Math Books

[Cleverbeans]

This is now the official thread for questions about math books at any level, from high school through advanced college courses.

I'm looking for a good vector calculus text to brush up on what I've forgotten. We used Stewart's Multivariable Calculus as a baseline but I was unable to purchase the text for financial reasons at the time. I figured some things may have changed in the last 12 years, so if anyone can suggest some good texts on this subject I'd appreciate it.

[ThomasS]

The textbooks go up in price and new pretty pictures appear. However, Calculus really hasn't changed all that much.

If you don't mind a certain lack of pretty pictures, you might try something like Widder's Advanced Calculus from Dover. it is much easier to carry around than Stewart. It is also written in a style that a mathematician might consider normal. If you think that you might want to move on to real math at some point, it might serve as an introduction to the associated style of writing.

[Cleverbeans]

The textbooks go up in price and new pretty pictures appear. However, Calculus really hasn't changed all that much.

If you don't mind a certain lack of pretty pictures, you might try something like Widder's Advanced Calculus from Dover. it is much easier to carry around than Stewart. It also is written in more like a real math book. If you think that you might want to move on to real math at some point, it might serve as an introduction to the style.

Well to give you an idea of the textbooks I like I think "Abstract Algebra" by Dummit and Foote is probably the best text I've used, and I also like "Elementary Linear Algebra" by Anton. I've found "Principles of Mathematical Analysis" by Rudin a little terse. Since I'm mainly self taught, so some pictures are useful but I don't require full color glossy shots, however I would like some meaningful diagrams so I can get a picture of what's going on rather then a pure string of proofs.

[ThomasS]

If you like Dummit and Foote, then I'd expect you to find an "Advanced Calculus" text (old terminology for multivariate calc) to be more interesting than Stewart and company. I've also heard excellent things about Spivak's Calculus on Manifolds.

Looking forward, if you really like vectors and curves and such, there is an entire field known as differential geometry. Spivak also wrote a multi-tome "introduction" to the subject. The first volume is most of what a normal differential geometry text would be and stands well on its own. Lee's "Introduction to Smooth Manifolds" is also pretty nice, and a single volume.

I've never liked Rudin, and it is hard for me to put my finger on why. I'm not sure if a lack of pictures are the real problem. Perhaps it is more a lack of motivation, or some other missing connectives. For introductory real analysis, I like Komogorov and Fomin (another easy to carry Dover book) much better than Rudin.

[Cycle]

If you're looking at Dover books, don't go for Kline. That's the book I used to teach myself Calculus. Once I took a Calc class, I realized how terrible the explanations were.

I don't know about Rudin's Calculus text specifically, but I've read other books by him. They suck. That may just be me though, I have a real hard time learning from classical books like that.

I haven't read Spivak's Calculus book either, but I have read his differential geometry books. I really like it. If his Calc book is as good, I imagine it'd be one of the best choices, especially if it's cheap.

[Cleverbeans]

Has anyone had experience with "Calculus with Analytic Geometry" by Simmons? It's what they're using for the MIT OpenCourseWare resources however I don't want to pick it up if it's a dud.

[Pathway]

Calculus, Volume II by Tom Apostol is what my class uses. I think it's good.

[Verdigris]

Hi all,

I'm looking for a good calculus book to refresh my knowledge of calculus (a good refresher on trig and analytic geometry, in the same book or another, would also be helpful). Ideally, it would be fairly compact/concise (not one of the doorstops typically used in classes) and not hideously expensive. Even though my last calculus classes were as a undergrad in 1991-1992, I've still got a fairly good grasp on the fundamentals, i.e. derivatives and integrals. My goal here is to advance on to other topics, such as linear algebra, probability and statistics, queueing theory and whatever else I find necessary/useful in my career. Specifically, I'm a Unix system administrator (with an MS in Computer Science, FWIW) who wants to move towards system performance analysis, modeling, etc. So I'm not necessarily looking for the most rigorous books since I doubt real analysis, proving theorems, etc. will be of much value given my goals, but I don't want an overly simplistic "Calculus for Dummies"-type book either.

Suggestions?

Thanks,
Verdigris

[Verdigris]

A lot of views so far but no recommendations? Actually, I happened upon one book at Amazon that I recalled seeing in stores several years ago, Modern Calculus and Analytic Geometry by Robert Silverman. It is more of a "doorstop" in size (1000+ pages), but as I recall, it's a trade-format paperback and not loaded with tons of useless pictures like most modern textbooks. Does anyone have any direct knowledge of and experience with this book?

