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Related but not duplicate: What courses require multivariable analysis?

By multivariable analysis I mean the rigorous version of multivariable calculus (something equivalent to Ch.9-10 in baby Rudin or topics cover in Analysis on Manifolds by Munkres).

My question is,

Why do advance analysis courses(at least those I took) emphasize so little in topics related to multivariable analysis, compare to topics like measure theory and Lebesgue integral? Is there any reason undergraduate/beginning graduate math education is designed this way?

Aren't they equally important prerequisite for different advanced topics(e.g. Measure Theory to Probability/Stochastic Process, Multivariable Analysis to Differential Geometry, etc.)?

My experience and observation,

  1. In my undergraduate institution, 1st semester real analysis is first few chapters of baby Rudin, 2nd semester is introduction to measure theory. 1st semester graduate level is point set topology/measure theory, 2nd semester graduate level(which I didn't finish) is closer to functional analysis.

  2. In the graduate program I'm currently in, 1 year sequence of graduate real analysis and qualifying exam focus almost exclusively on Measure theory and Lebesgue Integral. Topics in multivariable analysis are included in 3rd quarter of honor undergraduate/intro. graduate course where graduate students are only require to take it if not passing (preliminary) assessment exam.

  3. The focus of real analysis qualifying exam in many places seem to be more on Measure theory and Lebesgue integral, too.

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$\newcommand{\RR}{\mathbb{R}}$I am not really sure if I understood your question correctly, especially if we have the same understanding of what "multivariable calculus" is.

If by multivariable calculus you mean that topic where you do differential calculus for maps $f:\RR^n\to\RR^m$, i.e. introducing total differentiability, the Jacobian, the Hessian, Taylor's formula for such maps, paths and tangents, the implicit function theorem and such, then I don't understand your question as these are frequently taught in analysis courses (as far as I see, but this may not be correct).

However, if by multivariable calculus you mean the multivariable theory of integration then things get more complicated. There are different choices to order the content: Do integration of smooth functions over smooth structures with an ad hoc approach using Riemann integrals and Fubini type arguments without appealing to measure theory at all. However, when moving to integrals over surfaces one needs some theory to back up what the surface element is and I don't know any proper motivation without the change of variables formula. This suggests to introduce Lebesgue integration theory first. Personally, I find the approach with ad hoc integration not very satisfying. What I like about integration is that it can deal with very rough functions. Moreover, using Lebesgue's approach one can somehow "rectify" several things that seem a bit obscure in basic analysis (e.g. the fundamental theorem is much nicer for absolutely continuous functions, the notion of Lebesgue point shows how well defined measurable function really are and also there are Lusin and Egoroff who show what's behind continuity and uniform convergence). But, in essence, I agree that the choice "Lebesgue vs. differential forms" is somehow a matter of taste (or, agreement in the department).

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    $\begingroup$ (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 $\endgroup$ – user2139 Feb 2 '16 at 7:22
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    $\begingroup$ 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. $\endgroup$ – user2139 Feb 2 '16 at 7:39
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    $\begingroup$ @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. $\endgroup$ – Steven Gubkin Feb 3 '16 at 4:40

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