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Hello. I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you will).

We had an analytic geometry course in fall; now we are having a course in linear algebra and it seems like most of the facts from "analytic geometry" are proved in a much more systematic and easier manner (quote from wikipedia "Linear algebra has a concrete representation in analytic geometry"). Many of our progressive professors also think that analytic geometry should be eliminated from the curriculum to clear more space for a linear algebra course.

So I'm confused: 1) if analytic geometry is a "concrete representation" of linear algebra, then why is it studied along with calculus (and not along with linear algebra) in US universities? (e.g. textbooks like Simmons )

There were, however, interesting parts of the course that were not covered in linear algebra: synthetic high-school-style treatment of beautiful topics like non-Euclidian and projective geometries. Then 2) why is not there a separate course for such topics in US curricula? As I understand US freshman math majors study 2 basic subjects - real analysis and (abstract+linear) algebra (math 55 at Harvard, 18.100 and 18.700-702 at MIT). Are these geometric topics integrated into one of these courses or are not they considered worth studying for a modern math major?

Thank you

PS. This question is also important for me because it helps a lot to browse through US top universities for textbooks they use and notes. Unfortunately, Russian mathematical school is now in tatters and US textbooks are often significantly better. And since in high school geometry was among my favorite subjects I am particularly concerned about our geometry sequence and want to browse through best geometry syllabi.

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Math majors in the US don't just study algebra and real analysis! There's complex analysis, differential/Riemannian geometry, number theory, perhaps some point-set and/or algebraic topology... I think the main point of the statement (about "concrete representation") is that you could study linear algebra completely abstractly, but in fact one can very often illustrate/reformulate its statements as facts about linear maps on $\mathbb{R}^n$. But the methods and ideas in analytic geometry really do belong more properly to calculus. –  Aaron Mazel-Gee Mar 28 '11 at 7:35
Using Harvard's Math 55 as an example of what the typical American freshman math major is learning is sort of like using Buckingham Palace as an example of where the typical London resident is living. It's the reality for some people, but certainly not the majority. –  Pete L. Clark Mar 28 '11 at 9:04
It's just like the word "algebra", which is used today with very different meanings at the high school level and more advanced level. –  KConrad Mar 28 '11 at 16:39
@K: right. I still get a little embarrassed when I go to a cafe and place my copy of Jacobson's Basic Algebra II on the table... –  Pete L. Clark Mar 28 '11 at 17:17
@Pete, well, more generally, we should start a campaign to vanish the word Introduction from titles! –  Mariano Suárez-Alvarez Mar 28 '11 at 17:26
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4 Answers

To answer your first question, that the label "analytic geometry" is found in the title of a calculus book doesn't mean what you might think. The reality is that in the 1960s and 1970s most calculus books had a title like "Calculus with Analytic Geometry". My father was a high school math teacher and he had a lot of these books on his shelves at home. Nearly all of them had that title. The point was that analytic geometry = coordinate geometry and these books had preliminary sections on coordinate geometry before they jumped into discussing calculus. Thus they were titled "Calculus with Analytic Geometry" to emphasize the review aspect on coordinate geometry. This way a teacher could direct students to read over chapters on coordinate geometry which would be needed in calculus (if that material wasn't taught directly in the course.)

In recent years the buzzword to have in the title of a calculus book is "Early Transcendentals", which means the author includes a discussion of transcendental functions earlier than usual in the book. (The book you mention by Simmons has some interesting features, but it is not a widely used book anymore and in particular is not used in courses like the ones at Harvard and MIT which you mention as your "model" for a course you're perhaps interested in.) In any case, the style of calculus book like Simmons aren't the ones you should be interested in anyway. You want to look at genuine math books, like Rudin's Principles of Mathematical Analysis.

