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 Posted: Thu Oct 19, 2006 7:45 am
I know there are at least two, very knowledgeable math-peoples out there. I was wondering what prerequisites you would recommend for complex analysis, topology, and differential geometry. Of course, the course guide states the "required" prerequisites, but sometimes those can be severely underjudged. For example, there are many senior-level classes which supposedly "only" require linear algebra, despite being grounded in either analysis or number theory. So I'm basically asking, "What should I know going into [course]?"
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Posted: Sat Oct 21, 2006 10:34 am
I know what you mean I entered a quantum physics course only to find out that I didn't have the real pre-reqs so I had to drop the course sad
The best way I can think about this is to ask the professor through an email.
^.^*
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Posted: Sat Oct 21, 2006 12:30 pm
You know, I have that problem with the current Lagrangian mechanics course I'm in now. Technically, the course only requires a cursory knowledge of multivar calc, but the problem sets so far have required us to be able to work with a lot of complex numbers and lots and lots of differential equations. The only two people doing really well in the class are in/have taken complex analysis and partial differential equations.
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Posted: Sat Oct 21, 2006 11:28 pm
Well, I'm currently doing differential geometry armed with intro real analysis and intro linear algebra. So far it's been heavy use of Leibniz' law, a lot of differential forms, and the Levi-Civita connection. And the pullback operator. Goddamnit, I'm starting to hate that thing.
The rest is just equation manipulation with little bits of linear algebra, group theory, and geometry thrown in.
The course was listed as having no pre-reqs other than the intro real analysis/linear algebra that all freshmen math majors take.
My intro topo class doesn't really require anything either, although a bit of experience with point-set/metric topology helps; informal experience is good enough. But then, supposedly it's easier this year than it was in previous years.
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Posted: Mon Oct 23, 2006 10:32 am
Did you take single-variable or multi-variable real analysis, Layra? For some reason, they had me take both, so I'm just now in multivar as a sophomore. I think I could've just as easily done linear algebra then multivar...and there are some in my class who took single-var like I did, and then simply skipped to complex analysis this semester. Single-var seemed like a pretty useless class.
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Posted: Tue Oct 24, 2006 12:27 pm
Oh, definitely multivar. You definitely need multi-var for diff. geom. Or at least the machinery that multi-var gives you: connections, vectors/covectors, differential forms, pullbacks, exterior calculus.
And I forgot to mention something before: be ready for horrendous notational abuse. The notation gets abused more than a fat nerd in middle school.
Yeah, single var is fairly useless. They should start with lin. algebra, then do multi-var and tell people that single-var is just a special case of multi-var.
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Posted: Wed Oct 25, 2006 6:40 am
Layra-chan The notation gets abused more than a fat nerd in middle school. My roommate couldn't understand why I started laughing so hard at this. It feels kinda wierd, though. Single-var was my introduction to *real* math. Before then, I'd never seen a proof and hadn't even known there were other fields of math out there. I definitely could NOT have survived linear algebra that first semester. But on the other hand, the linear algebra-multivar track you're thinking about is what this year's freshman are doing. So it feels like I'm constantly behind now not only my peers (both you and my friends here) but also the incoming frosh. I'm also now slightly scared 'cos as far as I know, connections, covectors, pullbacks, and exterior calculus are not part of my multivar curriculum...the professor claims "I'm not an analyst, I'm a topologist". I think I'm going to end up in a course on special topics in algebra next semester. Out of complex analysis 1, topology, and differential geometry, which do you think is most simple/necessary?
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Posted: Wed Oct 25, 2006 4:19 pm
Topo is probably both the simplest and the most widely used. It will show up in analysis, although topo on the complex plane isn't that complicated, and it will show up in diff. geometry.
I'm CAing one of the lin-alg/multivar courses for the frosh, and the course assumes that the kids haven't seen too much proof so far. The first few problem sets were still fairly horrible. But they're improving.
I'd recommend that you find and read Spivak's Calculus on Manifolds. It's a short-ish, very good book on multivar. How much linear algebra have you done?
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Posted: Fri Oct 27, 2006 12:28 pm
I've taken the linear algebra course (using Hoffman and Kunze), and our textbook for this multivar course is Spivak's Manifold Calc. I've also read through Rudin's Principles (baby Rudin) on my own, though; while I like Spivak's treatment of manifolds and integration, it lacks the analytic rigor expected in this course. Not that Rudin is an easy book to just pick up and read, but I refuse to settle for less than an A in this course. But neither of them mention connections, covectors, pullbacks, and exterior calculus in those terms exactly. Maybe they're called something else, but I doubt it.
I s'pose I'm just frustrated. The people I'm competing with for top spots in the math department skipped multivar calc to take linear algebra and then complex analysis, along with other out-of-the-way math courses. In physics, the top dogs took electromagnetism, quantum I, and Lagrangian mechanics last year. It just feels like I'm behind everyone else.
Our courses just assumed that we knew the notation and understood the proofs. I mean, I don't think I've really done many problems in these courses that haven't been proofs. Of course, it doesn't help that our TA this year expects our proofs to be analytically rigorous, while our professor ("a topologist") doesn't display any type of rigor in the few proofs he does do.
