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 ℝR.
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?