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.
Best Answer
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.