Which Dover Math Book Should I Add to My Self-Study Collection?

[Matt_from_now]

I graduated with a BA in Classicals and didn't take any math in college. Now that I've been out for a few years I find that I have an overwhelming desire to learn how things work in a way I can describe mathematically (it's hard to explain, hopefully you understand) and am starting some self study. Once I've gotten a good grasp on math I would like to tie in some physics as well, but I'm taking one thing at a time. I'm choosing Dover books because I get a lot for the money and I'm hoping they are accessible for someone who doesn't have experience with calculus or beyond but is willing to put in the hours to learn. So far I've decided to buy these books:

Kline's Calculus,
ODEs by Tenenbaum
Linear Algebra
Topology

I would like to add one more book that would round out these subjects and that would build on them. I'm thinking about a book on Abstract algebra, but I don't know anything about the topic and am not sure if it is related. Should I think about a book on tensors or vectors? Maybe differential geometry? Real analysis? I'm interested in statistics too but don't see how this would fit in. I don't know enough to know how everything relates or builds off other topics. I'm well aware that to get through these books it will probably take me a couple years. I'm not in a race and I expect to be learning the rest of my life.

 

[Jorriss]

Hey! That ODE book is supposedly very good. However, Shilov and Mendelsen are almost assuredly too advanced at the moment.

Rather than a dover book I would get a used, old edition of Strang, Lay, etc for linear algebra (as I am not too familar with other LA dover books but there may be a gem I am missing).

Before Mendelsen, I would personally get a dover book on advanced calculus/real analysis, or, even better, Spivak Calculus. It's very challenging but infinitely rewarding. Infact, screw everything else, get Spivak lol.

 

[czelaya]

Springer publications has an undergraduate series of books on mathematics and some physics. A number of the "undergraduate series" books have complete solutions.

There's a book by Andrew Pressley titled Elementary Differential Geometry with complete solutions. However, it only deals with differential geometry up to R^3. If you want a more advanced differential geometry that goes into the theory of manifolds, topological spaces, lie groups, and so forth, I really don't know of any books that may help at an introductory level.

 

[Matt_from_now]

Thanks! The last thing I want to do is get a book that is so advanced I can't get past the first page. Don't worry, I know I'm being very ambitious. Would learning statistics help me at all with calculus, ODEs, or linear algebra? If not I may wait on that too so I can concentrate on those first.

 

[niklaus]

No, statistics won't really help you with those subjects.

 

[Matt_from_now]

Could a beginner like me handle abstract algebra or should I hold off on that as well?

 

[Jorriss]

Could a beginner like me handle abstract algebra or should I hold off on that as well?

For the average, or even above average student, probably not.

BTW, While reading Spivak I would also get a proof book such as Eccles. https://www.amazon.com/dp/0521597188/?tag=pfamazon01-20

 

[QuarkCharmer]

A professor at my school gave me (but I will return it to him) a copy of "Linear Algebra, and it's applications" by Lay. It's an older (2E) instructors edition and I really like it so far. I have the dover book posted above, and a copy of Lang's Linear Algebra that I got used on amazon for like 10 dollars. I plan to work through that after Lay.

Anyway, check out some older editions of known good books. Math doesn't really change all that much, and a solid book from the 1970's can cost a few dollars and I find the problems more difficult and the presentation more direct. Some of the dover books are hit and miss.

 

[lsaldana]

Dover books certainly look very appealing. I own three Dover math books, "An Introduction to Topology" by Mendelson, "A book of Abstract Algebra" by Pinter, and "Introduction to Partial Differential Equations with Applications" by Zachmanoglou and Thoe. Of these, I can only really comment on the later two since I have been teaching myself abstract algebra for a good while now and have used Z&T as a reference; planning to tackle topology after real analysis. Pinter is by all means a superb book. However, it is very succinct in its exposition of topics. The other book I'm using for abstract algebra is "A First Course in Abstract Algebra" by Fraleigh which is also very good for a beginner and certainly has more breadth. I feel that out of the two, Pinter serves more as a quick reference and second-look to what I mainly learn from Fraleigh. Z&T served me greatly as a reference to a math physics course when we covered PDEs but have heard from friends who used it for a class that it is not a very good "textbook".

