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I am currently teaching myself Mathematics at the moment, with my current focus being Pre-calculus mathematics however at some point I'll eventually reach undergrad material. As I do not have a great grasp of the subject and its many topics I'd like to know the best or the most well trodden sequence in which to study undergrad Mathematics, for example: Calculus, Linear Algebra, Differential Equations... . These are pretty broad topics, so if someone who's studied Mathematics could break this down into a comprehensive list of sub-topics or some kind of syllabus for me to self study (and also list any good books/materials/websites) that would be very much appreciated. I am placing emphasis on the "best sequence" part as I am certain many of you have studied mathematics and with the power of hindsight may have wished you had studied a particular topic before/after another as it would have made life just that little bit easier for you or it would have enhanced your understanding, i'd like to benefit from your experience. I hope I am not asking for too much, thank you for your help!

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Why not study in the order of college requirements? Most of them are constructed from careful reflection on the best order. –  toypajme Oct 21 '13 at 16:48
I am just giving you a suggestion, how about obtaining a calculus book "Stewart's Calculus". The current edition is 7th, but the 6th edition will also do fine. It can be had for little money. This book is the staple book for undergrad engineers in North America and it is a good book to study from if you want to do self study. –  imranfat Oct 21 '13 at 16:51
@EnjoysMath, "bookos.org"? is it a legit source? can you post it here? –  Vikram Oct 21 '13 at 17:56
@EnjoysMath You should probably have a look at this question on meta. It seems that it was asked (partly) because of you. –  Martin Sleziak Oct 22 '13 at 8:26
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5 Answers

up vote 2 down vote accepted

I am someone who did just as you are hoping to do -- I don't enjoy studying material without a strong background in the assumed prerequisites, so some would feel that I'm suggesting too much.

My suggestion is to go through Stewart's calculus front-to-back, doing typical problem sets. Whenever you struggle with a pre-calculus or algebra fact, make sure you review that material. Once you've done this, I recommend deviating from the typical routine and picking up a copy of the book "How to Prove it." With the maturity you have after going through calculus, you will understand the essence of what a proof is, and the importance of working with and thinking in terms of definitions. After this, for fun you might enjoy going through "Geometry: Seeing, Doing, Understanding" by Jacobs. Then, quickly, go back through your Algebra/Pre-Calculus textbook examples and work through any of them that do not seem immediately obvious how to do. Then, quickly go back through Stewart, doing the same thing. You will be very prepared at this point to do very well in your future studies.

From here, you would want Linear Algebra, Complex Analysis, Abstract Algebra, and Real Analysis. For Linear Algebra, I recommend Gilbert Strang's MIT opencourseware videos, and for Real Analysis, I recommend Francis Su on YouTube (he goes through the textbook Principles of Real Analysis by Rudin).

These, I think are essential. you can't plan too much further in advance because your interests will change as time goes on. But some other areas you could look at are probability and statistics, number theory, numerical analysis, and differential equations, or more advanced topics in what I've already mentioned.

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I was worried about the transition from Pre-calculus to undergrad mathematics however you've seemed to have addressed that, are there any other books that would help bridge material that you used? Also outside of the books you have just listed what others would you recommend? Thank you for you answer @David. –  Assad Oct 21 '13 at 17:50
I wouldn't worry about needing a bridge until you start working through calculus and are unable to succeed. Stewart is a very accessible book, so I expect that you will do just fine between the book and internet resources (like stack exchange) –  David Oct 21 '13 at 18:10
Oh okay, thanks! –  Assad Oct 21 '13 at 18:14
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I would go

  1. Calculus

    • limits and continuity
    • differentiation
    • sequences and series
      • taylor series
      • convergence tests
    • integration
    • applications of the above
    • intermediate value theorem
    • extreme value theorem
  2. Linear Algebra

    • vector spaces and linear maps
    • systems of linear equations
    • row-reduction operations
    • linear independence and span
    • invertible linear transformations
    • bilinear forms
    • geometry with vectors, lines and planes
  3. Algebra
    • there's too much to list
  4. Complex Analysis
  5. Real Analysis
  6. Functional Analysis
  7. Differential Equations
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Having self-studied (and continuing to study) a lot of undergraduate mathematics, I will tell you the order in which I did things (the list might be a bit ambitious, especially the jump from Apostol's Calculus to Artin's Algebra...you may want to give yourself a bit of background in linear algebra before trying to tackle Artin):

  1. Mathematical Logic
  2. Set Theory (introduction to axiomatic set theory)
  3. Apostol's Calculus Vol. 1
  4. Multivariable Calculus (used OpenCourseWare)
  5. Artin's Algebra
  6. Rudin's Principles of Mathematical Analysis
  7. Calculus on Manifolds (Spivak+Munkres)
  8. Topology (Munkres)

If I were to do it over again, I might put less emphasis on mathematical logic and set theory as I did (but that was what motivated me to self-study in the first place).

I would recommend taking linear algebra before studying differential equations--a lot of differential equations exploits linearity of the derivative, and this can be rephrased easily in terms of linear operators.

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I would recommend you read thoroughly go through a Calculus textbook (only doing the single variable sections). Then pick up an introductory linear algebra textbook (Poole is good) and work through that. After that I would proceed in the following order:

  • Multivariable Calculus
  • Differential Equations

At this point you can either pursue real analysis or move to abstract algebra. I'll flesh out the first rout first:

  • Introductory Real Analysis (Rudin or Abbot are recommended)
  • Complex Analysis (Saff & Snider for more computational approach, Stein and Shakarchi for more theoretical)
  • General topology (Munkres)
  • Measure theory (Royden is great for this)

For abstract algebra:

  • Literally just read through Fraleigh or Artin
  • If you want to continue, maybe consider reading Atiyah & McDonald's commutative algebra book or J.S. Milne's online notes on Representation Theory.

Do note, however, that differential equations isn't really necessary for any of the courses listed afterwards, its just a fairly essential part of the undergraduate curriculum.

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Thank you for your answer! A question, what material would you recommend for bridging the gap between pre-calculus and undergrad mathematics? –  Assad Oct 21 '13 at 17:57
I suppose Calculus is sort of the bridge between the two. It operates at a higher conceptual level than pre-calc but isn't nearly as rigorous or abstract as some of the more advanced courses an undergraduate would take. But before you tackle Calculus, make sure you are comfortable with polynomials, rational functions, logarithms, and trig functions. If you don't some of the calculations will be very difficult to follow. –  Ben Perez Oct 21 '13 at 19:08
thanks for the advice! –  Assad Oct 21 '13 at 21:50
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I would recommend these to be read at the same time:

  1. Euclid's ELEMENTS (translated by Heath), and

Proving theorems by construction is a wonderful way to do mathematics, and Hartshorne's book will point out and correct the logical mistakes Euclid made, extend his results, and provide a connection to contemporary mathematics by way of introductory modern algebra. Best of all, the prerequisites are zero, zilch, nada. By the time you have finished both of these books (which should take the better part of two semesters going it alone) you will know what you want to know more about.

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Thank you for this! Are there Any other books you recommend? Specifically Pre-calculus, calculus, and introduction to under-graduate study texts? –  Assad Dec 5 '13 at 16:03
In that case, I would recommend "Calculus Made Easy", by Martin Gardner. In fact, everything by Gardner is good. –  bwkaplan Dec 5 '13 at 18:32
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