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Posted: Fri Oct 27, 2006 1:36 pm
The exterior derivative is just the derivative that you're using in Spivak. A covector is just something of the form , where a is a vector and < , > is the inner product. In the case of diff. geom, the vectors are differential 1-forms in the directions of the coordinate axes, so if we have coordinates x_1, x_2, x_3...x_n in R^n, then the vectors are dx_1, dx_2, etc, and the covectors are partial/(partial x_1), partial/partial x_2, etc, where partial is the curly d in partial derivatives.
A connection is basically a way to map from the tangent space at one point to the tangent space of a nearby point, so that you can take derivatives of vector fields on a manifold. You'll learn more about them in diff. geom.
The pullback of f with respect to g, denoted as g*f, is just f circ g.
Don't worry about going too deeply into these topics on your own; just keep these definitions in mind because you'll be seeing them all over the place in diff. geom.
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Posted: Tue Oct 31, 2006 12:07 pm
Layra-chan The exterior derivative is just the derivative that you're using in Spivak. A covector is just something of the form , where a is a vector and < , > is the inner product. In the case of diff. geom, the vectors are differential 1-forms in the directions of the coordinate axes, so if we have coordinates x_1, x_2, x_3...x_n in R^n, then the vectors are dx_1, dx_2, etc, and the covectors are partial/(partial x_1), partial/partial x_2, etc, where partial is the curly d in partial derivatives.
A connection is basically a way to map from the tangent space at one point to the tangent space of a nearby point, so that you can take derivatives of vector fields on a manifold. You'll learn more about them in diff. geom.
The pullback of f with respect to g, denoted as g*f, is just f circ g.
Don't worry about going too deeply into these topics on your own; just keep these definitions in mind because you'll be seeing them all over the place in diff. geom.Oh...then nevermind, I know all of these terms. Why are they called anything different from "derivative, del, field derivative, and composition"?
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Posted: Tue Oct 31, 2006 4:02 pm
Swordmaster Dragon Layra-chan The exterior derivative is just the derivative that you're using in Spivak. A covector is just something of the form , where a is a vector and < , > is the inner product. In the case of diff. geom, the vectors are differential 1-forms in the directions of the coordinate axes, so if we have coordinates x_1, x_2, x_3...x_n in R^n, then the vectors are dx_1, dx_2, etc, and the covectors are partial/(partial x_1), partial/partial x_2, etc, where partial is the curly d in partial derivatives.
A connection is basically a way to map from the tangent space at one point to the tangent space of a nearby point, so that you can take derivatives of vector fields on a manifold. You'll learn more about them in diff. geom.
The pullback of f with respect to g, denoted as g*f, is just f circ g.
Don't worry about going too deeply into these topics on your own; just keep these definitions in mind because you'll be seeing them all over the place in diff. geom.Oh...then nevermind, I know all of these terms. Why are they called anything different from "derivative, del, field derivative, and composition"? The exterior derivative is called such because there is such a thing as an "interior" derivative, which has very different properties. Both are important, but the exterior derivative shows up a lot more. Del is a very specific version of what I'm getting at, equivalent to basically the vector (1, 1, 1,...). Oh, by the way, I screwed up on the vector/covector thing: dx_i are the covectors, and partial/(partial x_i) are the vectors. So a general field of vectors would be all elements of the form sum_i {a_i partial/(partial x_i)} where a_i are functions of (x_1,...,x_n) (del is thus a_i = 1). The connection is not actually the field derivative, but rather a way to find how to transport vectors in a "parallel" manner. A derivative can be derived from it (no pun intended) but the connection is more than just the derivative. The pullback I don't quite get, but there's some stuff with derivatives that looks better with pullbacks than with compositions, especially if there's multiple arguments. Also, the pullback g* is usually a function on cotangent bundles or tensors and should be thought of as a function of f rather than as a function on the domain of g. Suppose f has n arguments (x_1,...,x_n). Then g*f(x_1,...,x_n) = f(g(x_1),...,g(x_n)). Wikipedia actually has a good set of articles on all this stuff.
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Posted: Thu Nov 02, 2006 11:46 am
I'll have to look more into it. So what do you think I should be taking within the year? I figure if I keep a courseload of 2-3 math, 1-2 physics, and 0-1 humanities I should be able to double major, single minor. The physics track is pretty much set in stone, but math is almost completely open-ended.
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Posted: Thu Nov 02, 2006 10:34 pm
It really depends on where you want to go. I'd recommend abstract algebra and complex analysis and topology, in some combination.
I'm taking an absolutely useless intro topology course that I don't really need to take as well as a diff. geom. course that I absolutely love, and will be taking abstract algebra (rings/fields) and complex analysis (maybe) next semester. I'm planning to go into geometry or topology, so I'll be loading myself with geometry and topology.
But yeah. Complex analysis, abstract algebra, and topology are three courses you need to take as soon as possible. From there you can wander around.
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Posted: Sat Nov 04, 2006 5:28 am
Well, we have this four-course analysis sequence that I'll probably take all of eventually (partial diff. eq., complex analysis, integration theory, and special topics). I'm in an abstract algebra class now, which I love. It's s'posed to cover rings, fields, and Galois theory.
I really have no clue what kind of math I want to go into. I've also never had a course on probability or number theory, and I might want to do some of that. But recently I've been on a trip with my physics class, and I'm liking String theory ideas more and more (so diff. geometry, integration theory, and complex analysis are necessary) but outside of that, I don't know what I want to specialize in.
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