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Thread: Some advice on studying mathematics?

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    Some advice on studying mathematics?

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    HI, I'm a Korean undergraduate student from Top 3 Econ school in my country.
    I have read through lots of posts here and they all have been of tremendous help to me.

    Straight to the point, I am planning to take advanced calculus and differential equations next semester, but the problem is I have yet virtually no math background except mathematics for economics, college math, and general stuff like that. After conquering those courses, (hopefully) I will need to take more advanced courses like real analysis, and etc.

    Now, I strongly feel that I have to start digging into some math to build a solid foundation for the tough ones (in my view) coming up next semester and ahead, but I have noticed that calculus books alone were like 1,200 pages thick (e.g. James Stewart), so I'm wondering if it is a wise strategy to go through all of it. I'm not allergic to math or anything like that but I'm just curious what would be the most recommended way of building mathematical structure in my brain.

    I'm currently having an internship in one of the UN offices in my country so I need to spend my time wisely because I do not have a plenty of time to focus on my study.

    I would very much appreciate any kind of advice. Thanks

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    Re: Some advice on studying mathematics?

    I recommend "The Humongous Book of Calculus Problems" by Michael Kelley... he also has the same series in Trigonometry (refresher?) and Statistics... I'm looking to read them before starting grad school next fall... they've been in my bookshelf for a while now! Despite the seemingly daunting name, it's actually an effective way to study because these problems all have solutions with them so it's like immersion for math. I also recommend "3000 Solved Problems in Linear Algebra"... it looks good and I'm going to read it before next fall...

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    Re: Some advice on studying mathematics?

    If your goal is merely preparing yourself for the courses you are going to take next semester, I would focus on reviewing 1) differentiation and integration techniques, 2) some basic intuition and principles about limits, and 3) infinite series. If you've taken mathematics for economics, I'm assuming you have a knowledge of at least some principles in multivariable calculus (e.g. Lagrange Multipliers) and linear algebra, so you should be fine. As always, the more you understand calculus, the better. A review of linear algebra (this could even be a less difficult, not as proof heavy book) would probably be useful. A course in proof-based linear algebra is quite enjoyable, I would say.

    I'm assuming that by "Advanced Calculus" you are referring to a course in real analysis. Most of these courses are relatively self-contained. Some general calculus knowledge does help to "connect the dots" in your understanding but is not necessary. Reviewing an introductory proof book like "The Book of Proof" or "How to Prove it" or even notes here would be useful. But reading through proofs and developing mathematical intuition is going to be important, so yes, reading, thinking through, and writing proofs in your time leading up to and during your coursework is important.

    Differential equations is not a difficult course, so long as you know your calculus. It's a pretty elementary computation course. Keep that in mind.

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    Re: Some advice on studying mathematics?

    thanks for the recommendations! I will def be looking for them next time I go to the bookstore.

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    Re: Some advice on studying mathematics?

    What I meant by "advanced calculus" was actually literally the "advanced calculus". In my school, we have calculus I, II and then the next level classes are the "Advanced Calculus I, II", which are again prerequisites of "Real Analysis I, II". I haven't taken those courses yet, so I don't know what's covered in them, but this is how the classes are structured and labeled in my school.

    But since I'm going to take real analysis eventually, your post helped me figure out what I need to work on more to get closer to my ultimate goal of excelling in real analysis classes I, II. Huge thanks for the info. By the way, in what classes do I get to learn the contents in the PDF file you've attached? I'm a real novice in this league, so please be patient with me.

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    Re: Some advice on studying mathematics?

    Ah, okay, feel free to DM me the syllabus or course descriptions of Advanced Calculus I & II so I can give you better advice. If in fact "Advanced Calculus I and II" is more like "Calculus III and IV - multivariable calculus" - just brush up on your Calculus I & II. I do have a feeling this may be a more proof-forward course, in which case having some practice with upper-level proof-based math may be useful.

    As for proofs, if you take an introduction to higher math course, a proof-based linear algebra course (highly recommended) or analysis or a more advanced probability course, those techniques will come in handy. In general, you have been training your computational abilities; reading through proofs and doing proofs will help to build your mathematical maturity. Reading and doing proofs is the single best way you can do well in real analysis.

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