I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and engineers, most of them find some use for differential geometry nowadays. One theoretical physicist went as far as to say you could "do nothing serious without it." Yet at most schools (at least the few I've looked at) differential geometry is reserved for graduate students in math and advanced math undergraduates. No schools I looked at had an elementary differential geometry class in, say, a similar style as the calculus sequence. Some of the people I talked to also expressed a lot of difficulty in learning it for the first time on their own. I myself am taking an advanced graduate course in General Relativity, and a good portion of the difficulty of the students is in misuderstanding the fundamental concepts of differential geometry.
To cover differential geometry rigorously, of course one needs quite a bit of advanced mathematics, including topology and analysis. But universities teach elementary calculus classes, most of which are not terribly rigorous, but are sufficient for the purposes of non-mathematicians. Linear algebra, multivariate calculus, and a bit of differential equations would (in my mind) be sufficient to teach a course for engineers. You might argue that one needs to know the theory of manifolds first, but I see this as analagous to studying calculus without really knowing the structure of $\mathbb{R}$.
From my viewpoint, differential geometry is the logical extension of calculus. Based on it's huge (and growing) impact on applied disciplines, It seems logical to have a course in it for engineers and physicists, which I would put immediately after the final semester of calculus (assuming the students have also had linear algebra).
So my question is this: Are there specific instances, either textbooks or courses at a university, of differential geometry classes taught with the intent of being useful for engineers and scientists, which assume only basic calculus knowledge and linear algebra? (Obviously, there are books like "Differential Geometry for Physicists," but I really mean something that would be used by mathematicians teaching such a course). If so, how successful have these courses/books been? If not, or if the attempts have been unsuccessful, is there any particular reason as to why it is not feasable/common?
Best Answer
I think, one of the big problem is that aside from theoretical physics (string theory, general relativity), most mathematicians aren't terribly aware of what engineers and scientists use differential geometry for. This certainly makes it difficult to write/plan a course in that regard.
It was only recently when I heard a talk by Alain Goriely did I find out that biologists care about differential geometry too! But during the talk there were quite a few theorems about curves in three dimensional space that I've never heard of, and I do geometry PDEs and general relativity for a living. This at least provides an isolated data point to illustrate the above, on how mathematicians typically don't know what is or is not important for applications to other fields.
Ideally such a course/textbook should be prepared by someone with great interdisciplinary familiarity.
In terms of differential geometry "as a natural extension of calculus", I think you may have better luck going to older textbooks, where instead of calling it differential geometry, the subject is just called "advanced calculus". Quite a few books are written back then with an eye toward the applied mathematician (but of course, I am incapable of giving recommendations).
Let me add that I am currently supervising a third-year undergraduate course in University of Cambridge on differential geometry. It fits half of your bill: it does not assume more than basic calculus and linear algebra (partly due to the funny way the Cambridge maths curriculum is rather scant on analysis); the current set of lecture notes is written by Gabriel Paternain (if you are interested you can try asking him for a copy). Unfortunately the way the degree program works, the course won't attract much non-pure-mathematicians other than the future-theoretical-physicists. So I can't really comment on how well it works for engineers and other scientists.
The course is divided in essentially four parts:
One more note: I just remembered that Gary Gibbons is teaching a course titled "Applications to Differential Geometry to Physics". It is not necessarily elementary, but certainly has a lot of applications. Being taught from the point of view of a polymath, the examples given in the notes do cover some more ground than is typical.