Total and Partial Differentials as Algebraically Manipulable Entities

TL;DR

This paper explores the algebraic manipulability of differentials, proposing a framework that simplifies calculus understanding and reduces the need for specialized theorems by treating differentials as algebraically independent entities.

Contribution

It introduces a novel approach to differential operators based on Robinson's infinitesimals, enabling algebraic manipulation of differentials in calculus.

Findings

Differentials can be treated as algebraically manipulable entities.

This approach simplifies calculus teaching and understanding.

Reduces reliance on specialized theorems in calculus.

Abstract

Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to prevent contradictions. However, this is primarily a notational and conceptual problem. The work of Abraham Robinson has shown that there is nothing contradictory about the concept of an infinitesimal differential operating in isolation. In order to make this system extend to all of calculus, however, some tweaks to standard calculus notation are required. Understanding differentials in this way actually provides a more straightforward understanding of all of calculus for students, and minimizes the number of specialized theorems students need to remember, since all terms can be freely manipulated algebraically.