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.

Worked Example: Coarea Integral on a Quadratic Disk

Set radial power p = 2, disk radius R = 1, and level fraction α = 1/4. We compute the local level-circle geometry and compare the direct gradient-weighted area integral over the disk to the 1D integral of level-circle circumferences over t from 0 to R^p = 1:

Selected Level Circle (α = 1/4): The level value is t = (1/4)(1²) = 0.25, giving radius r = 0.25^1/2 = 0.5, circumference 2π(0.5) = π ≈ 3.1415926535898, gradient magnitude |∇f| = 2(0.5)¹ = 1, and inverse shell factor dr/dt = 1/|∇f| = 1.
Direct Weighted Disk Integral: Integrating |∇f| = 2r in polar coordinates yields ∫₀^2π ∫₀¹ (2r) r dr dθ = 2π [2r³ / 3]₀¹ = 4π / 3 ≈ 4.1887902047864.
Integral Over Level Values: Integrating the circumference 2π t^1/2 over t ∈ [0, 1] yields ∫₀¹ 2π t^1/2 dt = 2π [2 t^3/2 / 3]₀¹ = 4π / 3 ≈ 4.1887902047864.

Both calculations produce the exact value 2πp R^p+1 / (p + 1) = 4π / 3. Removing a small disk of radius ε > 0 around the origin avoids the critical point r = 0 where |∇f| = 0, and letting ε → 0⁺ confirms that the single critical center has measure zero and does not alter the total integral.

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:

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.
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.
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.
Set Disk radius R to 0 to observe the Collapsed disk where the domain shrinks to a single point and both integrals vanish.

Assumptions and Limitations

This visualization restricts attention to radially symmetric power functions f(x, y) = r^p on centered disks where |∇f| is constant along each level circle. For general Lipschitz functions on ℝⁿ, level sets f^-1(t) may have varying curvature, multiple connected components, or singular sets of positive codimension, and the coarea formula equates ∫_Ω g(x) |∇f(x)| dx to ∫_ℝ (∫_f^-1(t) g dℋ^n-1) dt using (n − 1)-dimensional Hausdorff measure rather than simple circle circumference.

References

The coarea decomposition of gradient-weighted area integrals into level-set arclength integrals follows standard geometric analysis notes:

Lenya Ryzhik, Stanford Math 221 Notes: The Coarea Formula (PDF)— derives gradient-weighted integration by level-set arclength and the Jacobian change-of-variables factor along normal directions.

Coarea Formula for Radial Level Sets