# Motivations for the term "jet" in the context of viscosity solutions for fully nonlinear PDE

My question is very direct:

What are the motivations for the name "jet"(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?

The definition of which can be seen in Crandall, Ishii, Lions:

Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., New Ser. 27, No. 1, 1-67 (1992). ZBL0755.35015.

• This question had a downvote. I would appreciate comments on why is this inappropriate for the site or how could it be better. Nov 23 '20 at 21:52
• I guess that the analogy between a jet, in the sense of a jet engine, and a jet, in the sense of an equivalence class of function germs by their agreement up to some order, arises from the case when the function germ is the germ of a map from one variable to many, so a path. Picture it as a moving object, but we can only see it for an infinitesimal instant of its motion. Similarly the air inside a jet engine is brought to a particularly motion by the engine, but only inside the short stretch of the jet engine itself, rapidly dissipating outside the engine. Nov 23 '20 at 22:12
• The term jet has been in use for many years in differential geometry, but I don't know its origin. Nov 23 '20 at 22:12
• To complement the previous comment, the answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus.
– YCor
Nov 24 '20 at 9:14

Strictly speaking, he answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus/geometry.

Still one can wonder about when and where it was introduced. I suspected it was initially introduced in French and possibly by Bourbaki, and asked J-P. Serre in an email, and got the answer:

Il me semble que la notion de jet, et la terminologie correspondante, sont dus à Ehresmann (mais pas à Bourbaki), vers 1950. Je vous envoie ci-joint un article de 1961 qui confirme ceci.

(It seems to be that the notion of jet, and the corresponding terminology, are due to Ehresmann (but not Bourbaki) around 1950. Here is an article from 1961 confirming this.)

The article is Une généralisation du calcul des jets et quelques prolongements généralisés de variétés différentiables (DOI link linking to ProjectEuclid, no paywall) by M. Kawaguchi. Its first two sentences are:

Le mot "calcul des jets" a fait son début dans l'article [1] de Ch. Ehresmann. En 1951, Ch. Ehresmann s'est intéressé surtout aux fondements de la geometrie differentielle. [1] Les prolongements d'une variété différentiable, Atti IV Congresso Unione mat. Italiana, Taormina Ott, (1951), 1–9.

(The phrase "jet calculus" first appeared in the article [1] by Ch. Ehresmann. In 1951, Ch. Ehresmann was especially interested in foundations of differential geometry.)

I should add that in French "jet" has nothing to do with plane or engine.

"Jet" is the substantive of "jeter" = throw, launch. I can see the translations: spurt, jet, squirt, throw, spray. Also a derived phrase is "premier jet" = "first draft".

• The good old time when many Japanese wrote mathematics in French :). One of my most cited paper, written in French, was published in a japanese journal. Nov 24 '20 at 12:52

As I understand it, the word "jet" is meant to evoke the idea of a "spray" of curves through a point, or more accurately, their equivalence classes up to kth order contact

The machinery of Jets were introduced by Ehresmann as an geometrically invariant way of describing PDEs and their solutions in analogy to the way that vector fields and integral curves invariantly describe ODEs and their solutions.

For example, there is a way of geometrically describing symmetries here and so Noether's theorem.

It's worth noting that sprays describe 2nd order ODEs and are a sort of midway house. It comes from iterating tangent bundles - but this doesn't appear to generalise well - at least I haven't seen any generalisations of such, except in passing. Whereas the notion of a jet generalises the equivalence class of curves of a certain tangent order.