To answer your second question, non-Euclidean and projective geometry can have a place in the curriculum, but they might not appear in courses titled "Non-Euclidean Geometry" or "Projective Geometry" if you're trying to find them in US course catalogs. For example, the topics might be in a course with a bland name like Geometry. Also, courses on algebraic geometry will certainly have discussions of projective geometry. [Edit: At Harvard, the course on non-Euclidean geometry is targeted at the students who do not know how to write proofs because their prior experience with math focused on computation more than conceptual thinking. The more experienced math majors there bypass that course. That the primary objects of interest in high school math seem to disappear in more advanced math makes mathematics different from most other sciences. Students of chemistry, say, would not encounter such an abrupt change.]

Concerning your PS, which I think is actually the most important part of your posting, go over to IUM (НМУ, mccme.ru) this week and speak to the faculty and students there. You will get practical and useful answers to your questions from them since they know first-hand the situation you are noticing as regards the curriculum situation and how to get a good math education in Moscow. In particular, you should look at the courses offered at IUM and consider attending them.

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Actually, I'm pretty sure I taught from a book called "calculus with analytic geometry" as recently as back in 2001. As for "Early Transcendentals", one may be tempted to take it to be a nice coded way of saying "we gave up on rigor"... –  Thierry Zell Mar 28 '11 at 9:43
I think that "early transcendentals" is motivated by fitting in better with introductory physics classes, since there are few physical systems described by polynomials (other than quadratics, which you don't need calculus for, anyway). –  arsmath Mar 28 '11 at 12:07
arsmath: calculus is still needed for quadratics in order to find their derivatives. Maybe you're thinking calculus is not needed to find max/min values of a quadratic, since that could be done just with algebra by completing the square. However, that is a very special method whereas differential calculus is an approach which solves a much wider range of optimization problems. (Also, try convincing students in calculus that completing the square is a better technique than derivatives to locate the max. value of a quadratic!) –  KConrad Mar 28 '11 at 16:43
You don't have to convince me of the value of calculus. :-) But non-calculus-based introductory physics classes are not that unusual, and they do rely on special properties of quadratics. I don't think higher-order polynomials arise much in intro physics, while sine and cosine do. –  arsmath Mar 29 '11 at 9:46
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I think you're essentially correct that analytic geometry is not considered a worthy topic of study for a contemporary US math major, or at least not worthy enough to be part of the standard curriculum.

I hear that things used to be quite different: in the first half of the 20th century math majors spent much of their career learning analytic geometry and only learned calculus in their junior or senior year. (I learned this from Paul Halmos's autobiography, which I highly recommend to all.)

Nowadays someone my age (I am in my early to mid thirties) or younger need not know what "analytic geometry" means at all: it is that much of a forgotten field. I feel like I got a sort of brief window into the past by virtue of a self-paced math course I took in the CTY program (as a high school student). In such courses you work through an entire textbook by yourself. It turns out that then when you actually read a high school math textbook from cover to cover you learn lots of things which are not covered in most actual high school courses: perhaps most notably, I assumed ever after that mathematical induction was part of the algebra two curriculum because it was in the textbook I read for that subject. (Well, not ever after. In the last ten years of teaching freshman calculus, I've found that maybe one student in 200 has encountered induction in their high school curriculum.) Eventually I got to "analytic geometry", which was ten times more about conic sections than I really wanted to know: foci, directrices, something called the latus rectum, and so forth. I wasn't that thrilled with it, to be honest. I especially remember that sometimes you got conics with "cross terms" and then you had to rotate axes in order to deal with them. This topic I did revisit later on in a linear algebra course, and I agree that it's kind of ridiculous to do it without linear algebra: it's both much harder and much less clear what's going on.

I think most of the "analytic geometry" that we do see nowadays is in the geometric applications that one often does a little bit of in either linear algebra or multivariable calculus. In fact, exactly why linear algebra is entirely sufficient for analytic geometry is not clear to me, since linear algebra is most directly concerned with linear subspaces and in analytic geometry one studies certain plane and space curves. I can see that a connection is provided by the theory of quadratic forms, but -- depending once again on the scope of what you mean by analytic geometry (I have never seen an intrinsic definition of the subject; to me it's just a collection of topics) -- I would want to use some non-linear algebra (e.g. basic projective geometry like Bezout's theorem) and multivariable calculus.

