Predictor–corrector method

2013-11-06T02:19:14Z

128.97.83.171: /* Example: Euler method with the trapezoidal rule */

In the [[mathematics|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 the integral of the [[level set]]s 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 '''R'''''n'' 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 problem]]s.

For [[smooth function]]s the formula is a result in [[multivariate calculus]] which follows from a simple [[change of variables]]. More general forms of the formula for [[Lipschitz function]]s were first established by [[Herbert Federer]] {{harv|Federer|1959}}, and for [[Bounded variation|''{{math|BV}}'' functions]] by {{harvtxt|Fleming|Rishel|1960}}.

A precise statement of the formula is as follows. Suppose that Ω is an open set in '''R'''''n'', and ''u'' is a real-valued [[Lipschitz function]] on Ω. Then, for an [[Lp space|L1]] function ''g'',

:<math>\int_\Omega g(x) |\nabla u(x)|\, dx = \int_{-\infty}^\infty \left(\int_{u^{-1}(t)}g(x)\,dH_{n-1}(x)\right)\,dt</math>

where ''H''''n'' − 1 is the (''n'' − 1)-dimensional [[Hausdorff measure]]. In particular, by taking ''g'' to be one, this implies

:<math>\int_\Omega |\nabla u| = \int_{-\infty}^\infty H_{n-1}(u^{-1}(t))\,dt,</math>

and conversely the latter equality implies the former by standard techniques in [[Lebesgue integral|Lebesgue integration]].

More generally, the coarea formula can be applied to Lipschitz functions ''u'' defined in Ω ⊂ '''R'''''n'', taking on values in '''R'''''k'' where ''k'' < ''n''. In this case, the following identity holds

:<math>\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</math>

where ''J''''k''''u'' is the ''k''-dimensional [[Jacobian]] of ''u''.

==Applications==
* Taking ''u''(''x'') = |''x'' − ''x''0| gives the formula for integration in spherical coordinates of an integrable function ƒ:
::<math>\int_{\mathbb{R}^n}f\,dx = \int_0^\infty\left\{\int_{\partial B(x_0;r)} f\,dS\right\}\,dr.</math>
* Combining the coarea formula with the [[isoperimetric inequality]] gives a proof of the [[Sobolev inequality]] for ''W''1,1 with best constant:
::<math>\left(\int_{\mathbb{R}^n} |u|^{n/(n-1)}\right)^{\frac{n-1}{n}}\le n^{-1}\omega_n^{-1/n}\int_{\mathbb{R}^n}|\nabla u|</math>
:where ωn is the volume of the [[unit ball]] in '''R'''''n''.

==See also==
* [[Sard's theorem]]
* [[Smooth coarea formula]]

==References==
* {{citation
| last = Federer
| first = Herbert
| authorlink = Herbert Federer
| title = Geometric measure theory
| publisher = Springer-Verlag New York Inc.
| location = New York
| year = 1969
| pages = xiv+676
| isbn = 978-3-540-60656-7
| mr= 0257325
| series = Die Grundlehren der mathematischen Wissenschaften, Band 153 }}.
* {{citation|last=Federer|first=H|authorlink=Herbert Federer|title=Curvature measures|journal=Transactions of the American Mathematical Society|volume=93|year=1959|pages=418–491|jstor=1993504|doi=10.2307/1993504|issue= 3|publisher=Transactions of the American Mathematical Society, Vol. 93, No. 3}}.
* {{citation|last1=Fleming|first1=WH|last2=Rishel|first2=R|title=An integral formula for the total gradient variation|journal=Archiv der Mathematik|volume = 11|year=1960|doi=10.1007/BF01236935|pages=218–222|url=http://www.springerlink.com/index/WV67N13464926501.pdf|format=PDF|issue = 1}}
* {{citation|last1=Malý|first1=J|last2=Swanson|first2=D|last3=Ziemer|first3=W|title=The co-area formula for Sobolev mappings|journal=Transactions of the American Mathematical Society|year=2002|volume=355|pages=477–492|url=http://www.ams.org/tran/2003-355-02/S0002-9947-02-03091-X/S0002-9947-02-03091-X.pdf|format=PDF|doi=10.1090/S0002-9947-02-03091-X|issue=2}}.

[[Category:Measure theory]]

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Predictor–corrector method