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KConrad
KConrad
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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.) But those stylesIn any case, the style of calculus booksbook 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 cancan 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.

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.) But those styles of calculus books 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.

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.

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.

Source Link
KConrad
KConrad
  • 50.6k
  • 9
  • 196
  • 277

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.) But those styles of calculus books 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.

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.