Exploring Algebraic Themes in Calculus

Abstract

This paper outlines two different approaches to studying the algebra underlying the ideas of differentiation and integration typically presented in analysis. In the first section we approach the notion of derivative by defining a derivative to be an endomorphism of an algebra over a commutative ring satisfying the product rule. We prove some general results classifying the structure of derivatives before proceeding to prove some general facts about differential algebras. We conclude this section by studying some of the properties of linear differential operators on differential algebras and their similarities and differences with polynomial rings studied in commutative algebra. In the second section we follow Christopher Heunen’s1 treatment of ℓ2 as a dagger preserving monoidal functor to a similar treatment of L2.