integral on $\mathbb{R}^{d-1}$?