Could a beginner like me handle abstract algebra or should I hold off on that as well?

I would say that there are really no pre-reqs to learning abstract algebra other than mathematical maturity. By maturity I mean abstract reasoning and, of course, proofs. Calculus, ODE's, statistics, etc are not required to tackle this topic in any way. If you're really starting from calculus I, I would say hold off on even buying books like topology or modern algebra and see if you can get used to other less abstracted topics like ODE's and calculus first, especially from Dover books. Another good first step would be linear algebra. If you're interested in differential geometry I highly recommend "Elementary Differential Geometry" by O'Neill as an introduction which I used as a supplement for a class assigned Do Carmo. For real analysis, I've enjoyed "Analysis with an introduction to proof" by Lay.

Some of the dover books are hit and miss.

True that.

 

[intwo]

I've bought a few Dover books, including the differential equations book that you listed. I thought they'd be excellent (and inexpensive) supplements to my required textbooks, but they mostly fell short of my expectations.

If you're short on cash, I recommend purchasing older editions of current textbooks. They're usually identical with the exception of renumbered or extra exercises and they're as inexpensive as Dover books. Alternatively, you can purchase one book at a time and save up money while you work through that book. I think that might be the best option.

Here is a sequence of books for the alternative route. :)

Basic Mathematics - Lang
Calculus, Volume 1* - Apostol
Calculus, Volume 2* - Apostol
First Course in Probability - Ross
Linear Algebra** - Hoffman, Kunze
Ordinary Differential Equations*** - Birkhoff, Rota
Mathematical Analysis - Apostol
Algebra - Artin

* If these books seem too difficult, supplement them with an older edition of Calculus - Stewart.
** If this book seems too difficult, supplement it with an older edition of Linear Algebra and Its Applications - Lay.
*** If this book seems too difficult, supplement it with an older edition of Elementary Differential Equations and Boundary Value Problems - Boyce, DiPrima.

 

[mal4mac]

Why to try Heath's translation of Euclid's Elements? It has incredibly detailed notes, and I wouldn't recommend it for anyone without a degree in classics... but, as you have a degree in classics, you might like it.

 

[Matt_from_now]

I have thought about Euclid's Elements. It would be fun, especially going back to look at the original. Right now I'm trying to get a foothold on calculus but I'm sure when I come back around to some geometry it might go for that.

 

[Sankaku]

I would say that there are really no pre-reqs to learning abstract algebra other than mathematical maturity. By maturity I mean abstract reasoning and, of course, proofs.

This is good advice. Your best bet is to study some Linear Algebra before trying Abstract Algebra. However, you can always give Pinter a try and see what you think.

There is a solid free Linear Algebra book here:
http://joshua.smcvt.edu/linearalgebra/

For intro analysis, I haven't found a really good Dover textbook. Rosenlicht is ok, but there are better books out there.

 

[Daverz]

Kline's Calculus,

You might try Silverman's more compact calculus book.

Also check out the charming book by George Owen, Fundamentals of Scientific Mathematics.

Dover doesn't really have a good introductory basic Physics text (I keep meaning to suggest that they obtain the rights to Chalmers Sherwin's Basic Concepts of Physics.) You're better off looking for an old edition of Halliday & Resnick.

I'm thinking about a book on Abstract algebra, but I don't know anything about the topic and am not sure if it is related.

Interesting in itself, not very helpful for Physics until you reach a very advanced level. I'd suggest a "math methods for physics" book. Dover has a ton of these, but I'm not quite sure which to recommend at the moment.

Some other recommendations:

http://store.doverpublications.com/0486234002.html
http://store.doverpublications.com/0486613887.html
http://store.doverpublications.com/0486661105.html
http://store.doverpublications.com/048660361x.html

When you have some more Physics under your belt:

http://store.doverpublications.com/0486661695.html

 

[jbrussell93]

I cannot say much about the other three books but I have read Calculus: An Intuitive and Physical Approach by Kline. I would DEFINITELY recommend this book as a self study calculus book... especially for the price!

