A-Differentiability over Associative Algebras

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.