# Why are differential forms called closed and exact?

It seems to me that "exact" relates to exact differential equation. So, why are they called exact?

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"Closed" might be related to the idea of a closed manifold. It is closed if the boundary is zero. Just speculation... –  Jim Conant Dec 6 '10 at 21:40
Because they are EXACTLY a differential? –  Igor Rivin Dec 6 '10 at 21:41
Duplicate? mathoverflow.net/questions/22910/… –  Qiaochu Yuan Dec 6 '10 at 21:43
@Igor: Hmm, and CLOSE to being 0? –  Cam McLeman Dec 6 '10 at 23:37

According to Hans Samelson's historical note "Differential Forms, the Early Days", both notions were introduced in Les Méthodes nouvelles de la Mécanique Céleste by Poincaré (vol. 3, Gauthier-Villars, Paris, 1899, pp. 9-15). Samelson notes

Given a p-form $\omega$ whose integral over any closed manifold is 0, then there is a (p - 1)-form, let's say $\psi$, that stands to $\omega$ in the relation described by Stokes's theorem (so that $\omega=d\psi$; he calls such an $\omega$ exact). Thus we have here the non-trivial half of what today one calls the Poincare Lemma: $\omega$ is $d\psi$ for some $\psi$ ($\omega$ is "exact") if and only if $d\omega = 0$ ($\omega$ is "closed").

Apparently, it had taken some time for the terminology to stabilize as, for instance, Goursat used the term "exacte" in his book for a form that today one calls closed (E. Goursat, Leçons sur le problème de Pfaff, Hermann, Paris, 1922).

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Thanks for that, Andrey, I didn't know that paper, it sure looks interesting. –  Theo Buehler Dec 6 '10 at 22:45
I've always thought that the term exact for differential forms was adopted as a generalization of an older existing term in the theory of complex variable (i.e., $f(z)$ is exact on $\Omega$ iff $f(z)=g′(z)$), and that the usage there had been borrowed from arithmetic, as well as the term residue: $f(z)$ (say a rational function) is exact iff it has no residue, like a quotient is exact iff it has no remainder (lat. residuum). Unfortunately I have no reference for this (which is not in contrast with the above hystorical note). –  Pietro Majer Dec 6 '10 at 23:00
Pietro, I think that work on integrating systems of first order linear PDEs (Pfaffian systems) predates the development of complex analysis by a long margin. Initially, holomorphic functions were studied using the tools from linear PDEs, cf the history Cauchy-Riemann equations, which had been first considered by D'Alambert (in Russian they are in fact called "D'Alambert-Euler conditions"). –  Victor Protsak Dec 7 '10 at 2:40
that is good, but why are they named such way? Did Poincaré be the first who studied exact differential equation? –  Nikita Kalinin Dec 7 '10 at 14:35

This is not an answer to the original question but closely related, so I leave this post here for the interested reader

There are many variants of the story but here's what I found by googling which is as good an account as any other:

At the urging of my host, Douglas Clark, I asked Edwards what he knew about the origin of the term "exact sequence." Edwards was sure the term was invented by Eilenberg and/or Steenrod. He re- called reading or hearing that, as Eilenberg and Steenrod were writing their book but before they de- vised a satisfactory term, they left a blank everywhere the term "exact" would later appear.

During the week of my return to Berkeley, Saunders Mac Lane turned up and gave a delightful colloquium talk on "Mathematics for sixty years: What has changed?" I cornered him before the talk and pumped him for information. He told me the same story as Edwards and said he heard it directly from Eilenberg.

Two days later Eilenberg phoned in response to my letter. Indeed, he related, during about the first year he and Steenrod worked on their book, they wrote "blank sequence" everywhere for Hurewicz's concept, with the intention of replacing the word "blank" by the "right word" once they found it. They refrained from using a provisional term in fear that would dis- tract them from their search for the "right" term. Once they hit upon the term "exact" they shared it with anyone interested. Eilenberg used it in a course at the University of Michigan in the spring of 1946.

I did not press Eilenberg on whether it was he or Steenrod who originally dreamed up the term. At the time it seemed a rude thing to ask, and the question seemed unimportant.

Copied from The Exact Answer to a Question of Shields by Donald Sarason, Mathematical Intelligencer, Vol. 12, No. 2, 1990.

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I thought the question was about exact forms, not exact sequences... –  Andrei Moroianu Dec 6 '10 at 21:54
Me too. And though I can imagine that the two uses (sequences, forms) of the word "exact" are related, I have no evidence for it. Anyone? –  Charles Rezk Dec 6 '10 at 21:57
You're absolutely right, I was too quick, but I still like this account of the origin of the term "exact" in algebra, so I leave the answer here. It certainly is related, but feel free to vote it down. –  Theo Buehler Dec 6 '10 at 21:59
In the appendix of his book "Abelian Categories," Peter Freyd says that the terminology "exact sequence" was suggested by "exact differentials" (pg. 157). He reports having heard about this from Eilenberg and Steenrod themselves. [But he also writes (pg. 155) "The origin of concepts, even for a scholar, is very difficult to trace. For a nonscholar such as me, it is easier. But less accurate."] –  Andreas Blass Aug 17 '11 at 16:46

Well, the notion of exactness lies in an algebraic background. Given a sequence of groups or R-Modules and morphisms given by arrows in the following way:

... -> A_n --f_n--> A_n+1 --f_n+1--> A_n+2 --> ...

we call it exact whenever Im f_n = Ker f_n+1. Normally forms in Global Analysis are related to De Rham Kohomology which is precisely the quotient of such sequences for certain R-Modules (or C-Modules). The de Rham Cohomology group of certain order is trivial whenever the short sequence is exact (exactness in the 3 modules involved), this occurs exactly when all the closed forms are exact.

About "close" I dont have an answer although I may think of some reasons I prefer not to comment hehe.

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But I think "exact differential" is much earlier than "exact sequence" ... –  Gerald Edgar Dec 6 '10 at 22:09

Because of homological meaning and the relation with simplicial cohomologies?

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Previous answers have shown that the use of "exact" for differentials came before its use in homology. –  Andreas Blass Aug 17 '11 at 16:36