Noncommutative Partial Derivative

TL;DR

This paper develops an axiomatic framework for defining point-derivatives in noncommutative algebras, extending classical calculus rules like the chain rule and Clairaut's theorem to this noncommutative setting.

Contribution

It introduces the axiomatic definition of noncommutative point-derivatives and generalizes fundamental calculus rules to noncommutative algebras.

Findings

Axiomatic definition of noncommutative point-derivatives

Extension of chain rule to noncommutative context

Extension of Clairaut's theorem to noncommutative context

Abstract

We introduce the axiomatic definition of the point-derivative for noncommutative algebras and present the counterparts of the ordinary multi-variable chain rule and Clairaut's Theorem in the context of partial point-derivatives.