# A course in multivariable real analysis?

I have a question about university course offerings.

This semester I'm taking a course called "Introduction to Analysis," which uses Edward Gaughan's Introduction to Analysis, and is basically just a more rigorous/proof-based coverage of the topics we learned in the first two semesters of (single-variable) calculus. Neither the book nor the course ever cover functions of more than one variable. (In contrast with, say, Rudin's Principles of Mathematical Analysis, whose last three chapters deal with multivariable topics, or Apostol's Mathematical Analysis, whose last five chapters deal with multivariable topics.)

My school has no other undergraduate class in real analysis (just one in complex analysis). So to go further in real analysis at my school, one must start taking analysis at the graduate level. The first PhD course in analysis, as far as I can tell, is still just about single-variable topics (basically, it seems like, the same kind of topics as my current course, just at a deeper level). In fact, I'm not sure if one ever covers multivariable stuff until two or three courses in at the graduate level!

This seems a little crazy to me. What I'd really like to do is study the topics that I learned in multivariable calculus a second time, at a deeper / more abstract / more rigorous / proof-based level (especially since I don't think I have a super-great understanding of that material, so I would love to deepen my understanding), but I'm not sure how I can do that. Maybe I'll just have to self-study it? If so, how would you guys recommend doing that? I also wanted to hear your guys' opinions on my university's situation --- don't you think it's strange that this kind of thing isn't offered in an undergraduate course? (Keep in mind we have a respectable math department that offers a wide range of courses.) Or is it normal for this to be lacking in the curriculum? Maybe this area of math isn't as important as I think it is? (It just seems strange to me to learn all sorts of nifty things in multivariable calculus, and then to just regress back to single-variable topics when taking the rigor up a notch, without ever re-covering multivariable stuff.)

(For the record, this is my junior year, and after I graduate I plan on getting a PhD in economics --- for which, in case you don't know, multivariable calculus/analysis is pretty freakin' important!)

i think the standard books are spivak's calculus on manifolds & munkres' analysis on manifolds. both need linear algebra & some basic topology & metric spaces like you'd do in that gaughan book. both culminate in stokes' theorem, which is the multivariable version of the fundamental theorem of calculus. i don't know that every university would have actual courses on that stuff though, & what it has to do with economics i have no idea, but see if it's what you're looking for:

I'll summarize basic analysis in R^n. Basic integration in R^n introduces almost nothing new, except perhaps the appropriate notion of measure zero so that one can prove the theorem that completely characterizes Riemann integration. Fubini's theorem and the change of variables theorem for R^n have uninteresting proof techniques. The extensions of the important differentiation theorems (Chain Rule, MVT, Taylor) taught in single variable calculus to R^n are uninteresting and there is no simple bidirectional relationship between the total derivative of a function and its partial derivatives. Unfortunately I haven't learned about differential forms, but I figure that's where most of the interesting theory is anyways.

The reason why the theorems are not particularly interesting is because the nature of the subject is computational. Probably the most important application of basic differentiation theory in R^n is to optimization, and here the big theorems (Taylor, Inverse Function, Implicit Function) are used to verify computational tools such as the second derivative test, usage of quadratic forms, and Lagrange multipliers. The proof techniques involved in these theorems are hardly different from those you learn in single variable analysis (as long as you have the appropriate metric topology and linear algebra background suggested by fourier jr). Therefore learning each of these theorems deeply will not necessarily enhance your understanding of optimization tools in economics.

From the economics courses I've taken, I've gathered that multivariable calculus is important, but basic multivariable analysis is not that important. Understanding the proofs associated with theorems about Lagrange multipliers has not allowed me to use the tool more effectively. Part of the reason is that you are always working with say a Cobb-Douglas function or the more general CES production function, and all the conditions that allow you to use Lagrange multipliers has already been verified for you. But really, the optimization tools I've already mentioned is really the limit of multivariable calculus as far as important applications go. The next step in optimization tools involves numerical methods and linear programming. Economic growth requires facility with differential equations. Econometrics requires facility with statistics and linear algebra.

This is not to dissuade you from learning analysis in R^n, but I think a very strong grasp of single variable analysis is more important for economic theory (especially at the graduate level). You should at least have a full grasp of the first 7 chapters of Rudin's PMA. Having said all this, if you want to learn analysis in R^n, I would also go with Spivak's Calculus on Manifolds for the basic theory (chapters 2 and 3 if you already know topology in R^n). Apostol's Mathematical Analysis has an excellent treatment of differentiation in R^n and its applications to optimization (chapters 12 and 13).

EDIT: For reference, here is preparation material for a graduate course in mathematical methods in economics at UChicago

http://home.uchicago.edu/~kovrijny/teaching.htm" [Broken]

Note that pretty much all of the real analysis material is basic undergraduate real analysis. There is however a separating hyperplane problem in the calculus problems.

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that's probaby a better answer than mine :tongue: