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Does anyone have any suggestion for books to study the following topics (self-study)?

Abstract Algebra, Linear Algebra, Calculus, Differential Equations, Analysis (Real and Complex), Topology and Differential Geometry. (So $\approx 6$ books are required). (Not online series).

Required topics are:

$\big( \bigstar \big)$ $\text{Abstract Algebra}$:

Some suggestions in this website are:

$\star$ "$\text{A First Course in Abstract Algebra}$" by "$\text{John Fraleigh}$".

$\star$ "$\text{Abstract Algebra}$" by "$\text{David S. Dummit and Richard M. Foote}$".

$\star$ "$\text{A Book of Abstract Algebra}$" by "$\text{Charles C Pinter}$".

$\star$ "$\text{Contemporary Abstract Algebra}$" by "$\text{Joseph Gallian}$".

$\big( \bigstar \big)$ $\text{Linear Algebra}$:

Some suggestions in this website are:

$\star$ "$\text{Linear Algebra}$" by "$\text{Friedberg, Insel and Spence}$".

$\star$ "$\text{Linear Algebra}$" by "$\text{Serge Lang}$".

$\star$ "$\text{Introduction to Linear Algebra}$" by "$\text{Serge Lang}$". (different from the previous one).

$\star$ "$\text{Linear Algebra}$" by "$\text{Kunze Hoffman}$".

$\star$ "$\text{Linear Algebra Done Right}$" by "$\text{Sheldon Axler}$".

$\star$ "$\text{Linear Algebra Done Wrong}$" by "$\text{Sergei Treil}$".

$\big( \bigstar \big)$ $\text{Calculus}$:

Some suggestions in this website are:

$\star$ "$\text{Calculus}$" by "$\text{Michael Spivak}$".

$\star$ "$\text{Calculus: Vol. 1 and Vol 2.}$" by "$\text{Tom M. Apostol}$".

$\star$ "$\text{Calculus}$" by "$\text{James Stewart}$".

$\star$ "$\text{Thomas' Calculus: Early Transcendentals}$" by "$\text{George Thomas Jr., Christopher Heil and Maurice Weir}$".

$\big( \bigstar \big)$ $\text{Differential Equations}$:

Some suggestions in this website are:

$\star$ "$\text{Differential Equations with Applications and Historical Notes}$" by "$\text{George F. Simmons}$".

$\star$ "$\text{Ordinary Differential Equations}$" by "$\text{Morris Tenenbaum and Harry Pollard}$".

$\star$ "$\text{Ordinary Differential Equations}$" by "$\text{Vladimir I. Arnold and R. Cooke}$".

$\star$ "$\text{Ordinary Differential Equations}$" by "$\text{Wolfgang Walter and R. Thompson}$".

$\star$ "$\text{Partial Differential Equations}$" by "$\text{Lawrence C. Evan}$".

$\star$ "$\text{Partial Differential Equations: An Introduction}$" by "$\text{Walter A. Strauss}$".

$\star$ "$\text{Partial Differential Equations for Scientists and Engineers}$" by "$\text{Stanley J. Farlow}$".

$\big( \bigstar \big)$ $\text{Analysis}$:

Some suggestions in this website are:

$\star$ "$\text{Principles of Mathematical Analysis}$" by "$\text{Walter Rudin}$".

$\star$ "$\text{Mathematical Analysis}$" by "$\text{Tom Apostol}$".

$\star$ "$\text{Introduction to Real Analysis}$" by "$\text{Robert G. Bartle and Donald R. Sherbert}$".

$\star$ "$\text{Understanding Analysis}$" by "$\text{Stephen Abbott}$".

$\star$ "$\text{Real and Complex Analysis}$" by "$\text{Walter Rudin}$".

$\big( \bigstar \big)$ $\text{Topology and Differential Geometry}$:

Some suggestions in this website are:

$\star$ "$\text{Introduction to Smooth Manifolds}$" by "$\text{John Lee}$".

$\star$ "$\text{A First Course in Geometric Topology and Differential Geometry}$" by "$\text{Ethan D. Bloch}$".

$\star$ "$\text{Elementary Topology and Applications}$" by "$\text{Carlos R Borges}$".

I have studied some of the above topics when studying chemical engineering a few years ago. However, I did not study them properly, and I need to study them again from scratch. So my question is about books satisfying the following criteria:

$(1)$ Easy to read for self learners.

$(2)$ Book should be as comprehensive as possible, but starting from scratch. And Includes many topics. For examples, calculus book should include all single-variable, multi-variable, vector calculus (Stokes' Theorem, Divergence Theorem, ...), and the analysis book should include both real and complex analysis. Also, it would be better to have a single book about topology and geometry together. In that manner, I will have less number of books. Of course not necessary, for instance, having one book in real analysis and another book in complex analysis, that will be fine in case those books are greater than a single book. Similar for other books like differential equation books, it will be fine to have one of ordinary and one for partial differential equations in case those books are more informative.

$(3)$ Contain proofs as much as possible.

$(4)$ Good number of examples.

$(5)$ Require least prerequisites.

I have already checked through MSE, but I see different answers from different people, that made me confused, and could not take a decision. Of course everyone has his own opinion, however, I believe, each one of the topic (listed above) has a book that everyone agrees it is great.

You can suggest me other than the listed books, if you think it is better. Keeping in mind the $5$ criteria mentioned above.

Possibly this post will be considered as duplicate. But I believe it is not as I am asking about books of specific topics with specific criteria.

