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I've seen "advanced calculus" as a prerequisite to some courses. What are people referring to when they say that?

  • Is it a particular set of topics?
  • What are some key examples of books that cover these topics?
  • What are the prerequisites to advanced calculus?
  • Is it normally an undergrad or grad-level subject?

P.S. There is a related question here but I wanted to ask a more focused one.

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    $\begingroup$ Typically it is just undergrad analysis. i.e. construction of reals, sequences, limits, derivatives, Riemann integration, series, etc. $\endgroup$ – user204299 Apr 9 '16 at 1:49
  • $\begingroup$ As the answer to your linked answer indicates, there isn't much agreement on what exactly "advanced calculus" entails $\endgroup$ – Omnomnomnom Apr 9 '16 at 1:50
  • $\begingroup$ You can get some idea of commonalities and differences by Googling course catalog "advanced calculus". For example, at Madison Wisconsin, at Rutgers, at the University of Akron, etc. $\endgroup$ – pjs36 Apr 9 '16 at 2:16
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Edit: It occurs to me I did not fully answer the question in the OP, so I have included some more details now.

Calculus is taught in America (and maybe in other parts of the world) in several "passes," with increasing rigor and scope. The exact order of the approaches varies from place to place, but these different trips through the theory of calculus all have a few things in common.

1) Naive calculus: This is calculus which is highly computation and application based. Students see limits in terms of tables of values, and the idea of "go close but don't touch." You compute so many derivatives and integrals your eyes bleed. You see things like related rates, applications to physics, arc length, volumes of revolution, etc. These are the "easy" applications of calculus that you can do without much theory.

Some standard books for naive calculus that I like are Calculus Deconstructed, by Nitecki (Hope I spelled that right), the book by Simmons, Calculus with Analytic Geometry. The former is aimed much more at mathematicians, while the latter is a giant book that is also useful in multivariable. It includes many applications. Proofs of key theorems are in appendices, which is nice.

2) Multivariable calculus: This is calculus where all the stuff from the "naive" section is generalized to several variables. You do more complicated integrals, learn about partial derivatives and a chain rule for several variables. There is some amount of emphasis on geometry in the sense of vectors, surfaces, and maybe some introductory linear algebra, but the bulk of the theory is left out.

Both these classes are the kind of things new students take in their early years in mathematics. They are likely aimed at engineers or physicists rather than aspiring mathematicians.

I don't know a book that I'm really happy with that's "naive" in the sense of (1) but covers multivariable other than Simmons book above. I suppose Larson is the standard reference.

What follows is material that is more likely to be "advanced calculus."

3) Elementary analysis - limits of sequences, series, topology of the reals, functions, continuity, etc. The bulk of the course is in these "foundational" topics in analysis. The idea is to figure out what makes the theorems tick, rather than proving the theorems, as by now, students have seen the intermediate value theorem before, but since they've probably never heard of a "supremum" they are not in a position to prove it.

Ross's book "Elementary Analysis" is pretty compact, covers everything important, in my opinion, if a little basic. Baby Rudin is the standard reference for people who are up for the challenge.

4) Multivariable analysis - A rigorous generalization of the material from (3). Certainly includes a substantial amount of linear algebra and topology of metric spaces, and things like a rigorous treatment of the Jacobian and its significance, Taylor's theorem in several variables, the inverse and implicit function theorems, Stoke's theorem, etc. This is what we call "advanced calculus" in my school.

The book they use at my school is Munkres "Analysis on Manifolds" which is actually not about manifolds all that much, but about the tools used in the study of manifolds, which you would then apply in a more sophisticated setting. I like this text. An honorable mention in the analysis category is the two-volume sequence by Zorich called "Mathematical Analysis" which covers both (3) and (4). In some sense, this is an interesting series because it assumes the reader knows nothing, and includes lots of applications, but is certainly rigorous, and includes key theorems in the development of the real numbers. Between both volumes, it goes pretty deep too - not as deep as Munkres, but it also includes a variety of topics you won't find in Munkres' book that's really interesting stuff in its own right, like stuff in differential equations and Fourier series.

5) Measure theory(?) - This could be included in that scope. It typically refers to the study of Lebesgue measure and integral, as a generalization of Riemann integration that people are familiar with.

The introductory book at my school is Capinski. I don't like this book. I feel it is highly unmotivated and perhaps hard to read... too many computations and not enough intuition for me, especially as analysis is a weak spot for me. I much prefer Royden's book. This is a nice text because (a) it includes all those missing pieces of intuition that I really need, (b) it starts with the Lebesgue measure but eventually passes to general measures all while (c) introducing the reader to a really nice collection of other cool things in analysis, especially the elements of functional analysis.

Measure is kind of the bridge between undergraduate and graduate work, so everything after here is at another level, and is probably not so much "advanced calculus" as "analysis" proper. We've been saying "analysis" all this time, but really what that means in undergraduate studies seems to be "the theory of calculus of one or more variables." A standard reference is something like the book by Folland.

Depending on context, it's not completely unreasonable to think that "advanced calculus" refers to calculus on manifolds, which refers to taking the ideas of (4) and transferring them to more abstract settings where more powerful geometric tools can be developed. But again, these ideas are really entering into graduate level stuff that's more "analysis" or even geometry than "advanced calculus."

In my studies, the path I took was (1) -> (3) -> (2) -> (5) -> (4), which is a little non-standard I think, but I do a lot of self-studying, so I picked things up kind of hodge-podge, and have been assembling them in my brain ever since. I think the order I listed them is probably a little more standard, although I think people probably study measure theory before worrying about some of the things I listed under (4). It all comes down to what the exact content of the courses are.

Here is my undergraduate bulletin. The courses correspond roughly to

1: MAT 131/132. 2: MAT 203 3: MAT 319/320 4: MAT 322 5: MAT 324

http://sb.cc.stonybrook.edu/bulletin/current/courses/mat/

I'm sorry if it seems like I ramble about books in places - I'm a book hoarder. Hope this helps more completely answer your question.

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  • $\begingroup$ Ah, "bulletin", that's another one of the magic words. Good thinking! (In addition to a very good overview) $\endgroup$ – pjs36 Apr 9 '16 at 2:27
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    $\begingroup$ Most universities, advanced calculus is a slightly more "accessible" version of real analysis, very rarely so sophisticated as a calculus on manifolds course. $\endgroup$ – Ted Shifrin Apr 9 '16 at 3:15
  • $\begingroup$ about 3, and 4 are they same as real analysis and complex analysis. $\endgroup$ – Muhammad Umer Oct 17 '17 at 2:41
  • $\begingroup$ math.stackexchange.com/questions/2475355/… $\endgroup$ – Muhammad Umer Oct 17 '17 at 2:41

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