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I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the mathematics and its phenomenal applications. I will be taking the theoretical linear algebra and multi-variable calculus on upcoiming semester and proceed with introductory analysis (Rudin's PMA) on next Fall along with the theoretical differential equations. I used the calculus textbooks called "Calculus and Analytic Geometry" by George Simmons and "A First Course in Calculus" by Serge Lang to pass the single-variable calculus. The theoretical linear algebra course (one I will be taking on next semester), teaches the proof-based linear algebra and the proof-methodology; its required textbook is one written by Friedberg & Insel & Spence. The multi-variable calculus is heavily on computational and use my university's course notes...In this case, should I self-study the Spivak/Apostol/Courant before advancing into the analysis like Rudin's PMA? I own the Apostol's two volumes and also Lang's Basic Mathematics but I do not think I will be able to study them fully befoe the end of Summer. Is it okay to jump toward the math analysis without completing the Apostol or Spivak? My current research utilizes a lot of differential equations, linear algebra, and numerical analysis. Is it okay to study for numerical analysis without taking the analysis courses like real, complex, and functional analysis? If so, could you recommend a good introductory/elementary textbook on numerical analysis?

Also about the theoretical linear algebra textbooks, which among those books complement well with Friedberg? Hoffman & Kunze, Axler, Lang's LA, or Lang's Introduction to LA? If Hoffman & Kunze better for a starter than Friedberg? I always love to use at least two textbooks on given topic. Also are books called "How to Prove It" by Velleman and "How to Solve It" by Polya good for learning the proof methodology and techniques?

Thank you very much for your time, and I apologize for asking too many questions especially on the books. I think it is important in mathematics to choose the right books to study for. I look forward to your advice!

Sincerely,

MSK

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closed as primarily opinion-based by Adam Hughes, user147263, Asaf Karagila, Matt Samuel, user149792 Mar 7 '15 at 21:02

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I think you will find Rudin's PMA extremely challenging if you have not already been exposed to delta-epsilon proofs and rigorous calculus in general. In my opinion, working through Spivak's Calculus would be the ideal preparation. I would strongly prefer Spivak over Apostol because Spivak is shorter, more user-friendly and fun to read, and it has great problem sets. Lang's Basic Mathematics doesn't even belong in this list: it is a good book, but you probably know almost everything in that book if you've studied calculus (rigorous or not). Lang's LA book is good; also consider Axler's. $\endgroup$ – Bungo Jan 6 '15 at 8:13
  • $\begingroup$ FWIW, I learned analysis from PMA and a video lecture series, with minimal analysis experience (we did two days of delta epsilon in my HS calculus class, but nobody understood it and the teacher knew it wouldn't be tested so wasn't interested in spending much effort there). I then failed an examination to place out of the course, so I might not be the best endorsement, but it's definitely doable. // Numerical analysists should probably know some analysis, but it's my [limited] experience that people find it only tangentially useful. $\endgroup$ – Eric Stucky Jan 6 '15 at 9:04
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There are several questions here.

The first is whether you can start with Rudin's analysis book now. I would say that for most people this would be difficult after only Lang's calculus book, although Lang is far better than most calculus books in use these days.

I don't recommend reading Spivak or Apostol (Vol. 1) now. It will take too long, as they're really intended for people who don't know the routine part of calculus yet. Instead, I would suggest reading an analysis book at an intermediate level of difficulty, such as Apostol's Mathematical Analysis, or even the easier analysis book by Ross. Once you've read several chapters of one of those, as a byproduct you'll probably find that you can just jump in anywhere in those calculus books and understand.

The second issue raised by your question is about how to approach multi-variable calculus at this point. I would say that without having studied either analysis or single-variable calculus at the level of Volume 1 of Apostol's Calculus, it will be difficult for you to follow much of the theory in Volume 2. In this situation, since you need multivariable calculus immediately, it's okay to study it at a slightly lower level and fill in the gaps later. (Parts of the theoretical material are so difficult that even Apostol is forced to omit some proofs in Volume 2 anyway.) Lang's book on multivariable calculus would be good in this situation. The amount of theory is intermediate in it.

For linear algebra books, I don't really know which is best. I imagine one would be sufficient. The two introductory chapters of Lang's Introduction to Linear Algebra could be good preparation if you haven't worked with vectors or matrices before.

On numerical analysis, I'm afraid I don't know.

Finally, you mentioned some books about proofs. Not everybody needs a book that teaches them how to write proofs specifically; some people just pick it up when they learn math with proofs for the first time. But if you would like to read a book like that, I would recommend that it be something that actually teaches you interesting math as you go along, rather than being too narrowly focused on proof techniques. My favourite in this regard is Journey into Mathematics by Rotman, which covers several interesting topics. Even if you don't need a book on proofs, you won't feel like you've wasted your time. It has a complete solution manual too.

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  • $\begingroup$ Thank you very much for the advice! Personally, I liked George Simmons' calculus book more than Lang's calculus due to his clear exposition and balance in both the theories and applications. What do you think about the vector calculus textbooks written by Wendell Fleming, Richard Williamson, Marsden, and Hubbard? Are they on the same level as Lang's multivariable calculus book or more on Apostol Vol.II book? I am planning to purchase Lang's book and another one from the list above to use along with my etextbook (very computational multivariable calculus). $\endgroup$ – MathWanderer Jan 7 '15 at 20:53
  • $\begingroup$ Also do you mean that I do not necessarily have to study Apostol or Spivak and can directly go to analysis texts like Ross and Rudin? If so, could you recommend as many analysis textbooks as possible? I have a scholarship for purchasing the textbooks, so the cost is not an issue for me. $\endgroup$ – MathWanderer Jan 7 '15 at 20:55
  • $\begingroup$ @MathWanderer I don't know about the vector calculus books you mentioned. If you're planning to do differential geometry at some point, the vector calculus part will be taken care of as a special case. I think Rudin's book might be a bit difficult at this stage. I would recommend either Apostol's Mathematical Analysis or Zorich's book instead. The second volume of Zorich, incidentally, treats vector calculus at a higher level than does the second volume of Apostol's Calculus, and vector calculus is absent from the analysis book by Apostol. Lang's multivariable book does vector calculus $\endgroup$ – user204305 Jan 7 '15 at 22:06
  • $\begingroup$ at a non-rigorous level. Considering that you will be taking multivariable first, you don't have much time for that, so I'd say start with Lang's multivariable book. $\endgroup$ – user204305 Jan 7 '15 at 22:07
  • $\begingroup$ Thanks again! I will buy Lang's multivariable book then. After completing the multivariable course and theoretical linear algebra in the level of Friedberg or Hoffman, am I safe to jump toward the analysis books like Zorich, Apostol (not Calculus), Ross, Rudin, Pugh, etc.? Or do I need to still study the Spivak or Apostol? I have both volumes of Apostol but I only took a look at volume 1, which is not hard so far.. $\endgroup$ – MathWanderer Jan 7 '15 at 22:28

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