2. Differential calculus on applications of several variables

2.1 Introduction

Once the notion of the derivability of an application defined on an interval of the real line has been acquired, it seems natural to extend it to functions whose sources are more general sets. Arbitrary normed vector spaces could be proposed. In reality, such a framework is both too restricted and almost unnecessarily general:

• too restricted, since the appropriate environment for dealing with, for example, the calculation of variations, would be that of differentiable varieties;

• of superfluous generality, since the elementary cases only occur when the source of the application is an open of a real finite-dimensional normed vector space.

This propaedeutic...