The coarea formula is derived from mass considerations and proved by building a gradient-flow diffeomorphism whose Jacobian is found geometrically.
Federer, Geometric measure theory, Die Grundlehren der mathematischen Wis- senschaften, Band153
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Physical derivation of the coarea formula and an elementary proof via gradient flow
The coarea formula is derived from mass considerations and proved by building a gradient-flow diffeomorphism whose Jacobian is found geometrically.