The coarea formula is derived from mass considerations and proved by building a gradient-flow diffeomorphism whose Jacobian is found geometrically.
Ding, A natural geometric proof of the length-curvature identity, Math
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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.