area formula

Let m denote the Hausdorff measureMathworldPlanetmath. Let mn and consider a Lipschitz function f:mn. If Am is a Lebesgue measurable set, the equality


holds, where


is the Jacobian determinant of f in the point x and represent the m-volume of the image of the unit cube under the linear map Df(x).

If uL1(m) then one has


Notice that this formulaMathworldPlanetmathPlanetmath is a generalizationPlanetmathPlanetmath of the change of variables in integrals on n.

Title area formula
Canonical name AreaFormula
Date of creation 2013-03-22 14:27:36
Last modified on 2013-03-22 14:27:36
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 12
Author paolini (1187)
Entry type Theorem
Classification msc 28A78
Related topic ChangeOfVariablesInIntegralOnMathbbRn