I'm trying understand the proof of the lemma $1.1$ of this paper (see page $366$), but I didn't understand how coarea formula was applied to obtain the following equality:
$$\int_K |\nabla v| + v |\nabla u| = \int_a^b |\partial^* F_t \cap K| - \int_K \int_a^b \chi_{\{ v(x) < t \}} |\nabla u| + b \int_K |\nabla u|$$
I would like to understand how the equality above is related to the coarea formula. The coarea formula used is presented here on page $70$ (it's the page $82$ of the PDF).