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Academic Guidance Academic Guidance - Academic guidance for those pursuing a college degree... what college? Grad school? PhD help?


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June 14th, 2010, 11:57 PM   #1
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Having trouble, have hit a wall in my learning?

Ever since I got into math this past january I've been moving at a really fast pace. I blazed thorough learning the ins-and-outs of single variable calculus (mostly because I spent so much of my free time with i), and more recently I've been learning aspects of Linear Algebra up to eigenvalues and eigenvectors, diffirential equations prior to Laplace Transformations, basic number theory, intermediate set theory, and the most recently vector and multivariable calculus. I've almost completed the book I bought on multivariable calculus. I will be going into college as a freshman this fall and I want to get ahead on as much mathematics as I can ovet this summer, but I'm having trouble. I've been reading different e-books to teach myself even higher mathematics, but I always seem to have the trouble of understanding the topics. I seem to be stuck. I cant seem to find a book that can transition me from the computational world of calculus to the axiomatic and proof-neccesary world of real and complex analysis! Am I stuck for now? Must I wait till I reach these clases in college? Set thoery seems to be the closest I can get to these other topics, and only an intermediate understanding of set theory. Is there some book or books anybody has in mind as a good transition between these two different mathematical approaches? Cheaper is better, gota save my money as much as possible. E-books are best. I welcome any suggestions from anybody about this issue, because I'm starting to feel slowly but surely trapped at the mathematical level that I am. Thanks in advance!
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June 15th, 2010, 07:06 AM   #2
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Re: Having trouble, have hit a wall in my learning?

If you are as fluent in calculus as I should assume from your post, you should consider a book on Real Analysis. And if you want it cheap, you should look for a Dover-published book on Real Analysis (also called Advanced Calculus). At least, that seems reasonable to me.
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June 15th, 2010, 09:27 AM   #3
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Re: Having trouble, have hit a wall in my learning?

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Originally Posted by mfetch22
I cant seem to find a book that can transition me from the computational world of calculus to the axiomatic and proof-neccesary world of real and complex analysis! Am I stuck for now? Must I wait till I reach these clases in college?
Frankly? Yes, most likely. You can try reading Enderton's introduction to logic (or his introduction to set theory), but you probably won't properly understand mathematical proof until you take several college courses.

I don't think that a real analysis text would be helpful; it may be the same material as calculus in some sense, but it's much harder and much easier to lose your way without a teacher. Perhaps a complex analysis book would be useful to you, though; that's somewhat easier to self-study (and much easier overall).
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June 15th, 2010, 11:30 AM   #4
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Re: Having trouble, have hit a wall in my learning?

The transition from computation to proof is something everybody struggles with, and it's very difficult to properly make without instructors. A book on discrete math may seem basic to you, but it will likely help a lot. The material covered will be rather basic, but the way it's covered is typically as an introduction to proofs and higher math.

I wouldn't second CRG's suggestion to try reading Enderton's book... it covers the material in an understandable way, but the material is rather high level for someone without previous experience with proofs.

Either way, you're certainly ahead, in terms of material and-- it would seem-- mathematical maturity, so I wouldn't worry too much.
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June 15th, 2010, 11:56 AM   #5
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Re: Having trouble, have hit a wall in my learning?

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I wouldn't second CRG's suggestion to try reading Enderton's book... it covers the material in an understandable way, but the material is rather high level for someone without previous experience with proofs.
I quite agree, but [s]he asked...
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June 15th, 2010, 05:17 PM   #6
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Re: Having trouble, have hit a wall in my learning?

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Quote:
Originally Posted by cknapp
I wouldn't second CRG's suggestion to try reading Enderton's book... it covers the material in an understandable way, but the material is rather high level for someone without previous experience with proofs.
I quite agree, but [s]he asked...
Just offering my two cents.
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June 15th, 2010, 06:17 PM   #7
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Re: Having trouble, have hit a wall in my learning?

Has anybody else experienced a large reduction in the rate of learning mathematics at the level that I described, as in, is anybody fimmiar with this slow down between these two levels of mathematics that I'm experiencing, and is it something that is common?
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June 15th, 2010, 09:59 PM   #8
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Re: Having trouble, have hit a wall in my learning?

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Originally Posted by mfetch22
Has anybody else experienced a large reduction in the rate of learning mathematics at the level that I described, as in, is anybody fimmiar with this slow down between these two levels of mathematics that I'm experiencing, and is it something that is common?
I would say it happens to ~100% of people going from lower-level math (calculus, linear algebra) to upper-level math. It *will* pass if you keep up, but there's a transitional period while you change from one to the other (because different techniques are used and different skills are needed).
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June 15th, 2010, 10:22 PM   #9
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Re: Having trouble, have hit a wall in my learning?

CRG nailed it-- different techniques and skills. There will probably be another slow-down again, when you're expected to be doing more "original" math, roughly when you start grad school (+/- a year), for similar reasons.
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