Thanks,
V

[Yesila]

It's only 70 pages, and it comes in a free .pdf download.

http://www.math.umn.edu/~garrett/calcul ... /notes.pdf

I glanced through it and it seems to cover most topics from a first year calculus class with the notable exception of avoiding "general" series , and only covering Taylor polynomial material.

[raymondo]

i don't know if this has been posted before so if it has sorry and can you link to the post please

As a college student hoping to do (pure) maths at university it is expected that i read around the topic. This means learning about previous maths, mathematicians and current maths and mathematicians. I was hoping that the people here, being of such high IQ and mathematical skill , would be able to help with ideas of things to read or... whatever .

all help will be greatly appreciated

[Ended]

I would recommend two books in particular:

• A Mathematician's Apology: A classic defence of the value of pure maths. Somewhat outdated and stylistically old-fashioned maybe, but something that every mathematician should read.
• Mathematics: A Very Short Introduction: An absolutely excellent introduction to mathematical thinking. Doesn't require any specialist knowledge beyond, say, pre-GCSE (age 12/13), but is not dumbed-down at all (it's written by a Fields medallist) and explains subtle concepts well. I can't recommend this book enough.

Also, Simon Singh's 'Fermat's Last Theorem' and 'The Code Book' are good, and there's a bunch of other popular mathematical books like 'The Music of the Primes' and 'The Book of Nothing' which do a good job at giving historical perspective and describing various areas of maths. Also, I think it's obligatory to recommend 'Gödel, Escher, Bach'.

If you're really hardcore and want some real maths, I'd suggest Beardon's Algebra and Geometry.

i don't know if this has been posted before so if it has sorry and can you link to the post please

There might be some useful suggestions here or here.

[Fafnir43]

New user reporting for duty, sir!

First off, there's a massive list of books here. They're mostly very good, but you might want to take the last section or two with a generous grain of salt since it's UK-oriented and we get to do a lot more maths before university than you.

If you want to get straight into some serious maths, then my recommendations vary depending on what you're interested in. Bryant's Yet Another Introduction to Analysis is a wonderful book for real analysis (basically the rigorous foundations of calculus) - it's not actually a textbook, so the author's free to be a bit less formal in style. If you're feeling more adventurous, Spivak's Calculus is good as well if a little harder - it covers more stuff, and it has a few amusing little detours like proving that e is transcendental. (It's also invaluable as a textbook.) If you're at all interested in group theory or abstract algebra in general, try to get hold of Fraleigh's A First Course In Abstract Algebra - it gets quite hard quite fast, but it covers a lot - the insolubility of the quintic is proved in the last chapter. Anton's Elementary Linear Algebra is an easy introduction if you're interested in matrices, although it might be less easy with a US background - I'm not sure how much you cover.

For lighter, more general stuff, I'd highly recommend Ball's Strange Curves, Counting Rabbits and Other Mathematical Explorations. It has things like a space-filling curve, Pick's Theorem (which will make you gape in awe the first time you see it), and Stirling's approximation to n! - a very fun read. If you get one book, make it that one. Darbyshire's Prime Obsessions is probably the best book out there on the Riemann Hypothesis in that it actually goes into a little of the maths, while not actually assuming any maths background. And I second the recommendations of Godel, Escher, Bach and anything by Simon Singh. (If you're interested in the intersection between maths and computer science, I'd also recommend you pick up Dewdney's New Turing Omnibus - quite a friendly book with a very intimidating name.)

And if you're feeling really adventurous, get Concrete Mathematics by Knuth, Graham and Patashnik. Yes, that Knuth...

And finally, a piece of advice that will stand you in good stead: always always always buy maths books second hand if you can. You'll get things for £5 instead of £40.

[Qoppa]

The Mathematical Experience is a very good book giving an overview of the history of maths, some open problems, and many of the problems that were faced (or are still faced) by mathematics and mathematicians today. A light read, but it also gets quite technical at times, which as an aspiring mathematician, I'm sure you'll enjoy.

[Cycle]

This may be a tad elementary, but I'd strongly recommend Burger and Starbird's "The Heart of Mathematics". It doesn't really assume any math (need algebra, no calculus). It goes over a very wide variety of topics, including topology, infinite cardinals, fractals, alternate geometries, and discrete math. Like I said, it's a tad elementary (goes over Pythagoras' theorem, for example), but it's the best introduction to mathematics in general I've ever seen. Plus, it includes 3-d glasses (they have 3d pictures in the book). 3-D glasses!

