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Studying Studying mathematics post-single-variable calculus ?

  1. Jan 4, 2009 #1
    Studying mathematics "post-single-variable calculus"...?

    My apologies - I posted this in the Abstract and Linear Algebra section earlier, but thought it might be more appropriate in this section...

    Hi everyone,

    I'm in Grade 11 this year (in Australia), currently studying from Apostol's first volume "Calculus". I have just recently started working on the theory of integration of trigonometric functions (just giving some information on my background). I am thinking that, perhaps once I've finished the calculus section of Apostol's text, I could move on to either;

    (a) Multi-Variable Calculus
    (b) Real Analysis
    (c) Abstract + Linear Algebra.

    I cannot really make a decision at this point in time, as I am unsure about abstract algebra (hence my post here). My interests are in physics, mathematics, computer science and philosophy, and so, I want to make a decision based on relevance to these interests.

    A short time back, I e-mailed Professor Apostol himself (giving feedback on his textbook and asking for advice regarding his Mathematical Analysis and Calculus Volume 2 books). He suggested that his second volume isn't necessarily a pre-requisite for his more terse exposition in the Mathematical Analysis text. He also suggested, since I am seriously considering majoring in mathematics at university, that I take a look at his Intro to Analytic Number Theory text.

    I would assume this step would come one after that of Real Analysis. But, may I ask, if I decide to pursue pure rather than applied mathematics, how useful would abstract algebra be to this field? What ability does a solid knowledge of abstract algebra present to you: i.e. what type of problems does it allow you to solve, and what is the topic's main idea/motivation?

    How difficult is the topic (abstract algebra) in comparison with calculus: is it less visual and geometrically intuitive? My favourite aspect of mathematics is calculus at this stage, and also sequences/series, although I don't understand them as well as many of the integration/differentiation topics and theorems I have covered.

    While I may not be the most sophisticated mathematically, I am interested in a rigorous but understandable presentation of the various topics. I'm not afraid to see proofs or try and prove things myself (although for the most part, I'm unsure of how to do that).

    Would it be possible to pick up linear algebra as I go, if I decide to study abstract algebra? I'm aware that linear algebra has great importance in computer science and physics, but I've also heard that it's not a greatly interesting branch of mathematics. I'm aware that this is an entirely subjective question, but how does abstract algebra rate on this "interest scale"? Do people find it more fun than calculus?

    My ultimate goal, after university, is to become a theoretical physicist, and I've heard that both calculus (analysis?) and algebra are very important to this field. Any information on the basics of abstract algebra, what it is, how it's used, and if possible, recommended texts on this field would be useful, and greatly appreciated.

    So, based on all the information I have provided, what would you all suggest for me to move on to? Remember, analytic number theory is potentially an option too, but I fear it may be too advanced for me at this stage. Would abstract algebra be too advanced?

    Any enlightenment would be grand.

  2. jcsd
  3. Jan 4, 2009 #2
    Re: Studying mathematics "post-single-variable calculus"...?

    If you want to do pure mathematics, you pretty much need to know analysis, abstract algebra, topology and linear algebra (and plenty more as well).

    As a theoretical physicist, that should also be the case too. However, you might want to learn physics along the way and see what parts are needed as you go along. You can of course just dive into some formal math texts if you wish, as many people do when they are interested in theoretical physics (probably at the graduate level though). But there is no reason why you wouldn't want to start early.

    A word of caution though, given the sheer amount of material, you can get burnt out really quickly. I suggest looking at interesting part of each subject and chip it bit by bit, since it is not so much of a priority at the undergrad level (that is assuming you want to be a physicist).

    Also, in case you are wondering what working theoretical physicist actually know in terms of mathematics... I would say for the good ones, their understandings are no less than an average mathematician (maybe not rigor wise but conceptual wise) (that's coming from the theoretical high energy part and from my friend's advisor). You pretty much need to master group theory (algebra stuff), Lie groups, topology (cohomology stuffs) and many other board subjects... In general, the more the better. This is no kidding... sometimes I wonder if some high energy theorists are actually mathematicians from their lectures (and I do take plenty of upper physics and math courses).

    I hope I don't deter your interest in the subject but instead motivate you to start early and learn a board spectra of things.
    Last edited: Jan 5, 2009
  4. Jan 4, 2009 #3
    Re: Studying mathematics "post-single-variable calculus"...?

    Hi Tim,

    I have fairly recently developed quite a substantial interest in mathematics, so I don't find this a deterrent. I also agree that it is a good idea to learn a broad spectrum of topics. So, I gather, you are suggesting that I try abstract algebra, or should I stick with calculus at this point in time?

    Naturally, I am unsure, as I do not have a great deal of experience with mathematics.

    Thanks for enlightening me however.

  5. Jan 5, 2009 #4
    Re: Studying mathematics "post-single-variable calculus"...?

    First, I suggest you finish your calculs sequence before moving onto analysis. That means do Vol 1+2 of Apostol. Real analysis is better postponed as one of its uses is a calc 1 refresher.

    Second, I suggest you do linear algebra before abstract. You won't get far in abstract without linear. Get Axle'rs Linear Algebra Done Right, a good rigorous book (despite its title). Any abstract algebra you do at this point should only be a geometric introduction. Once you have a firm grounding in calc and linear, you are ready for analysis and abstract algebra.

    Your plan sounds like you are trying to learn trigonometry without knowing what a triangle is. Calc and linear are 99x more important than anything in theoretical physics, not to mention they are requirements for upper level math.
  6. Jan 5, 2009 #5
    Re: Studying mathematics "post-single-variable calculus"...?

