A -Differentiability over Associative Algebras

The unital associative algebra structure A on R n 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 ) = x 2 , we obtain d f x ( v ) = v x + x v , and for f ( x ) = x − 1 , d f x ( v ) = − x − 1 v x − 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.