I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go. I'm in AP Calc BC. What should I study to eventually learn manifolds? Linear Algebra? What else?
What math is necessary to learn manifolds?
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manifolds
self-learning
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3Multivariable calculus and general topology. – 2012-01-05
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6You need to learn linear algebra independently of what ulterior interest you might have, really. – 2012-01-05
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4I agree with Mariano's answer. In fact, manifolds are quite far away from freshman calculus -- at least two solid years of study away, in my opinion -- and in between lies a bunch of things you will have to take anyway, especially multi-variable calculus, linear algebra and basic real analysis. Really I think it's too soon to be worrying about this, and I don't mean this in the discouraging sense, I literally mean that you needn't worry about it yet... – 2012-01-05
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3...If you want advice, here it is: first, enroll in a university (or very good liberal arts college -- i.e., in the Amherst-Swarthmore-Williams class) with a very strong math department. Second, concentrate now on fully mastering the material up to and including BC calculus. Really having this down will serve you well, even unto your study of manifolds. Good luck! – 2012-01-05
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5@PeteL.Clark, what's «BC Calculus»? *Before Cauchy* :) – 2012-01-05
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0@Mariano: I'm not completely sure if you're seriously asking the question (I also don't know why someone unfamiliar with the American high school system would possibly know this, so...) there are two varieties of "Advanced Placement Calculus", one called "AB" and the other called "BC". If there is some actual meaning to these letters I have never been able to divine it, but the essence is that the "BC" course is more rigorous and covers more material.... – 2012-01-05
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1... Roughly speaking "AB" would place you out of the first semester of "freshman calculus" (which I am now worldly enough to understand is itself a peculiarly American institution), whereas the highest possible score on "BC" really should be good for two semesters. Or something. I am actually more describing my own memories of my student experiences than any recent information. You could scarcely pay me enough money to look at a contemporary AP calculus exam: I fear that I would be deeply traumatized by what I saw. – 2012-01-05
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1(@Mariano: if in fact you are unfamiliar with the American high school system, that is. You are such a knowledgeable guy that I probably shouldn't assume that just because you live in South America you wouldn't know these things...) – 2012-01-05
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0An interest in general relativity really helps too! – 2012-01-05
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1@Pete, I knew about the AB variety only! Thanks for the info :) – 2012-01-05
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0@Pete L. Clark: During 1956-1968 there was just one exam. Then, beginning in 1969, both the AB and BC exams were given. Originally, A referred to certain precalculus topics, B referred to certain topics that mostly involved derivatives, and C referred to certain topics that mostly involved integration techniques and applications of definite integrals. The AB exam tested A and B, the BC exam tested B and C. Most of the A material was omitted by the 1990s, but the names were too entrenched by then to change. – 2012-01-05
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0@Dave: thanks for this explanation. As it happens I took the "BC" exam in 1993, so this explains my relative fuzziness on the nomenclature. – 2012-01-06
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0A side note: I picked up Tu's Introduction to Manifolds well before I was really ready to get much out of it. At first, it was terse and foreign to the point of being traumatizing. As I keep revisiting it, it broadened my horizons tremendously. And it tied together ideas had been loose threads for years. (eg. the chapter on functors that answered a question I'd had since AP calculus: "Is something -- the space we're in, maybe -- changing when we take derivatives?" Answer from my calc. teacher: "Not to my knowledge." Tu's answer: "Yep, you're changing categories." I like insights like that.) – 2017-04-29