Math Books

For the discussion of math. Duh.

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vector
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Re: Math Books

Postby vector » Sat Dec 25, 2010 6:07 am UTC

I'm asking for opinions on Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables and what books one thinks one should read in preparation for its study. Unfortunately, my introductory complex analysis course will be using it as a text next semester, so I'm desperately flinging myself through Topology (Munkres), Baby Rudin, Algebra (MacLane), Algebra (Artin), Naive Set Theory (Halmos, soon to become Set Theory by [Hausdorff]) and the professor's lecture notes he put online through the next three weeks.

Obviously I'm not going to get super-far through any of those books, but I've been making some good progress thus far and it'll be even easier once Christmas is past. The professor in question seems to be in love with category theory, order theory, and weird filter-net interactions, so I'm brushing up on those. Mostly, I'm working on building up a certain intuition with a number of structures, which are coming much easier nowadays but are still not exactly friendly.

I've taken linear algebra (I really don't know this well, though), abstract algebra, metric differential geometry, an incompleteness/undecidability course, discrete math, game theory, calculus of all stripes, and real analysis. Most of those subjects are somewhat less than solid at this point, but then... what can one do. I'm about to be a second-semester sophomore (long story involving withdrawing mid-semester last time I tried second-semester sophomore-hood).

Anyway, any advice is helpful =/
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Re: Math Books

Postby GyRo567 » Thu Dec 30, 2010 4:53 am UTC

I think you might be panicking prematurely, especially with the background you already have.

Edit: If you want a slower paced book for reference, Brown & Churchill's Complex Variables with Applications is excellent, though it has different ordering of topics.
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Re: Math Books

Postby vector » Sun Jan 02, 2011 2:21 am UTC

GyRo567 wrote:I think you might be panicking prematurely, especially with the background you already have.


I think I agree with you. Studying has calmed me down significantly.

Anyway, thanks for this.
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Re: Math Books

Postby ++$_ » Sun Jan 02, 2011 3:10 am UTC

I guess this is a bit late, but I looked at the book on Google Books. I have to say that I think it is kind of strange, and I don't really like it very much. Judging from the pages I looked at, I do not think that there are many prerequisites for reading it (except a willingness to fight through Cartan's unnecessary and obfuscating Bourbakicity, and to look up the algebraic and topological concepts when necessary).

But it's possible that I looked only at the easy pages.

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Re: Math Books

Postby vector » Sun Jan 02, 2011 12:16 pm UTC

Yeah, the professor in question is kind of... weird... and Bourbaki's works have a certain impenetrability I've stumbled into before (albeit briefly). I can't understand anything they write without writing it out myself, along with copious sidenotes and explanations of where they're going with each step.

That laconic French style... *brrr*

Anyway, thanks for your input, as well. As I've been studying topology and algebra with some intensity over the last bit, I suppose I'll just hold onto my confidence and go for it.
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Re: Math Books

Postby existential_squirrrel » Wed Jan 12, 2011 11:11 pm UTC

Has anyone on the forum had experience with using A Transition to Advanced Mathematics, 7th Edition?
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Re: Math Books

Postby doogly » Wed Jan 12, 2011 11:12 pm UTC

I have no experience with it at all, but I'm guessing from the "7th edition" it's crappy.
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Re: Math Books

Postby existential_squirrrel » Wed Jan 12, 2011 11:21 pm UTC

doogly wrote:I have no experience with it at all, but I'm guessing from the "7th edition" it's crappy.



or re-written / re-worded to keep up with the times.
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Re: Math Books

Postby doogly » Thu Jan 13, 2011 12:15 am UTC

You don't appear to be cynical enough yet. Let me introduce you to my friend, the publishing industry!

That book purports to introduce one to things like analysis and algebra. For comparison, the standard algebra text, Dummitt and Foote, has had three editions in over twenty years. The standard analysis text, Rudin, has had three editions in something like forty years. Seven editions in ten years? This is a warning sign.
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Re: Math Books

Postby Mike_Bson » Tue Jan 18, 2011 9:06 pm UTC

Is Elementary Linear Algebra by Howard Anton a good book for beginning Linear Algebra?

