The term “differential analysis” has well-known meanings in the contexts of statistics and financial accounting, but it has different (ha!) meanings in other areas like mathematical analysis. In this note, I try to understand what these meanings in math are.
First, to give an overview of the setting: while I have seen this term discussed within analysis/calculus, I haven’t yet seen a Wikipedia page (or MathWorld or Encyclopedia of Math article) that clarifies its meaning. I’m not sure how standard of a term it is, but some popular sources (like MIT) do talk about it.
According to the textbook Differential Analysis by T M Flett (https://www.cambridge.org/core/books/differential-analysis/465F363DE78B9A6B6561686A287141D7), differential analysis deals largely with the generalization of results about derivatives from single-variable calculus to abstract settings (specifically, normed spaces.) The text cites an example of the power of such a construction: a single differential equation on a normed space could be equivalent to an infinite system of ordinary differential equations.
In the textbook Differential Analysis on Complex Manifolds by Raymond O Wells (https://link.springer.com/book/10.1007/978-0-387-73892-5), differential analysis seems to refer mainly to just analysis with some emphasis on differential geometry aspects as well as partial differential equations. The real crux of the subject matter seems to be the setting of complex manifolds, and differential analysis seems to be used as a header for a collection of topics in analysis involving differential notions, without a strict definition of the term.
In MIT OCW’s lecture notes on their differential analysis courses (18.155 and 18.156), it seems that this subject is defined to be the application of Fourier analysis to partial differential equations. Both the textbooks in the previous paragraphs mention applications to differential equations as well, but as far as I know now (without having gone deeper and seen any possible equivalencies that show up), it seems that the usages of the term are different among the various resources/authors.
Hence, it seems that the term “differential analysis” is used inconsistently by different authors, and thus it would be best to avoid using this term for now, and instead to use other language to clarify our intended meaning.
