Coarea Formula for Radial Level Sets

The coarea formula expresses the gradient-weighted integral of a function over a planar or spatial domain as an integral over scalar level values of the lengths (or Hausdorff measures) of its level sets. On a disk of radius R with the radial function f(x, y) = r^p (for p = 2 or p = 4), each positive level value t = α R^p defines a concentric circle of radius r = t^1/p and circumference 2πr. The gradient magnitude |∇f| = p r^p-1 weights the area integral, while its reciprocal dr/dt = 1/|∇f| measures the radial thickness of a level shell at regular values.

Interactive Coarea Verification

Use the controls in the Coarea Formula for Radial Level Sets module to compare regular level circles, critical center behavior, and collapsed-disk boundary cases:

1. Select Radial power p = 2, Disk radius R = 1, and Level fraction alpha = 1/4 to verify Level value = 0.25, Level radius = 0.5, Gradient magnitude = 1, Inverse shell factor = 1, and both integrals equal to 4.1887902047864.
2. Change Radial power p to 4 and Level fraction alpha to 1/16 so that r = (1/16)^1/4 = 0.5 remains at half-radius while Gradient magnitude becomes 4(0.5)³ = 0.5 and Inverse shell factor doubles to 2.
3. Set Level fraction alpha to 0 on the unit disk (R = 1) to inspect the Critical center: the inverse shell factor is unavailable because |∇f(0, 0)| = 0, yet both whole-disk integrals remain finite.
4. Set Disk radius R to 0 to observe the Collapsed disk where the domain shrinks to a single point and both integrals vanish.