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Re: Math Books

Postby B.Good » Wed Jan 19, 2011 2:19 am UTC

Mike_Bson wrote:Is Elementary Linear Algebra by Howard Anton a good book for beginning Linear Algebra?

I've looked through it a little bit, and it is not the best. It's an expensive book but I really liked the book that I used for my Linear Algebra class (Elementary Linear Algebra by Larson and Falvo). That said, other users in this thread have suggested texts on linear algebra which are more affordable and probably better than the book by Larson and Falvo.

For example, this one which was recommended by Other_Calvin: http://joshua.smcvt.edu/linearalgebra/

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Re: Math Books

Postby Marbas » Sun Jan 30, 2011 7:29 pm UTC

So, after doing Measure Theory and such things, I want more analysis stuff. I have already gone through Royden, now I don't know where to go next. What kind of book would be good if I want to learn about things like Sobolev spaces? And other things in analysis? I know functional analysis is a place I can possibly go, what else is out there?
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Re: Math Books

Postby forgetful functor » Wed Feb 02, 2011 11:56 am UTC

Marbas wrote:So, after doing Measure Theory and such things, I want more analysis stuff. I have already gone through Royden, now I don't know where to go next. What kind of book would be good if I want to learn about things like Sobolev spaces? And other things in analysis? I know functional analysis is a place I can possibly go, what else is out there?


You have a few options. If you want more "pure" analysis, I would get a text on functional analysis. Rudin's Functional Analysis is excellent but lacks any sort of motivation. Lax's book covers an incredible range of topics, but can be tough as a first exposure. I've heard Conway's Course in Functional Analysis is great, but I haven't read it myself.

A second option in pure analysis is Fourier analysis (or harmonic analysis). If you feel comfortable enough with real analysis, Duoandikoetxea's Fourier Analysis is excellent. Nothing is required beyond basic real and complex analysis, but you get into some very deep material rather quickly.

If you really liked the measure theory aspect of analysis, there is the subject of geometric measure theory, which has applications ranging from PDEs to the Plateau problem (see Wikipedia). The book by Morgan is very good, as is the book by Krantz (Geometric Integration Theory).

You could also try a different area of math in which you would use the analysis you learned. Obvious choices are ODE, PDE, and differential geometry. My personal specialization is analytic number theory, which uses analysis to study problems in number theory.

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Re: Math Books

Postby MidsizeBlowfish » Sat Feb 05, 2011 6:19 pm UTC

I'll elaborate on forgetful functor's suggestions about functional analysis, because, well, I'm a functional analyst.

While it's probably possible to learn functional analysis from Rudin's book, I feel like that's a tough route. Conway's book is a good suggestion; while I may not always love his writing, he takes an interesting approach to presenting functional analysis. He begins by developing the basic ideas of functional analysis in Hilbert spaces, where they're simplest, then generalizes those ideas to Banach spaces, where they're slightly more complicated, then generalizes to locally convex spaces, where they're again slightly more complicated. He finishes off with a decent treatment of the basic theory of operator algebras (which appeals to me bc I'm an operator algebraist). After you feel comfortable with a topic, Rudin is a great place to read about it in more depth. He generally deals with topological vector spaces from the very beginning, and proves a lot of the same theorems you'll find in Conway for Hilbert or Banach spaces in a great deal more generality.

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Re: Math Books

Postby B.Good » Sat Feb 05, 2011 6:49 pm UTC

I took calculus 3 (multivariable and vector calculus) two semesters ago and I didn't quite get all that I wanted out of it as the professor was kind of soft on us. Is there a book that covers calculus 3 stuff with quite a bit of rigor? Preferably a book that just covers multivariable and vector calculus as my knowledge of single variable calculus is sufficient. I still remember a bit from the class so it doesn't need to be a textbook that spoon-feeds me things and I think I should brush up on the subject before I take an analysis class at my university where I would have to prove things I've either never learned or don't know in detail.

