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. formula_119, where formula_120 and formula_121 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 formula_122 on an open rectangle is closed, i.e. formula_123, if and only formula_116 has the form formula_23 for some function formula_13 in the disk. The solution for formula_13 can be written by Cauchy's integral formula