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I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to an integral over level sets.

Application of the "vector-valued" coarea formula (3rd equation in the Wikipedia entry) to $\nabla u$ is not possible, because it only holds for functions from $\mathbb{R}^n$ to $\mathbb{R}^k$ with $n>k$, which is violated by $\nabla u$.

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