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 written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on Rn is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems.
A precise statement of the formula is as follows. Suppose that Ω is an open set in and u is a real-valued Lipschitz function on Ω. Then, for an L1 function g,
where Hn−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies
and conversely the latter equality implies the former by standard techniques in Lebesgue integration.
More generally, the coarea formula can be applied to Lipschitz functions u defined in taking on values in where k ≤ n. In this case, the following identity holds
where Jku is the k-dimensional Jacobian of u whose determinant is given by
ApplicationsEdit
Taking u(x) = |x − x0| gives the formula for integration in spherical coordinates of an integrable function f:
Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN978-3-540-60656-7, MR 0257325.
Federer, Herbert (1959), "Curvature measures", Transactions of the American Mathematical Society, Transactions of the American Mathematical Society, Vol. 93, No. 3, 93 (3): 418–491, doi:10.2307/1993504, JSTOR 1993504.
Fleming, WH; Rishel, R (1960), "An integral formula for the total gradient variation", Archiv der Mathematik, 11 (1): 218–222, doi:10.1007/BF01236935
Malý, J; Swanson, D; Ziemer, W (2002), "The co-area formula for Sobolev mappings" (PDF), Transactions of the American Mathematical Society, 355 (2): 477–492, doi:10.1090/S0002-9947-02-03091-X.
October 25, 2023
coarea, formula, mathematical, field, geometric, measure, theory, coarea, formula, expresses, integral, function, over, open, euclidean, space, terms, integrals, over, level, sets, another, function, special, case, fubini, theorem, which, says, under, suitable. 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 written as the iterated integral over the level sets of the coordinate functions Another special case is integration in spherical coordinates in which the integral of a function on Rn is related to the integral of the function over spherical shells level sets of the radial function The formula plays a decisive role in the modern study of isoperimetric problems For smooth functions the formula is a result in multivariate calculus which follows from a change of variables More general forms of the formula for Lipschitz functions were first established by Herbert Federer Federer 1959 and for BV functions by Fleming amp Rishel 1960 A precise statement of the formula is as follows Suppose that W is an open set in R n displaystyle mathbb R n and u is a real valued Lipschitz function on W Then for an L1 function g W g x u x d x R u 1 t g x d H n 1 x d t displaystyle int Omega g x nabla u x dx int mathbb R left int u 1 t g x dH n 1 x right dt where Hn 1 is the n 1 dimensional Hausdorff measure In particular by taking g to be one this implies W u H n 1 u 1 t d t displaystyle int Omega nabla u int infty infty H n 1 u 1 t dt and conversely the latter equality implies the former by standard techniques in Lebesgue integration More generally the coarea formula can be applied to Lipschitz functions u defined in W R n displaystyle Omega subset mathbb R n taking on values in R k displaystyle mathbb R k where k n In this case the following identity holds W g x J k u x d x R k u 1 t g x d H n k x d t displaystyle int Omega g x J k u x dx int mathbb R k left int u 1 t g x dH n k x right dt where Jku is the k dimensional Jacobian of u whose determinant is given by J k u x det J u x J u x 1 2 displaystyle J k u x left det left Ju x Ju x intercal right right 1 2 Applications EditTaking u x x x0 gives the formula for integration in spherical coordinates of an integrable function f R n f d x 0 B x 0 r f d S d r displaystyle int mathbb R n f dx int 0 infty left int partial B x 0 r f dS right dr nbsp dd Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for W1 1 with best constant R n u n n 1 n 1 n n 1 w n 1 n R n u displaystyle left int mathbb R n u frac n n 1 right frac n 1 n leq n 1 omega n frac 1 n int mathbb R n nabla u nbsp dd where w n displaystyle omega n nbsp is the volume of the unit ball in R n displaystyle mathbb R n nbsp See also EditSard s theorem Smooth coarea formulaReferences EditFederer Herbert 1969 Geometric measure theory Die Grundlehren der mathematischen Wissenschaften Band 153 New York Springer Verlag New York Inc pp xiv 676 ISBN 978 3 540 60656 7 MR 0257325 Federer Herbert 1959 Curvature measures Transactions of the American Mathematical Society Transactions of the American Mathematical Society Vol 93 No 3 93 3 418 491 doi 10 2307 1993504 JSTOR 1993504 Fleming WH Rishel R 1960 An integral formula for the total gradient variation Archiv der Mathematik 11 1 218 222 doi 10 1007 BF01236935 Maly J Swanson D Ziemer W 2002 The co area formula for Sobolev mappings PDF Transactions of the American Mathematical Society 355 2 477 492 doi 10 1090 S0002 9947 02 03091 X Retrieved from https en wikipedia org w index php title Coarea formula amp oldid 1035119189, wikipedia, wiki, book, books, library,