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WillJ

- 13

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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.)

I'll of course talk to my adviser and other professors about this, but I also wanted to hear your input.

(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!)