Introduction to $\mathcal{A}$-Calculus

Abstract

Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is right-$\mathcal{A}$-linear. We discuss the basis-dependent correspondence between right-$\mathcal{A}$-linear maps and the regular representation of real matrices in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of $\mathcal{A}$ gives $n^2-n$ generalized $\mathcal{A}$-CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra. The Wirtinger calculus is shown to generalize. We find the $\mathcal{A}$-CReqns are equivalent to the condition that the partial derivatives in all $n-1$ conjugate variables vanish. Our construction modifies that given by Alvarez-Parrilla, Fr\'ias-Armenta, L\'opez-Gonz\'alez and Yee-Romero in a 2012 paper. We also discuss how this conjugate technology gives us a method to convert real PDEs into differential equations over $\mathcal{A}$. Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. Taylor's Theorem and a Tableau for $\mathcal{A}$-differentiable function are derived. We prove many of the usual theorems of integral calculus including Cauchy's Integral Theorem for $\mathcal{A}$ and the Fundamental Theorems of Calculus part I and II. Certainly we do not claim originality in some of what we present, however, we hope this paper adds something useful to the existing literature.