About: 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 and , the multiplicity function , is the (possibly infinite) number of points in the preimage . The multiplicity function is also called the Banach indicatrix. Note that, . We will denote by the n-dimensional Hausdorff measure. whereis the Jacobian of .