In the middle of the 18th century, the theory of differential forms was first studied in the simplest case of 1-forms in the plane, i.e. A d x + B d y , where A and B are functions in the plane. The study of 1-forms and the differentials of functions began with Clairaut's papers in 1739 and 1740. At that stage his investigations were interpreted as ways of solving ordinary differential equations. Formally Clairaut showed that a 1-form ω ω = A d x + B d y on an open rectangle is closed, i.e. d ω ω = 0 , if and only ω ω has the form d f for some function f in the disk. The solution for f can be written by Cauchy's integral formula