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Coarea formula

For smooth functions the formula is a result in multivariate calculus which follows from a change of variables.

Version
v1 · 2026-09-28 · History
Domain-specific #
8511
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Geometric Measure Theory, Multivariable Calculus → Mathematics

Core Idea

Coarea formula is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: For smooth functions the formula is a result in multivariate calculus which follows from a change of variables. In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be.

How would you explain it like I'm…

Adding Up Hill Rings

Think of a hill drawn on a map with rings showing each height. To add up everything on the hill, you can add up what's on each ring, one ring at a time. But the rings are not all equally far apart: where the hill is steep they crowd together, and where it's flat they spread out. The coarea formula is the rule that says exactly how to fix that when you add the rings up.

Adding Up by Contour Slices

Mathematicians often want to add up a quantity spread over a region, which is called integrating. The coarea formula gives a clever way to do it: pick a second function, like height on a hill, and slice the region into curves or surfaces where that function has the same value, like the contour lines on a map. You add up along each slice and then add up all the slices. Because slices are packed tightly where the function changes fast and spread out where it changes slowly, the formula includes a correction for how steep the function is. Slicing a box into flat layers, or a ball into shells like an onion, are special cases.

Integrating Over Level Sets

The coarea formula rewrites an integral over a region of space as an integral over the level sets of another function, the sets where that function takes a fixed value, like contour lines or surfaces. You integrate over each level set, then integrate those results over all the values. The size of the function's gradient enters as a weight, which accounts for level sets bunching up where the function changes quickly. Two familiar special cases are Fubini's theorem (slicing a box along coordinate lines) and spherical coordinates (slicing space into spherical shells, the level sets of distance from the center). For smooth functions it follows from a change of variables in multivariable calculus.

 

The coarea formula in geometric measure theory expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function f. In its standard form, the integral of g times the magnitude of the gradient of f over the region equals the integral, over all values t, of the integral of g over the level set where f equals t, measured with the appropriate lower-dimensional (Hausdorff) measure. The gradient factor accounts for how tightly level sets are packed. Special cases include Fubini's theorem, where f is a coordinate function and the level sets are parallel slices, and spherical coordinates, where f is the radial function and level sets are spheres. For smooth f it is a consequence of a change of variables; more general versions hold for Lipschitz functions and were first established by Herbert Federer. The formula plays a decisive role in the modern study of isoperimetric problems.

Scope of Application

  • Applications. Taking u(x) = |x − x 0 | gives the formula for integration in spherical coordinates of an integrable function f.

  • Documented setting. In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level.

  • Documented setting. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated.

  • Documented setting. Another special case is integration in spherical coordinates, in which the integral of a function on R n is related to the integral of the function over spherical shells: level sets.

  • Documented setting. For smooth functions the formula is a result in multivariate calculus which follows from a change of variables.

Clarity

A clear use of Coarea formula names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is For smooth functions the formula is a result in multivariate calculus which follows from a change of variables. The strongest recognition evidence in the frozen account is: In particular, by taking g to be one, this implies.

Manages Complexity

Coarea formula compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—more general forms of the formula for Lipschitz functions were first established by Herbert Federer , and for ' functions by .—and the practical consequence—\int{\R^n}f\,dx = \int0^\infty\left{\int{\partial B(x0;r)} f\,dS\right}\,dr.

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: For smooth functions the formula is a result in multivariate calculus which follows from a change of variables.
  3. Check operation and conditions. and conversely the latter equality implies the former by standard techniques in Lebesgue integration.
  4. Demand recognition evidence. In particular, by taking g to be one, this implies.
  5. Test variation.

Knowledge Transfer

Within the home domain. Knowledge about Coarea formula transfers literally when a new case preserves the same carrier type, relation, and recognition test. Taking u(x) = |x − x 0 | gives the formula for integration in spherical coordinates of an integrable function f. In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. Beyond the home domain. No canonical parent is asserted for Coarea formula.

Neighborhood in Abstraction Space

Coarea formula sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Named Analytic Theorems & Operators (39 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08