Differential forms ? Cartan to de rham

Abstract

ω = Zf«v«2...«k dxai dx^.^dx^. They have played a major role in such fields as differential equations, algebraic topology, and Lie groups during the twentieth century. In an earlier paper [Ka] I discussed their early history from their origins in the 18th century through the work of PoiNCARE and Volterra at the end of the 19th century, exploring particularly the notions of line and surface integrals and the Poincare lemma and its converse. In this article I continue certain aspects of the story during the first half of the 20th century. For the convenience of the reader, I have also included some material already discussed in the earlier paper. Though they had been used for over a century, differential forms were first defined (albeit "purely symbolically") by Elie Cartan in 1899. 1 will therefore concentrate on the influences which led Cartan to his definition as well as the