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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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  • $\begingroup$ In my graduate program, theory of differentiation & integration, including topics like the implicit function theorem, differential forms, and the generalized Stokes' theorem, rigorously proven, is in the 1 year sequence of pre-graduate (background) material. $\endgroup$
    – Opal E
    Nov 27, 2022 at 18:43

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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, 2016 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, 2016 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. $\endgroup$ Feb 3, 2016 at 4:40
  • $\begingroup$ Hello, Dirk. Is there any undergraduate text that builds the theory of multivariable integration on Lebesgue integrals? $\endgroup$
    – user1551
    Dec 8, 2022 at 10:32
  • $\begingroup$ Some German books do it: Forsters "Analysis 3" and "Analysis 2" by Barner and Flohr for example. Hewitt and Stromberg define the Lebesgue integral (and the measure comes later) in "Real and abstract analysis". $\endgroup$
    – Dirk
    Dec 8, 2022 at 11:17
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My understanding of the reason (I am in a university in USA ranked 50-100):

1, No motivation

A faculty does not have right to open a new course freely to his/her discretion. A new course has to be applied for to the college with pretty much paper work. After approval, the contents of the course are fixed and no way to extend.

If an instructor teaches more to students, s/he (and the TA) will not be paid more.

In some culture, being eager to open and teach a hard course but failed in the end is embarrassing. Colleagues will laugh at it.

So, why bother?

2, Dangerous

If an instructor teaches more, often harder contents like multivariate calculus to the students, s/he is taking the risk that students will have a hard time understanding it, causing a low GPA. The students may be unhappy about it and complain to the department that the syllabus is too hard, or they have to spend more hours per day on study so that they can't play or stay with family. According to Natwest's most recent "Student Living Index", students of University of Leeds spend only 3.7 hours a day on study, which is however the longest in UK. This is particularly the case in USA. Recently, Dr. Maitland Jones Jr., a famous chemist, was fired by New York University because his course "Organic Chemistry" is too hard. 82 of his 350 students complained to the university due to their low scores in the exam. This affair was reported early last month (Oct. 3rd) by New York Times.

So, why bother?

3, Qualification

Well, admit it, some faculties themselves don't understand multivariate calculus very well. How can they teach students?

To sum up, I still think multivariate calculus is sufficiently emphasized in top universities (like Munkres' university). But top universities like MIT account for only a very small portion of all universities in the world, which gives people an impression that multivariate calculus is often omitted. After all, the course design has to match the background of students admitted as well as of existing faculties. And money is more important than education in many institutions. So, why bother?

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[Speculative, but too long for a comment.]

I think the main purpose of calculus (multi or single) is to learn how to solve calculus problems and to follow derivations, primarily to support physics, chemistry and engineering classes. Real analysis or theoretical calculus doesn't help you solve more calculus problems. It teaches rigor that underlies calculus. And of course it is a program for math majors only.

RA is a very hard course and gets people to scrunch their brains and try to create proofs. A few tools from RA are needed for follow-on courses (e.g. topology, theoretical statistics), but in general, not even most of the standard RA course.

Finally, life is full of limits (in time, money, IQ) so the "why don't we do this extra thing" (many questions on this site are like that) needs to consider what would be pruned to make way for the addition. Or, if this hypothetical course were an elective, we need to find the market for it (like enough students wanting it).

I would also add, that there's very few anecdotes of people saying they were lacking in multivariable real analysis and that it held them back in some other course. Even you don't talk about needing it for some specific purpose.

Probably the few people that really need it are in grad school (tiny market, dispersed). And even here, we have evidence from your school that it doesn't seem like a huge demand for all students even in grad school to do MV RA.

IF you need it, take that third quarter that is offered. You're lucky that you even have that option, because at a certain point, you will be too specialized and there will be too few people near you physically to justify a course. And you will have to learn it on your own. (Which is not as good as having an instructor. But that's life.)

But in any case, why push to want everyone in grad school (or undergrad for gosh sakes) to take it. Life is hard and we have limits. You can't just add requirements without considering the cost (not only financial).

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