The differential: Nineteenth and twentieth century developments

Abstract

In this essay I shall be concerned principally with the way in which mathematicians have dealt with the concept of the differential and the meaning of differentiability of functions of more than one real variable or of a vector variable. For the nineteenth century the principal concern is with realvalued functions of several real variables. In the twentieth century attention shifts to the differential in functional analysis, and specifically to the Frechet differential for functions from one normed vector space to another. The classical function of several real variables is subsumed under functional analysis. The exposition is arranged in three parts: In Part I, I recount briefly how the concept of the differential for ordinary analysis was given essentially its modern formulation by Stolz in 1893 and how Frechet, in 1911 and 1912, pointed the way to the concept of the differential appropriate to functional analysis. In Part II, I review the treatments of differentials in the nineteenth century. In Part III, I survey the conceptual developments of the differential in the twentieth century.