I want to read the statement and proof of the co-area formula for Riemannian manifolds given in Isaac Chavel's Eigenvalues in Riemannian Geometry (pg. 85).
Assume all the manifolds we talk about are compact.
The section begins with the following:
Let $M$ be an $n$-dimensional Riemannian manifold. In all that follows, we let $V()$ denote the $n$-dimensional volume of submanifolds of $M$, and $A()$ denote the $(n-1)$-dimensional volume.
I don't know what is meant by the $n$-dimensional and the $(n-1)$-dimensional volumes of a submanifold of $M$.
I could guess the definition of the $(n-1)$-dimensional volume of an $(n-1)$-dimensional orientable submanifold $N$ of an orientable Riemannian manifold $M$: Since $N$ is itself a Riemannian manifold after borrowing the metric from $N$, and since $N$ is orientable, it has its own Riemannian volume form, and thus it has a volume.
But it seems that the author has a more general notion in mind. I flipped through the book but could not spot the definition. I also tried googling but couldn't find anything.