Differentiability

Abstract

This fourth chapter establishes the notion of derivative in such a way that both the real and complex derivative can be obtained from it. The first section of this chapter deals with derivations, which are probably the most general differential operators. As a curious fact, the quotient rule and a polynomial chain rule are obtained directly from the product rule. The second section of this fourth chapter serves to construct the notion of derivative in topological modules over topological rings where the invertibles approach zero (practical rings). Particular interest is placed on uniformly differentiable functions because the chain rule is accomplished for this kind of functions. This allows, in the third and final section of this chapter, to construct differential manifolds over topological modules and to extend the basic results of Differential Geometry to this new setting.