On maps whose distributional Jacobian is a measure

When the dimensions of domain and co-domain are the same, the Jacobian of a map is the determinant of its gradient. It was noticed long time ago that this nonlinear operator can be extended in a natural way to maps in certain Sobolev classes. This extension, known as distributional Jacobian, does not always agree with the Jacobian according to the standard (pointwise) definition, in the same way the distributional derivative of a function with bounded variation does not agree with the classical one. In particular, for map that take values in a given hypersurface, the pointwise Jacobian must vanish while the distributional one may not. If this is the case, the latter has an interesting interpretation in term of the (topological) singularity of the map. In this lecture I will review some of the basic results about maps whose distributional Jacobian is a measure, focusing in particular on maps with values in spheres