Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some recommendation from some veterans of math,please?:

1.How prove it by velleman.

2.A Concise introduction to calculus by W Y Hsiang.

3.Calculus By Michael spivak or Courant and fritz's calculus?

4.Linear algebra Hefferon or Hoffman Kunze?

6.Calculus Vector Calculus, Linear Algebra, and Differential Forms by Hubbard
or Calculus on Manifolds by spivak
or Advanced Calculus: A Geometric View by James J. Callahan?

Introduction to topology and analysis by simmons

Mathematical Analysis by either Apostol or Zakon or Schröder ?

Is there any recommendation on what to skip,or some better books?

P.s I don't think this a duplicate. Background:

Principles of Mathematics by Carl Barnett, Allendoerfer

Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry George F. Simmons

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    $\begingroup$ I don't like Shilov's book on linear algebra. I prefer Axler or Hoffman Kunze. I'm not really a fan of Hubbard, though I suppose it works for your purposes. I also have bones to pick with your later choices, but really, if you're actually reading and learning from these books, you'll be lucky if you make it to item 4 within the next year, so planning this far ahead is really a waste of time. $\endgroup$ – Potato Aug 31 '13 at 23:13
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    $\begingroup$ Then why don't you spend your time reading the books instead of writing up lists of books you plan to read? $\endgroup$ – Potato Aug 31 '13 at 23:16
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    $\begingroup$ I've taught using Hubbard & Hubbard. I think it is a fine book because it integrates the linear algebra into the multivariable calculus. It's also accessible and has lots of examples. I'd also recommend Munkres' Topology. $\endgroup$ – John M Sep 1 '13 at 0:30
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    $\begingroup$ @DanielFaustMontana : you should tell us your background. $\endgroup$ – Stefan Smith Sep 1 '13 at 1:33
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    $\begingroup$ If you want something to replace Hubbard consider using Ted Shifrin's book Multivariable Mathematics. It really is superlative. $\endgroup$ – treble Sep 4 '13 at 21:02

I have debated many times as to whether to send you this reply. I have noticed that you have asked several questions regarding recommendations for study material.

Maybe I can share my personal experience with you. I don't know how extensive your background is. But I started studying math rather late in life (66 to be precise). I was somewhat fortunate in that I had always wanted to study math, and one day came across a small interview of Vaughan Jones discussing his favorite math books. I had no idea who he was at the time. But I checked out the books on Amazon and clearly they were way over my head.

However I did learn that Prof. Jones was a Fields Medal recipient (aka Noble Prize for math). So I periodically checked his website. One day I hit the jackpot. THere was a set of beautifully transcribed notes from his real analysis course.

As you may know, real analysis is quite often the foundational course in studying (let's say) real math.

So I really didn't have the problem of deciding amongst an array of otherwise good to great books.

I studied those notes day and night many times over. In essence I didn't read them, I lived them.

So on the one hand, it's important to find a book that you can deeply engage with. But likewise it is most fruitful if you can fully commit to it. That entails writing down key concepts, going through proofs inch by inch, drawing pictures. Testing yourself if you can convincingly present the material. And, the best of all worlds, if you can prove things before reading on in the text or with just catching a glimpse or two of what lies ahead.

Hopefully I have painted a good picture of what studying a math text entails for me.

I would not necessarily try to line up an array of future texts. Even though you say you read a lot; this isn't quite like reading. This goes at a pretty slow pace and if you miss a word or phrase it can produce difficulties.

Rather focus on one text - making it yours is what I would suggest. That does not mean you should not consider other reference material on the same topic. For example in addition to Vaughan Jones's notes:


I also went through the first four chapters of Pugh's "REal Math. Analysis."

This worked very well on a complimentary basis.

In conclusion, I would most sincerely suggest that you pick your level (if possible real analysis). Pick a primary text. (Keep in mind that there are some texts that are more what I would call teacherly - where the intention is to explicitly open your eyes to the material. And some that are more "opaque" where you have to supply a lot of the details. They both have their place.)

Best of luck,

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    $\begingroup$ +1 This is an excellent answer; get a good textbook, stick with it and learn it inside out. $\endgroup$ – seeker Aug 25 '14 at 21:41

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