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Pete, that's latus, not lattice. Butt you spelled rectum right. :) –  KConrad Mar 28 '11 at 9:38
@K: thanks. I'm 70% sure that was an insomniac typo: I guess "lattice" is not a latin word. –  Pete L. Clark Mar 28 '11 at 9:58
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I guess one way to explain the situation is in terms of a pedagogic problem with geometry, that doesn't correspond to any professional-level problem with geometry.

As reading MO would convince people quite quickly, there is geometry based on the concept of manifold, and geometry based on the concept of algebraic variety (scheme, if you insist), and these are both "geometry" and closely related. There is material on the level of equations of degrees 1 and 2 in n variables that can be treated either way, or both, and without some of that the student will not meet some of the basic language and examples. "Analytic geometry", i.e. the use of Cartesian coordinates, is not a research subject; it is a legacy from some of the traditional ways of teaching geometry. You can use algebraic manipulation and calculus to get a hands-on feel for it. While that may be helpful you may also have to "unlearn" the attitude, which is why the material may not be regarded as suitable for undergraduates now. In the UK I think it would be unusual to see a course called "analytic geometry".

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Thank you for great replies.

I'm satisfied with the answers to my first question (thank you Pete, KConrad and Charles) and I want to clarify a little bit the second one:

KConrad, there is a course with similar topics to ours at Harvard - 130. Classical Geometries and at Berkeley Math 130 The Classical Geometries , but these courses are not often recommended for math majors, for example at Harvard "Mathematics 113, 114, 122, 123, 131, and 132 form the core of the departments more advanced courses" - 130 is not even mentioned. MIT, for example, doesn't have a similar course at all. And most of the internet math discussions for undergraduates are concerned with Algebra, Real Analysis and sometimes Topology, but Geometry isn't mentioned at all. So I guess either topics like non-Euclidian and projective geometries are integrated into those three major subjects or are considered obsolete for a modern mathematician?

As a recent high school graduate that seems rather odd, because we spent most of the time studying it. Maybe part of my confusion is explained by this discrepancy between high school math and college math. Also, thanks for pointing to UIM, a am planning to enroll the next september.

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From my limited experience, these types of geometry courses are not generally taught for math majors. Typically, they are more directed towards math education majors or engineers who are trying to get a math minor. Non-euclidean geometries are certainly mentioned occasionally, but are not always explored with any depth. The reason for this is that geometry isn't totally necessary anymore. It would be difficult to do anything serious without, say, knowing what a group is, but geometry has become a niche topic. Graduate students typically learn some advanced geometry, but undergrads rarely do. –  Logan Maingi Mar 28 '11 at 12:08
I spent a semester studying at NMU and would recommend it highly. The faculty I dealt with were uniformly excellent. (As an American I have no idea how it would interact bureaucratically with your degree at MGU, but the mathematics was fantastic.) –  Tom Church Mar 28 '11 at 16:13
Dmitry: you're new here, so I'll point out that it is not standard to write comments to your own question as an answer. If you can edit your question, then it's better to add your comments to your question, perhaps at the end of it with "Edit:" placed in front. –  KConrad Mar 28 '11 at 16:17
Tom, IUM is not officially authorized to grant degrees, so the local students go there in their free time while enrolled somewhere else as their main institution. To make this possible, the classes at IUM for local students are at night. At mccme.ru/ium/FAQ.html is more information (если Вы читаете по-русски). –  KConrad Mar 28 '11 at 16:37
Logan Maingi: thanks for clarification. Then I won't bother too much about geometry and will concentrate on analysis and algebra as most undergraduates do, thanks! KConrad: thanks for kind help! Yeah, I didn't want to post it as an answer but I messed up with authorization the first time, and accidentally got the second account. As for the IUM, I didn't study at a math school, so this september I felt I didn't have the nessesary preparation. By the way, this is one of the reasons I prefer us textbooks to russian ones - they are not so aggressive and terse for an unprepared student –  Bananeen Mar 29 '11 at 7:53
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