The motivation for the area formula is very clear. Given a set A in $\mathbf{R^n}$, and a Lipschitz function f $: \mathbf{R^n} \rightarrow \mathbf{R^{N}}$, what is the measure of the set f(A) in $R^{N}$. The area formula gives you the area f(A) in terms of the Jacobian associated with f.
But what is the co-area formula? WHat is coarea even? And what can we even do with the coarea formuöa?
