The Differential Algebraic Structure of the Levi-civita Field and Applications

Abstract: It is shown that the non-Archimedean field introduced by Levi-Civita [7, 8] admits a derivation and hence is a differential algebraic field [1, 2, 4]. The differential algebraic structure of the Levi-Civita field is utilized for the decision of differentiability of functional dependencies on a computer, as well as the practical computation of derivatives. As such, it represents a new method for computational differentiation [5] that avoids the well-known accuracy problems of numerical differentiation tools. It also avoids the often rather stringent limitations of formula manipulators that restrict the complexity of the function that can be differentiated, and the orders to which differentiation can be performed. Examples for the use of the method for typical pathological problems are given. AMS Subject Classification: 26E30, 12J25, 33F99.