Polarizations and differential calculus in affine spaces
Abstract
Within the framework of mappings between affine spaces, the notion of -th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules, are shared -- at the finite level -- by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order , and are based on combinatorial identities. The infinitesimal limits of the -th polarizations of a function will yield its -th derivatives (without resorting to the usual recursive definition), and the above mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.
Cite
@article{arxiv.math/0510368,
title = {Polarizations and differential calculus in affine spaces},
author = {Margherita Barile and Fiorella Barone and Wlodzimierz M. Tulczyjew},
journal= {arXiv preprint arXiv:math/0510368},
year = {2007}
}