    Hi Khemix,

    I've come to that conclusion too. I think, by the sounds of things, abstract algebra would be too difficult for me, at this stage. Perhaps I should still take a look at some analysis topics (such as Fourier series etc) in my Understanding Analysis book (by Stephen Abbott), but I approach this, perhaps as only an offside, while I am working through Apostol volume 1 and 2 (I do have both, so no issues there!).

  7. Jan 5, 2009 #6
    Re: Studying mathematics "post-single-variable calculus"...?

    The standard route would be learn all the calc, linear algebra, analysis, then abstract algebra and some topology. However, I think it is much better if you just briefly look over each book, see what interests you and start learning. That way, you'll be more efficient and the material will be interesting (instead of just crank crank... and crank).
  8. Jan 5, 2009 #7
    Re: Studying mathematics "post-single-variable calculus"...?

    I bet there are a whole lot of people out there who would disagree with this statement.

    In the meantime of all the advanced mathematics you are intending on doing....I might recommend also picking up a foundations book or something geared towards formal proof writing as it will probably come in handy to know.
  9. Jan 5, 2009 #8
    Re: Studying mathematics "post-single-variable calculus"...?

    Abbott is a very good book from what I've heard. I studied Apostol myself in grade 11 and found it unbearable. I wish you luck in that endeavour. I really loved linear algebra from Axler. People said it was boring and whatnot, but I didn't find that to be the case. Everything was crystal clear, and the proofs were very direct and formulaic. This is why a lot of schools use linear algebra as a first exposure in proofs.

    I personally don't know how you've found so much time to do all this. In grade 11, I was finishing up Gelfand books and geometry and began using Apostol. My last piece of advice is to not get ahead of yourself. People like to make plans they never go through with. Stick to apostol for now and perhaps a proof book like "How to Prove it" by velleman. Its good to work slow when first developing your proof methods instead of looking up solutions 5 minutes after you gave up. Once you get good at proof writing questions, you will learn math a lot faster.
  10. Jan 5, 2009 #9
    Re: Studying mathematics "post-single-variable calculus"...?

    Hi Khemix,

    Interesting to hear you used Apostol in year 11. The reason why I've had the time is that I'll be in Year 11 this year (just finished Year 10), and at the end of the year I was not very busy, so I had quite some time to work on it. Admittedly, I'm only part way through Apostol, but I'll stick with it.

    I suppose the reason you found it unbearable was because of the language used?

  11. Jan 6, 2009 #10
    Re: Studying mathematics "post-single-variable calculus"...?

    You can learn everything in Dummit and Foote's Abstract Algebra with nothing but a good understanding of proofs. In fact, I just took a course on it in conjunction with my first proofs course and my first linear algebra course and it went perfectly smoothly. Linear algebra and Abstract algebra hardly intersect on a basic level. You don't consider vector spaces in abstract much until modules, but rather groups and rings and fields, which have similar type constructions but different axioms, e.g. a group only requires one associative binary operation, an inverse for each element and an identity. A ring is an extension of a group in that it adds a second binary operation and set of inverses corresponding to them. A field is essentially a ring in which the binary operations are commutative.

    Honestly, if I hadn't taken a linear course I would have done just as well in th course and linear algebra is not a prerequisite, official or otherwise.

    I found Algebra to be much more fun than the calculus courses due to how fundamental and axiomatic it was. You start with basic axioms and derive all of the consequences. This is true in analysis too, but there is definitely a different flavor to algebra. I would say that if you like philosophy it would indicate that you might like algebra. I say this because most people I have met with an interest in philosophy and mathematics seem to enjoy more abstractions and generalizations in their mathematics. Algebra is indeed very abstract, it is more of a logical system, its logic is more explicit than that of analysis(if this even makes sense). I don't know that I could necessarily say that algebra is more fun than analysis, they are both interesting, they can both be used to investigate one another as well as other things physical and mathematical. It is almost like a preference of genres of music or art. You might not like algebra, but you might enjoy topology and analysis or you might love them all.
  12. Jan 6, 2009 #11
    Re: Studying mathematics "post-single-variable calculus"...?

    Hi Bourbaki,

    Admittedly, I do enjoy some abstraction and generalisation in mathematics, but I also like to be able to derive properties that have some geometric intuition (I like calculus: I can visualise the results, and therefore gain a conceptual understanding).

    I also like to have some application stressed - doing pure mathematics would be fine, provided I come across greatly interesting or applicable results once in a while (if only to gain a deeper understanding). Again, some pure mathematics may be inherently interesting, even if there is absolutely no application (for example, topology does sound like an interesting and very active field, but I'm unsure of applications outside string theory and mathematical physics).

    As far as building everything up axiomatically, I do not know whether this would be to my taste or not. From the outset though, analysis does appeal to me, perhaps only because I truly do like calculus. I do appreciate "high school algebra" almost purely due to the notion of symmetry, although I do not appreciate it as deeply as calculus.

    I also hear that abstract algebra in no way resembles high school algebra (which I would say is fairly concrete rather than abstract). But I'll take a look at Dummit and Foote's book, and then decide. It might be that by the time I'm done with the first volume of Apostol (exc. linear algebra) I'll want to move on to algebra. Perhaps not, but naturally, I am unsure.

    Thanks for enlightening me, anyway. Nice to hear I could work some abstract algebra without the requirement of linear algebra (in fact, in my Year 12 Mathematics Specialised course next year, I will be learning some linear algebra).

    Perhaps this would be the time to study abstract algebra, just before I go off to university, but as I said, if I require a change in scenery, then I will take the initiative and do so (as I have done with calculus).

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