Differential Calculus as Part of Commutative Algebra

Abstract

This is the central chapter of the book. At the beginning of the chapter, it is shown that the classical definitions of the calculus or of differential geometry, say that of the derivative or tangent vector, are unsatisfactory, being of descriptive nature, and conceptually correct definitions are needed. The latter are provided by the differential calculus over commutative algebras. These conceptual definitions are of course equivalent to the classical ones, but appear in more general situations and so have meaningful applications in situations (e.g., in the presence of singularities) where the classical tools cannot be applied. The notions of tangent (co)vector and c(o)tangent manifold, of smooth map, of smooth (co)vector field, linear differential operator, and jet space, together with their basic properties, are described in the classical and in the general algebraic situations.