The Coarea formula is an important result in differential geometry and geometric measure theory. It relates integrals over a manifold to integrals over the level sets of a smooth function defined on that manifold. Specifically, it provides a way to express the volume of the preimage of a set under a smooth function, in terms of integrations over its level sets.
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For a scalar smooth map , integration can be decomposed into its level surfaces with the gradient weight displayed in the title. Near a regular level, using as the normal coordinate gives the volume element , which explains the formula by integration by substitution. In particular, when the zero level is regular and the integrals are well-defined. General versions allow Lipschitz functions and maps into higher-dimensional spaces.