I really hope to get the help from you so that I can conclude and take my decision.

Your help would be appreciated. Thanks!

Cornman
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Hussain-Alqatari
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  • Spivak's Calculus is a great and comprehensive book, but not sure how it would be for self learning. It's practically an intro to Analysis, a great on-ramp into those topics. Linear Algebra Done Right is amazing as a 2nd text, I wouldn't recommend it as a first entry to anyone except those who had a strong theoretical math background – Alan Sep 01 '22 at 02:46
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    It's a bit of a tall order expecting people to comment on that giant list of books. Even experienced mathematics educators might not have read them all. – Suzu Hirose Sep 01 '22 at 05:30
  • @Alan Thanks for commenting. Since you are not sure if Calculus by Spivak is a good one for self-learning or not, which book of calculus, are you sure about, for self-learning, that is easy to read, as comprehensive as possible, contains proofs, examples and exercises, and requires least prerequisites? Also, if Linear Algebra Done Right is 2nd text, what is first, in your opinion? – Hussain-Alqatari Sep 01 '22 at 16:34

1 Answers1

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I think you should take your time digesting the fundamentals of mathematics first, and then read a topic-related math book.

That means pick a book on elementary set theory, and a book on basic proof writing and mathematical methods. For elementary set theory, I highly recommend Halmos' book Naive Set Theory. You only need the first 43 pages. It is very well written, but contains not so many exercises, which might be positive at the beginning.

For proof writing the book by Jay Cummings Proofs: A Long-Form Mathematics Textbook seems great. Or the freely-available Book of Proof found here.

You can study them parallel to each other, and of course any other book. But learning the fundamentals first will help you get more quickly into the advanced topics, because you already have the knowledge of proof writing and mathematical reading required.

The problem with what you are asking is, that studying books is normally too much effort. Some of these books have hundreds, or even thousands of pages.

Abstract Algebra by Dummit and Foote is a great, great book. But it has roughly 1000 pages, and about 3000 exercises. You will never finish this. It is still great, to learn a topic, and I highly recommend it.

In my opinion, it makes no sense studying calculus and analysis separately. The book Calculus by Spivak is great, but again it has 600 pages, and 'only' contains calculus. Normally there is no separation of these topics. So in many parts of the world, calculus and analysis are the same. Unfortunately, I do not know any book on analysis which is good. I feel they are all mediocre. Maybe give the books by Terence Tao a shot. He is a gifted mathematician and a gifted mathematical writer.

As far as Linear Algebra goes, I like Shelon Axler's book. It is the only English book on the topic I know. Linear Algebra is also contained in the book by Dummit and Foote. In general, I would advise to stay away from books by Serge Lang. I personally do not like what I have seen so far written by him, but have never studied his books in depth. Many of his books do not contain exercises.

For topology and differential geometry, I would recommend the book Introduction to Manifolds by Tu. It recalls the fundamentals of topology in an appendix, and then gives an introduction to differential geometry.

Two more things:

First of all, you should not care too much about which book you pick. It is completely normal to look into two or three books of the same topic in parallel and just get started on something. When Analysis, Calculus or Linear Algebra is written on the book, then this is what it is about and you will be amazed how similar these books are. The differences are there, but minor. You will find that you like the proofs in book X better as in book Y, but the exercises in book Y suit you more then in book X. But then book Z comes in and has more and clearer examples then X and Y, so you just take all three, and never finish any of them completely.

Last, I would recommend to take lecture notes instead of books. As I said, it is very unlikely (and pretty expensive) to finish math books completely, and you should not read 600 pages of calculus before moving on. You can read books parallel to that, but lecture notes contain in 120 pages, what books tell you in 300 or 600 (because they normally contain more topics, but also stuff you do not really need to advance).

Due to Covid, there is a pretty good chance that you will find well-written lecture notes online, and the according problem sets. Maybe just consult the mathematical faculty that you trust and ask.

Good luck.

J W
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Cornman
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    Serge Lang's books intended for undergraduates contain mountains of exercises. Some of the books contain solutions. I have never found an error in those books. I don't remember whether his graduate texts contain exercises. – Suzu Hirose Sep 01 '22 at 05:28
  • I know Langs books on Algebraic Number Theory and Abelian Varieties, and his Algebra, but I like Dummit and Foote better, so I rarely use it. The first two books do not contain exercises. – Cornman Sep 01 '22 at 05:47
  • Thanks for the detailed answer and suggestions. I have some points; (i) I have some knowledge in the topics I mentioned, and I only want to go deeper but "go from scratch". So about the fundamentals, I have it. (ii) for Abstract Algebra by Dummit and Foote, no problem with the number of pages as long as it is good for self-learner, so do you still suggest it? (iii) For calculus and analysis, I know they are very much related, however there must be separate books, do you suggest any? (iv) What about DE? Any suggestion, whether a single book or ODEs and PDEs separated whichever better. Thanks. – Hussain-Alqatari Sep 01 '22 at 16:14
  • @Hussain-Alqatari I only know one german book on Differential Equations, and yes I can highly recommand Dummit and Foote for selflearning. As I said, I have tried to find a good book on Analysis, but was until now never satisfied. There is no particular book I can recommand. However Calculus by Spivak is very much recommandable. As I said, you might try the books by Terence Tao Analysis I and II. For ODEs and PDEs I have never really studied these topics. – Cornman Sep 01 '22 at 16:27