A-differentiability over associative algebras

Abstract

The unital associative algebra structure 𝔸 on ℝ𝑛 allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the 𝔸-calculus. Thus, we introduce 𝔸-differentiability. Rules for 𝔸-differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are 𝔸-differentiable, and their 𝔸-derivatives are the power series defined by their 𝔸-derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For 𝑓(𝑥)=𝑥2, we obtain 𝑑𝑓𝑥(𝑣)=𝑣𝑥+𝑥𝑣, and for 𝑓(𝑥)=𝑥−1, 𝑑𝑓𝑥(𝑣)=−𝑥−1𝑣𝑥−1. Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations.