Area formula (geometric measure theory)
In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the integral of the Jacobian of the map. It is one of the fundamental results of the field that has connections, for example, to rectifiability and Sard's theorem. Definition: Given formula_1 and formula_2, the multiplicity function formula_3, is the (possibly infinite) number of points in the preimage formula_4. The multiplicity function is also called the Banach indicatrix. Note that, formula_5. We will denote by formula_6 the n-dimensional Hausdorff measure. Theorem : If formula_1 is Lipschitz, then for any measurable formula_8, formula_9 where formula_10 is the Jacobian of formula_11. The measurability of the multiplicity function is part of the claim. The Jacobian is defined almost everywhere by Rademacher's differentiability theorem. Theorem was proved first by Herbert Federer .