1 - Differential Calculus

Abstract

We could cite various forerunners of differential calculus, including Descartes, Fermat and Cavalieri, but Newton and Leibniz should be remembered as the true pioneers of the field. This dual paternity created terrible priority disputes where the only certainty is the complexity of the controversy. Newton’s “fluxions” and Leibniz’s “vanishing quantities” are analogous to our modern concepts of derivative and differential, respectively. The extension of these ideas to functions of several variables was due to Euler (who introduced partial derivatives) and Clairaut. After contributions by many other mathematicians, including Volterra and Hadamard, the concept of a derivative of arbitrary order (Lemma-Definition 1.4) was ultimately introduced by Fréchet between 1909 and 1925. The inverse mapping theorem was proved by Lagrange in 1770, and the simplest case of the implicit function theorem was proved by Cauchy around 1833, followed by the case of vector-valued functions of several variables by Dini in 1877. Gateaux then extended Fréchet’s earlier ideas to develop his concept of differential, which was presented in a posthumous publication in 1919. The “convenient” form of differentials, one of their most recent incarnations, was introduced by Frölicher, Kriegl and Michor in the early 1980s (p. 73). These differentials were intended for mappings taking values in locally convex non-normable spaces, in particular nuclear Fréchet spaces (see, in particular, section 5.3.2 on manifolds of mappings). Wherever possible, this chapter therefore chooses to present differential calculus for mappings which take values in locally convex spaces rather than the normed vector spaces considered by the standard approach (nothing essential changes). Nonetheless, for the inverse mapping theorem and the implicit function theorem (Theorems 1.29 and 1.30), we will restrict attention to the Banach case (for a more general context, which uses yet another concept of differentiability distinct from any of those mentioned above). Both theorems are proved in full detail, including the Banach analytic case, by explicitly filling in the hints from. Together with the Carathéodory theorems mentioned below, these two theorems are the most profound results of this chapter.