Edit: added link

[raymondo]

Cheers for all that. a couple of the books i've heard of or had suggested to me but its good to have them confirmed.
I think i'll definitely read a couple of the introductory books since it's always good to know where you stand .
The list of books in that pdf was especially helpful, it had all the books suggested plus some i've already read plus a load more so thats great

i think i'm going to have my head down for quite a while now
plus the tip about buying used is good, i had a look at a couple of the ones suggested and new versions are really expensive! haha and 3-d glasses!
also did you think i'm from the US? if so y? im from the UK

any other tips on uni applications?

[Qoppa]

I knew you were from the UK because you called it maths.

[Ended]

And finally, a piece of advice that will stand you in good stead: always always always buy maths books second hand if you can. You'll get things for £5 instead of £40.

Also, in general, it's a good idea not to buy any textbooks before you've had a chance to check out the maths library at your uni. (The exception is if, like now, you want to read them before starting). It's frustrating to pay even £5 for a book and then find out there are multiple copies lying around in the library which you can take out on loan for the entire term / holiday.

[paragon12321]

Maybe Flatland? It might help you get the "fourth dimension" thing down a little. Plus, its a half-decent read.

[Fafnir43]

also did you think i'm from the US? if so y? im from the UK

Ooops! I guess I just tend to assume everyone on the Internet is from the US... In that case, if you've done Further Maths at A Level, there's a book by Riley, Hobson and Bence called "Mathematical Methods for Physics and Engineering" that I'd strongly recommend. It's not terribly friendly for reading ahead - almost impossible right now without Further Maths, really - but it's extremely good as a reference - it covers pretty much everything you need for applied maths in the first two years of university, and a lot of what you need for the rest of the course. It's a heavy book... Also, this might just be a matter of taste, but I'd tend to recommend Ian Stewart's Flatterland over Flatland - I find the writing more accessible, and it covers more exotic concepts.

As for uni applications, it's a very good idea to put a few maths books you've read on your Personal Statement, but (obviously) make sure you've actually read them! Some universities will ask a bit about them at interview - they won't expect you to know any given point in the book, but they will ask about the book in general. Also if you've done well in any maths competitions (or got a gold in the Maths Olympiad), make sure to mention that. As for interviews, most maths interviews will consist of (or at least contain) a long string of difficult questions. They're often of the sort knowledge of the syllabus either doesn't help for or isn't sufficient for - if you're doing AEA or STEP, you'll know what I mean. As such, the best way to prepare is to practice general mathematical thought - Paul Zeitz's The Art and Craft of Problem Solving is very good, and Siklos' Advanced Problems in Mathematics is great if you're doing STEP.

Which universities are you thinking of going for?

[raymondo]

My schools library is very small, and most of the content is to do with more wordy subjects such as history and religous studies so there arent many maths books. But i get the point about not paying for a book when you dont have to . I think i might be able to ask the library to order books in so i might ask them to get some in

it's a very good idea to put a few maths books you've read on your Personal Statement, but (obviously) make sure you've actually read them!

haha yeh thats what our careers teacher and all the upper sixth said, which is why i thought id start early . And i think i'll be able to understand most of the books being one of the best at maths in my year , if not our teachers are really helpful.

umm well im thinking of applying to Imperial college london, bristol, cambridge but im not sure which college yet, and something else with lower grade boundries just in case, not sure which yet. good/ bad choices do you think? i think i might check to find out what the lowest erdos number is at each

[Fafnir43]

I feel your pain about the library... It might be worth checking whether your Maths department has their own stash of books. My school's did, it had far more high level/interesting stuff in than the main library, and as far as I can tell I was the only one who ever used it.

Cambridge is incredible - I'm studying there. If you apply (which you should if you're one of the best in your year), you might want to consider applying to Trinity college. I'm at Peterhouse myself, but for various reasons we've been taken under Trinity's wing for maths so I've had a bit of a taste. More specifically, two of my supervisors were actually the lecturers for the courses they were supervising, and one of them was a Fields medalist (Prof. Gowers) who also supervised me for another course. Considering I'm a first year, I'd call that pretty bloody impressive! It's a very difficult course, but if you can make it through STEP you should be alright.

Imperial and Bristol are both good - I'd recommend you look into Warwick and Southampton as well, especially Warwick (I've heard it's second only to Oxbridge).

[SimonM]

Imperial and Bristol are both good - I'd recommend you look into Warwick and Southampton as well, especially Warwick (I've heard it's second only to Oxbridge).

I heard Warwick > Oxford for maths.

