Physical derivation of the coarea formula and an elementary proof via gradient flow

TL;DR

This paper provides an elementary derivation and a rigorous proof of the coarea formula using gradient flow and geometric Jacobian computation.

Contribution

It introduces a simple derivation of the coarea formula and constructs a diffeomorphism via gradient flow for the proof.

Findings

Elementary version of the coarea formula derived

Rigorous proof using changing variable formula presented

Diffeomorphism constructed via gradient flow with Jacobian computed geometrically

Abstract

In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density $g (x)$. Then we present an rigorous proof using the changing variable formula. To this end we construct the diffeomorphism $\Phi$ via the gradient flow and compute its Jacobian determinant via geometric method.