I'll tell you a little bit about my experience with the book. Last summer I was between my senior year of high school and my first year of college, planning to major in engineering. My high school did not offer a calculus course so I decided that I wanted some exposure before my first college level math class. Well to my surprise, I literally could not put the book down! I did nothing but read this book for hours a day for a few weeks and then slowed down a bit, but it was very enjoyable and clear. The exercises are great and if you email Dover they will email to you a solution key. I ended up getting the top grade in my calc I class of 500 and currently have the top grade in my Calc II class... So there is some proof that it works! In fact, I often use Kline instead of our standard Stewart text. Anyway, sorry I can't comment about the other three but definitely do not hesitate on purchasing Kline..

 

[Matt_from_now]

I cannot say much about the other three books but I have read Calculus: An Intuitive and Physical Approach by Kline. I would DEFINITELY recommend this book as a self study calculus book... especially for the price!

All the reviews say Kline gives a very intuitive understanding. I've very much looking forward to it.

 

[Robert1986]

It might, perhaps, be good to visit a university's website and see what course a typical math major would take, and in what order they are taken. Then, you can sort of plan out a course of study. Of course, this doesn't help with which books to buy, but it might help with the order in which the books should be bought and studied.

 

[jbrussell93]

All the reviews say Kline gives a very intuitive understanding. I've very much looking forward to it.

Yes, very intuitive. He really does offer a "physical approach". For instance, he introduces derivatives by physical example rather than abstract. I had no idea what a derivative was before I started reading but after just a couple of chapters I felt that I understood better than my friends who had already taken calc in high school. Also, he jumps right into differential calculus which I really enjoyed because I found the Stewart text to be too slow and often frustrating due to the order in which the information was presented. Kline introduces topics in a much more logical way and doesn't "beat around the bush" so to speak. If you read this entire book cover to cover you will basically cover the key topics of Calc I, II and I believe most of III for most universities. As far as independent studying in general (especially calculus), don't forget that the internet has many great resources such as Khanacademy.org, ocw.mit.edu, etc if you get stuck on a particular topic... often times, just seeing the information presented in a different way or even from a different perspective helps to really solidify concepts.

Anyway, good luck with your studies!

 

[autodidude]

Yeah, Kline and Tenenbaum's books are good. I think it's a good idea to read through an 'easier' linear algebra text before tackling Shilov's book.

If you want to learn abstract algebra, I like Pinter's book (Dover).

 

[Matt_from_now]

Kline seems to cover a lot. Once I finish would there be a need to follow it with an "advanced" calculus book? Does advanced mean beyond what calculus III would cover? I know Dover sells some that go by the title "advanced" calculus.

 

[autodidude]

In my opinion, Kline is a bit verbose sometimes which can be a good or bad thing depending on how you like your math texts. I don't think Kline's book covers vector calculus (calc 3?)

I think the Dover advanced calculus texts covers rigorous multivariable calculus...check out the table of contents if you can (Amazon or Google Books) and compare it to a more standard m.variable calculus book like Stewart's

 

[Daverz]

Dover has at least half a dozen "Advanced Calculus" books.

http://doverpublications.ecomm-search.com/search?keywords=advanced%20calculus

They all emphasize different things, since the phrase is sometimes used to mean multi-variable calculus and sometimes to mean general analysis. I think the one by C. J. Edwards, Jr. is pretty good as a rigorous but very readable multi-variable calculus book.

 

[mathwonk]

That is the same C.H.(Henry) Edwards Jr. as in the elementary calc book by Edwards-Penney. He writes well.

The book by Widder is a famous classic that was a standard when Iw as an undergrad at Harvard in the 1960's. At $15 it is a bargain, but you should look at a library copy first to see if you like to read that old style of exposition.

I notice the infamous Steenrod , Spencer and Nickerson is in that catalog, but i do not recommend it much to anyone.