Just stumbled upon an open tab with
Betancourt's "Geometric theory of higher order automatic differentiation"
which I started reading the new year night but quickly got distracted from.
I remember my first feeling was that it's slightly more verbose
than actually needed and that I'd have used some different wordings.
While this might be true, I'm finding the survey in its intro very clear and explaining.
I must remind my imaginary interlocutor that I have not as of yet
went through any course of differential geometry, only skimmed some textbooks and wikis.
I've been really struggling. Mainly because of terrible notation and language
established by physicists, I believe, and employed in most classic texts.
There are exceptions of course.
Some of seemingly good texts include e.g. works of Lee.
I think I would've solved my problems if I read carefully Lee's monographs
and walked through exercises therein. I'm not yet ready to make this effort
(not ready to do anything at all).
As for higher-order differential geometric structures,
I have only encountered jets when reading Arnold's "lectures on PDEs",
where they were in fact treated (if my memory doesn't deceive me)
as "things" that appear in Taylor expansions, without actually specifying
Now, here's what I actually wanted to remember when I started typing this note:
a good survey rapidly introducing principal concepts before verbose and detailed body of a text
makes a lot of difference. Lee, say, pours on you quite an amount of information
that makes use of terms that haven't been yet made concrete. And that information
might give you a good intuition if you already got some very basic framework of concepts
and notations to add new nodes and connections to. Betancourt on the other hand
throws "pullbacks" and "pushforwards" at you in the very beginning. He throws them
pretty concretized, almost tangible, in the sense that he defines domains
and ranges of the mappings, and essential properties of their actions.
He doesn't spend too much time on it, doesn't overload the reader
with questions like existence. These questions are important later for rigorous analysis,
but not for sketching the initial map of the field, not for initial understanding
of connections to other concepts.
Brain is a terribly lazy thing. At least mine is. The art of writing
(when the purpose of writing is to explain a subject and convey a message)
is in hacking reader's brain so as to leave it no chance to object
comprehending the message. That's an obvious thing that I always knew too.
Yet I tend to forget this when actually writing.