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Feb 3 '16 at 4:40 comment added Steven Gubkin @fmlin This is my experience as well. Even many established mathematicians I have met (who work far away from analysis or differential geometry) lack an understanding of the derivative as a linear map, and the chain rule as composition of linear maps. Especially lacking is an understanding of the higher order derivatives as higher order symmetric tensors.
Feb 2 '16 at 7:39 comment added user2139 It might surprise you, but even the stuff in your 2nd paragraph is not something I learned (beyond lower division calculus) until the summer before my graduate study and some of my fellow graduate students share similar experience.
Feb 2 '16 at 7:22 comment added user2139 (Copy from Jim Belk comment) In the United States, "multivariable calculus" would refer to a course that covers partial derivatives and multiple integrals, as well as vector fields, divergence, curl, line and surface integrals, Green's theorem, the divergence theorem, and Stokes' theorem for surfaces in ℝ3R3 (but probably not differential forms). The students would be a mix of math majors, science majors, economics majors, and possibly engineers, and they wouldn't write any proofs--the focus would be on computation. Such a course would typically be taken in the first or second year of college
Feb 2 '16 at 7:11 history answered Dirk CC BY-SA 3.0