Area and the area formulas

The concept of area of a surface, as well as the related questions concerning representation formulas for the area or the change of variable formula, are of central importance in many instances. An up to date historical analysis of the different contributions seems to be missing, though it would surely be of interest. To quote only a few books we refer to [23] [20] [21] [6]. Eventually the most satisfactory approach was formulated by Federer [9], compare also the references there. Still there seems to be a kind of break when looking to those questions from the point of view of geometric measure theory of weak maps, as Sobolev maps, or of a parametric or of a non–parametric approach. Also some problems seem still to remain open. Here we would like to discuss briefly some of those questions. We refer to the monograph [11] for a more detailed treatment.