The validity of infinitesimals as mathematical entities was demonstrated in the 1960s by Abraham Robinson. However, the most widespread application area of infinitesimals, differentials, has continued to be plagued by inconsistencies. Typically, these inconsistencies are papered over by forcing practitioners to never use differentials in isolation. However, new work from The Blyth Institute shows that the problem is actually a notational problem. Making straightforward modifications to the notation allows for free use of total and partial differentials.
This work was previously established with total differentials in “Extending the Algebraic Manipulability of Differentials.” The latest paper, “Total and Partial Differentials as Algebraically Manipulable Entities,” extends this work to partial differentials as well, and provides more theoretical background for what makes differentials work in the first place. This paper will be included in an upcoming volume from IntechOpen, titled Operator Theory – Recent Advances, New Perspectives and Applications, edited by Abdo Abou Jaoudé (Amazon Link).
This paper is also available on arXiv.