The coarea formula and level sets

Let f map Euclidean n-space to Euclidean k-space, with k no larger than n, and assume it is Lipschitz. The k-dimensional coarea Jacobian is the square root of the determinant of $DfDf^{\mathsf T}$. This matrix acts in the target, unlike the source Gram matrix in the area formula. When k equals one, the Jacobian is the norm of the gradient. A rank-deficient derivative gives zero Jacobian, so critical regions cannot simply be divided away.

For a nonnegative measurable weight, integrate that weight over each fiber using normalized $(n-k)$-dimensional Hausdorff measure, then integrate over the target using k-dimensional Lebesgue measure. The result equals the source integral of weight times the Jacobian. This is a measure-theoretic statement for almost every level; it does not say that every Lipschitz fiber is a smooth manifold. Smooth regular-level theorems require their own differentiability and noncriticality assumptions.

For a coordinate projection, the gradient norm is one and coarea reduces to the familiar slicing intuition behind Fubini. Multiplying that coordinate by a constant enlarges its target range and multiplies the source Jacobian by the same factor. Omitting either change breaks the identity. This provides an elementary test of a proposed formula before applying it to more complicated geometry.

The radial coordinate is another useful example. Its fibers in the plane are circles, whose normalized one-dimensional measures are circumferences. For radius itself, the gradient norm is one away from the origin, so integrating circumference over radii reconstructs area. For squared radius, the level parameter changes and the gradient norm becomes twice the radius. The same fibers are being labeled differently, but their integration weights must change accordingly.

Using a reciprocal Jacobian as a weight can recover an unweighted source integral only where that reciprocal is meaningful and where the omitted critical region is accounted for. One cannot set the reciprocal to zero on a positive-measure critical region and then claim the whole source volume has been recovered. In the annular example, the radius stays positive, so this particular obstruction is absent.

On countably rectifiable source surfaces, coarea has a related version using the tangential derivative and the surface's intrinsic dimension. An ambient gradient can contain normal components that do not change values along the surface. We restrict this lesson's calculations to Euclidean source domains, with their declared dimensions, so the displayed Jacobian has no hidden tangential projection. This distinction becomes important when slicing currents or measuring boundaries of finite-perimeter sets.