Also, by Gold in the Maths Olympiads, do you mean in the SMC or in BMO 1? I'm intending on mentioning a Distinction in BMO 1, despite not placing in the medals, I still believe that this is an achievement which the universities should know about. (I also mention my participation (but not score) in BMO 2)

[Fafnir43]

Imperial and Bristol are both good - I'd recommend you look into Warwick and Southampton as well, especially Warwick (I've heard it's second only to Oxbridge).

I heard Warwick > Oxford for maths.

Also, by Gold in the Maths Olympiads, do you mean in the SMC or in BMO 1? I'm intending on mentioning a Distinction in BMO 1, despite not placing in the medals, I still believe that this is an achievement which the universities should know about. (I also mention my participation (but not score) in BMO 2)

Yup, I did in fact mean the SMC - sorry for the ambiguity. So yeah, definitely mention the Distinction and getting through to BMO2 - both are exceedingly good, pretty rare among applicants, and will impress admissions tutors no end. That goes double if you weren't coached in the extra maths needed for the BMO (like the number theory), so mention that too if it applies. It's also quite a pretty good sign for being able to get in - I know someone who dropped out at BMO1 who still got S1 at STEP, far above his offer.

Warwick may well be better than Oxford, actually - I know very little about their maths course, since you're forced to choose between Cambridge and Oxford and for maths Cambridge is near-universally considered the better of the two. All the more reason to take a closer look then!

[Robin S]

If you are applying to Cambridge, I would actually recommend against Trinity unless you are very confident. It's extremely competitive. I applied there, having done well enough in BMO 2 to be invited to the course held annually at Bath for about a dozen students (and done better than most of those students, many of whom ended up at Cambridge, in the mock IMO paper), and done fine in all my STEP mocks despite being completely self-taught for almost all of the Further Maths modules. Yet despite all that, due to the stress of the exams I only got grades 22 in STEP 2 and 3, just missing my offer from Trinity. As a result I am now at Imperial which, while a respectable institution, does not compare with Cambridge. Oxford, likewise, does not compare, though it's still very good - I ended up helping someone doing Maths there with their holiday coursework. I've got a feeling it'd be easier to get into, especially compared with Trinity, Cambridge.

[SimonM]

Firstly, congratulations on making the Trinity training weekend, that is an impressive achievement (something which is my biggest target for next year (comparable with obtaining university offers and A levels)).

Trinity is indeed very competitive, but IMO/BMO experience does not affect ones ability to go there. I have several friends there and holding offers (which they are almost certain to achieve) who achieved similar or lower marks than me in the BMO 2 this year. However, it is actually (once you do the statistics) no harder to get into that many of the other colleges (since they accept far more maths applicants). Very few colleges would accept a 22 anyway, although with your history I am suprised that they weren't happy to "bend the rules" (I've heard them do that in the past for IMO-ers (and those which make the training weekend seem ideal).

However, most of this is said based on what I can ascertain from the experiences of others (I will not be applying until next September (and definately wont be applying to Trinity (The college just doesn't seem right for me))

Finally, a word of advice to anyone else doing the STEP (I'm taking some of it early). Try and find some help with it, STEP I and II are fine without extra input, but STEP III is extremely difficult and spending some time on it is essential in order to get a reasonable grade.

Final words, Gold in SMC is not that great an achievement and is pretty much standard for maths applicants although is still worth mentioning. Making BMO 2 is probably the biggest achievement of any of those, no matter what the score is. It is almost impossible to do well without some kind of training (which I demonstrated this year and hope not to do next year)

Edit

Forgot to mention something. Achievement at SMC, BMO etc does not show whether or not you will get into Cambridge or not. There are many people who didn't even get into the BMO who excel at Cambridge with Firsts etc. It is just a good indicator of mathematical ability. Those people who dislike competitive mathematics often give similar arguments to those against IQ tests. "It is a test of ones ability to do maths competitions"

[raymondo]

wow you guys really know your stuff
im pretty sure i know most of what your talking abou but nothing a little look on wikipedia wont fix...


i'll try not to get distracted...

[Various Varieties]

New user reporting for duty, sir!

First off, there's a massive list of books here.

I see that Ian Stewart's "From Here to Infinity" is mentioned in that pdf. I'm not a mathematician, but having read that book over the past few weeks, I'd like to recommend it - even though I didn't completely follow anywhere near all of it!

[raymondo]

should i be pleased that my thread has been converted to an official thread?
ah well i guess it'll be able to give more help for people who need it

[McHell]

Maybe Flatland? It might help you get the "fourth dimension" thing down a little. Plus, its a half-decent read.

