I want to prove the coarea formula $ \operatorname{Vol}(M) = \int_M d\operatorname{Vol}_M = \int_{-\infty}^\infty \frac{1}{|\nabla f|} \operatorname{area }(f^{-1}(t)) dt $ where $f\colon M \rightarrow {\Bbb R}$ is a smooth function.
Here I have a question. The above formula says that $|\nabla f|$ is constant on $f^{-1}(t)$. So how can we prove this ? Thank you in advance.
Correction : I checked that the following equality is right (See 85 page in the book Eigenvalues in Riemannian geometry - Chavel)
$\operatorname{Vol}(M) = \int_M d\operatorname{Vol}_M = \int_{-\infty}^\infty \int_{f^{-1} (t) } \frac{1}{|\nabla f|} d\operatorname{area}(f^{-1}(t)) dt$
I believe that this formula is the generalization of arc-length parametrization. But (1) I cannot explain clearly and (2) I cannot prove the above inequality. So if you have an interests in this please help me in (1) and (2).