The coarea formula is derived from mass considerations and proved by building a gradient-flow diffeomorphism whose Jacobian is found geometrically.
Willem, Minimax theorems,Progress in Nonlinear Differential Equations and their Ap- plications
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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.