Nooowwww, not that atrocious morality play... It goes nowhere. I always heard it extolled, but then read it. If I want to read an argument for the existence of god, I'll go look in another department than the maths section. It's mercifully short, though.

Now, to textbooks again (as the topic started, not math olympiad discussions): a very useful one is "all the mathematics you missed", with brief descriptions of the why/what/how of a host of unrelated fields and trickses. Very useful for general modelling, and if you know what you kind of need but have no idea how to call it and where to search.

[ensanglant]

Could someone recommend a good calc book that goes from around calc 3 to as far as possible? Preferably a while into the calc 3 year.

I am also looking for some abstract algebra book, or just some book in the advanced algebras that continues on past eigenvalues and eigenvectors.

Your help is very much appreciated.

[Fafnir43]

A First Course in Abstract Algebra (Fraleigh) is great for group theory, ring theory and Galois theory and doesn't really assume any prerequisites. (It moves reasonably fast, though, so try the exercises - they're easy, so there's no real reason not to.) I'm not sure what you mean by "the advanced algebras" - do you mean a book for linear algebra that covers things like Jordan Normal Form, the Cayley-Hamilton Theorem, infinite-dimensional vector spaces etc.?

[ensanglant]

something like that, I wasn't quite sure what to call it, but I just finished a basic linear algebra course and I wanted to find something to continue on, as I really enjoyed the subject. Thanks for the recommendation.

[Fafnir43]

I'm afraid I don't really have enough linear algebra myself to recommend something for that (Anton is quite good, but only for a first course). I'd be interested in a good book on the subject myself after exams end, actually - does anyone have any ideas?

[ThomasS]

something like that, I wasn't quite sure what to call it, but I just finished a basic linear algebra course and I wanted to find something to continue on, as I really enjoyed the subject. Thanks for the recommendation.

You might try "Linear Algebra" by Lax. Roughly speaking it covers normal linear algebra but with an abstract bent. If you want even more abstract studies involving eigenvector and eigenvalues and the like, the next step is functional analysis. Lax is intentionally abstract enough to be a decent warm up for it. However, you usually you do want a bit of real analysis/measure theory before you dig into functional analysis. If you want to dig into it anyway, you might try "Functional Analysis" by Yoshida. It is inexpensive as such things go and tends to develop all the theorems that it needs. It is however, rather dense and might be hard to follow coming out of a Calc II/III course. You'll learn a lot if you get even partway through though.

The whole real analysis/measure thing is itself one of the standard things to study after Calc. Royden is a reasonably well known and I like him better than Rudin. I also like Kolmogorov & Fomin, which as the added advantage of being not obnoxiously priced.

[Luthen]

Not sure if this quite on topic. It's not a textbook but it is a book about maths.

As a mathematics interest book I'd suggest "Zero: the Biography of a Dangerous Idea" by Charles Seife. It's a really good read and I just got through its section on Calculus. Newton's reasoning was very iffy (he divided by a number that was both zero and not zero). Also learnt that l'Hopital didn't discover his rule, his teacher did.

Anyone else have non-textbook maths books to share?

[daryuu]

Am I the only person that's finding "Analytic Number Theory" by Iwaniec and Kowalski to be absolutely painful to try to muddle through? It's not for a class, I got it because I wanted an introduction to the subject (they only list calculus and complex analysis as prerequisites), but even in the first chapter I have no idea where they're getting half of the things they're asserting. Nothing lowers your self-esteem more than thinking a book is just listing results, only to see the phrase "less obvious is the following formula of Jacobi" sitting in the middle of a page...

[ThomasS]

A lot of math books don't have strong prerequisites but do have a sort of underlying requirement for "mathematical maturity" which is alleviated somewhat if you've seen some of the material before in other contexts. Federer's "Geometric Measure Theory" is a book like this which I have only muddled through parts of.

Number theory in particular almost has a tradition of trying to understate the difficulty in book tittles. e.g. There is Serre's "A Course in Arithmetic", which is not nearly as elementary as it might sound.

[daryuu]

Thank you, that actually made me feel quite a bit better about the whole thing. I managed to do the first problem in the first chapter a while ago, but it took me so long that I got completely turned off the subject... Maybe now I'll try again while keeping that in mind.

[Teseract]

i'm searching a good book about "Discrete maths", for semi-begginers (i had that assignature in the college, but i want to know more about Graphs, Groups, and the Language Theory!).

What could you recommend me?

(In e-bay, or something similar, i'm an outsider! =P )