Courses Which version of Real Analysis to take?

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1. May 10, 2017

gstroot

Hello,

I am a mechanical engineering student that loves mathematics and fluid mechanics. My school offers three different analysis courses and I’m not sure which to take. I took honors Fundamental of Mathematics, where we covered Abstract Linear Algebra, Set theory (along with rings and fields), morphisms, construction of the reals, and analysis up to Heine Borel and Bolzano Weirstrass. I also want to be challenged and push my thinking.

Which of the below courses will challenge me, but not be overwhelming? Or how much harder is one versus the other, just from the list of topics of course.

Math 424: Honors Analysis

A rigorous treatment of basic real analysis via metric spaces recommended for those who intend to pursue programs heavily dependent upon graduate level Mathematics. Metric space topics include continuity, compactness, completeness, connectedness and uniform convergence. Analysis topics include the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations. As part of the honors sequence, this course will be rigorous and abstract.​

Math 447: Real Variables (http://www.math.uiuc.edu/Bourbaki/Syllabi/syl447.html)

Careful development of elementary real analysis for those who intend to take graduate courses in Mathematics. Topics include completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration.​

I would like to take the more rigorous one if possible, but I don’t want it to take away from the classes in my major. I will also be taking a lab course, doing research, and taking a graduate level fluids course. What would you all recommend for me?

2. May 10, 2017

Staff: Mentor

They both don't sound much differently. Whereas 424 stresses the rigor of methods presented, you added Bourbaki to 447, which in itself is rigor and abstraction. In any cases I would chose the one that will spend more time on differentiation and integration rather than on completeness of the reals or connectedness. Unfortunately, this cannot be deduced from the descriptions above, but maybe from additional information about who will hold the lecture. I would tend to 424 not knowing the teacher, which would be a major criterion if I had to chose.

3. May 10, 2017

gstroot

That would be Math 424 that focuses on integration and differention, as the honors pre requisite, that I took, covered Bolzano-Weirstrass and the construction of the reals through Cauchy sequences already. Also it should be noted that Math 424 will have a better teacher, as is the custom with honors courses. Do you think Math 424 is worth the extra work for a guaranteed good professor?

Also how would it be relative to a course cover this? I should get an A or A- in this class, but it was very hard. I also loved every moment of it though. I know you can't give a definitive answer, but any help is greatly appreciated

Edit: Here is the Math 424 syllabus from last semester

4. May 10, 2017

Staff: Mentor

This is impossible to answer for several reasons. Too many unknowns in this equation. I tried to assess your question from an engineer's point of view, which means that differentiation and integration are valuable tools you will probably have to learn anyway. Also "interchange of limiting operations" sounded good, without knowing what it means in detail. If you can handle the additional work according to your schedule, it's probably worth it. (I suppose it's ahead of you anyway.) However, it's always better to learn fewer things well, than more stuff worse. Usually the amount of exercises is the time eating aspect here.

I'm not quite sure, how Bourbaki presents analysis, but usually their style is rather formal, not to say boring. But this is a personal opinion. Some find it easier to learn like that, others prefer a type of learning, that is closer to real life problems.
I don't see a major overlapping here, except perhaps a bit about metrics and convergence. What's behind your link are fundamental concepts, which are needed in all further realms of mathematics or physics. In my experience, the difference between (linear) algebra and calculus is, that students are more used to analytic concepts as they occurred at school more often, whereas algebraic concepts tend to be a bit "new" to them. So the difficulties arise from a new way of thinking, rather than the content itself. But this again, is a personal statement which might not be true in all cases.

5. May 10, 2017

gstroot

Ahh that makes sense. Thank you for the advice. I'm not really taking this for any engineering reason, but for my love of mathematics and to study topology eventually. Does this change anything?

6. May 10, 2017

Staff: Mentor

No. You can always go further. And 424 also covers topological basics, if not more. All you'll have to keep in mind is, that not all topological spaces come from a metric. Dimension is of secondary interest in topology (@all purists: concerning the basics), why I found the emphasis of $\mathbb{R}^n$ in the second description a bit odd.

7. May 10, 2017

onoturtle

This is all speculation that you'll need to figure out if it applies to you or not. I'm guessing if you're in the honors track with your current foundations course you probably have to take 424 next. If you're not in honors, I presume you'll need to get instructor permission to register, assuming there is a free seat to go to a non-honors student. If your current class isn't honors and you're already finding it very difficult, do you have the time to deal with the increased difficulty of an honors course? It doesn't sound like it since you're pursuing this on the side out of self interest.

You can also check what the prereqs are for the topology course you want to see if there is some requirement either way.

I'm guessing either is fine. And probably a better learning experience than trying to learn it by yourself like I attempted some years ago with baby Rudin haha

8. May 10, 2017

gstroot

I am currently in the honors and I find it hard, but I still had an A going into the final. That's a good point; I need to set limitations since this is a "for fun" course.

Math 424 is the pre requisite for Math 535, which is a graduate level topology course that covers point-set topology and algebraic topology. Oh definitely, learning from Rudins is hard enough by itself.

9. May 10, 2017

onoturtle

Oh, since you are taking foundations in honors, I'm less worried about you taking 424. I just wanted to make sure you knew what you were getting into!

No undergraduate topology course? My university had one using, depending on the prof, Munkres' book (not sure how much of that it covered though; I didn't take the course) and the only prereq was the foundations course not unlike your's. I only read parts here and there (for point-set topology, I didn't touch the algebraic stuff), but Munkres' book seems really well written if you're curious to dive into self study. I thought it complemented baby Rudin's topology chapter that went over my head and it gave me another take on the concepts. I don't have much formal math background. Beyond my comp sci degree requirements, I only took courses in abstract algebra "for fun", as you put it, which probably was the most difficult courses I took as an undergrad and really had to fight for the A- I got. Felt really rewarding though. So I get your sentiment for taking these extra math courses.

10. May 11, 2017

Dr Transport

other than for knowledge sake alone, I do not see where a course in real analysis would help you as a mechanical engineer at all..

11. May 11, 2017

gstroot

I've always had a dream of taking Real Analysis and I wanted to major in math, but I did not see it as a practical without a PhD. Besides that, one can take topology after analysis and then utilize manifolds and topological surface for fluid mechanics (at least this is what my research professor has told me).

12. May 11, 2017

Staff: Mentor

Exactly. Navier-Stokes for compressible and incompressible fluids, Euler differential equations for currents in frictionless elastic fluids, Cauchy's law of movement and everywhere else where material derivatives (alias Euler operator, alias advective derivative, alias convective derivative, alias derivative following the motion, alias hydrodynamic derivative, alias Lagrangian derivative, alias substantial derivative, alias substantive derivative, alias Stokes derivative, alias total derivative) are used. The many names alone indicate the broad applications of it.