TITLE: Motivations for the term "jet" in the context of viscosity solutions for fully nonlinear PDE
QUESTION [5 upvotes]: 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.
see also https://arxiv.org/pdf/math/9207212.pdf.

REPLY [3 votes]: 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.