![]() Linear algebra, multivariate calculus, and a bit of differential equations would (in my mind) be sufficient to teach a course for engineers. But universities teach elementary calculus classes, most of which are not terribly rigorous, but are sufficient for the purposes of non-mathematicians. To cover differential geometry rigorously, of course one needs quite a bit of advanced mathematics, including topology and analysis. 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. Some of the people I talked to also expressed a lot of difficulty in learning it for the first time on their own. No schools I looked at had an elementary differential geometry class in, say, a similar style as the calculus sequence. 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. I've found that in talking to professional physicists and engineers, most of them find some use for differential geometry nowadays. It's related to research in math education, but not directly to math. Lee's: Manifolds and Differential Geometry.I have to admit I'm not sure if this is an appropriate question.
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