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Differentiation arithmetic is a principal and accurate technique
for the computational evaluation of derivatives of first and higher order.
This article aims at recasting real differentiation arithmetic in a formalized
theory of dyadic real differentiation numbers that provides a foundation
for first and higher order automatic derivatives. After we set the stage by
putting on a systematic basis certain fundamental notions of the algebra of
differentiation numbers, we begin by setting up an axiomatic theory of real
differentiation arithmetic, as a many-sorted extension of the theory of a
continuously ordered field, and then establish the proofs for its consistency
and categoricity. Next, we carefully construct the algebraic system of real
differentiation arithmetic, deduce its fundamental properties, and prove that
it constitutes a commutative unital ring. Furthermore, we describe briefly
the extensionality of the system to an interval differentiation arithmetic
and to an algebraically closed commutative ring of complex differentiation
arithmetic. Finally, a word is said on machine realization of real differentiation arithmetic and its correctness, with an addendum on how to compute
automatic derivatives of first and higher order.
Hend Dawood, Nefertiti Megahed. (2019) A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives, Punjab University Journal of Mathematics, Volume 51, Issue 11.
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