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Re: Math Books

Postby doogly » Sat Feb 05, 2011 7:56 pm UTC

If you want to brush up on things, something like Div, Grad, Curl and All That covers the computation. But for rigor, you approach things from a different sort of perspective. It becomes important to distinguish between differential forms and vectors, and it's generally done in the context of manifolds rather than just R^n. Munkres and Spivak have some great books on this subject. (or at least, in the case of Munkres, canonical. I haven't used it myself so should reserve personal endorsement as great.)
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Re: Math Books

Postby B.Good » Sat Feb 05, 2011 11:26 pm UTC

doogly wrote:If you want to brush up on things, something like Div, Grad, Curl and All That covers the computation. But for rigor, you approach things from a different sort of perspective. It becomes important to distinguish between differential forms and vectors, and it's generally done in the context of manifolds rather than just R^n. Munkres and Spivak have some great books on this subject. (or at least, in the case of Munkres, canonical. I haven't used it myself so should reserve personal endorsement as great.)

I've seen Div, Grad, Curl and All That mentioned in this thread before but I can't find the table of contents and it seems to be a book more geared towards scientists and engineers which might make it worth picking it up but it's not what I'm looking for right now. In my calculus 3 course we used McCallum, Hughes-Hallett, et al. but I didn't really like it as the examples weren't good and now that I remember, the theory in it is quite light. I also have Stewart's book which is very good for learning it but the theory in it is even lighter than McCallum's text. Basically what I'm looking for is a book that is more rigorous than Stewart's and McCallum's book (in terms of theory) but less rigorous than a real analysis text. From the few that I've seen so far, Lang's text seems to fit the bill but I'm not entirely sure about that. I don't quite think I'm ready for differential forms stuff as of yet or at least Spivak's as I've read that it's dense reading to say the least.

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Re: Math Books

Postby doogly » Sun Feb 06, 2011 4:45 am UTC

Yeah, I don't think a level of rigor exists between calc 3 and calc on manifolds. Maybe take a look at the Munkres book? Same sort of stuff as the Spivak but decompressed.
You're probably ready for those things if you've also had some linear algebra. It's tricky but fun stuff.
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Re: Math Books

Postby curtis95112 » Wed Feb 16, 2011 5:35 am UTC

Could you please help me find a precalculus book with the most difficult exercises available? Or a good online resource for difficult problems.
The precalculus course I'm taking at my high school is extremely competitive and I can't find any books with problems at the level of the quizzes/exams we'll be taking.

Here are some sample problems to give you a feel for the problems.

1. Suppose the lines ax+by+c=0 and dx+ey+f=0 intersect at the point (p,q). Show that every equation of the line that passes through (k,l) can be expressed as m(ax+by+c)+n(dx+ey+f)=0 where m,n are constants.

2. Let the distance between two points (x1,y1) and (x2,y2) be defined as d = (x1-x2)^2 + |y1-y2|.
Graph a circle centered at the origin with a radius of 3.
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Re: Math Books

Postby ahazaq2 » Thu Mar 17, 2011 12:38 am UTC

Unfortunately, I don't know of any good pre-calculus books. Maybe just try working through math competition problems to get a grasp on problem-solving?


Does anybody know of any good introduction to combinatorics books? All that I know of combinatorics is basically fuzzy notions from algebra 2 of what a combination and permutation is.

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Re: Math Books

Postby vector » Thu Mar 17, 2011 12:44 am UTC

To be honest, looking at those problems makes me think that you aren't quite understanding how they work--that is, by applying principles you should understand from other problems in new ways.

Try going through the process you'd use for the "usual" version on those problems, and see what happens. You may find that your main problem is anxiety vis-a-vis new things, rather than actual lack of knowledge.

I hope I'm being helpful, rather than trivializing your troubles in some sense =/


ahazaq2 wrote:Does anybody know of any good introduction to combinatorics books? All that I know of combinatorics is basically fuzzy notions from algebra 2 of what a combination and permutation is.


Rosen's book on discrete mathematics is a nice introduction, I thought. If you're more interested in graph theory, there's always Bollobas and so on.
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Re: Math Books

Postby Kurushimi » Thu Mar 24, 2011 4:29 pm UTC

In a few weeks, I'm going to attending a math team state competition. One of the tests I'll be taking is on the History of Prime Numbers. Does anyone have any good books, or internet sources, that I could read that covers prime numbers well?

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Re: Math Books

Postby skullturf » Thu Mar 24, 2011 5:45 pm UTC

Kurushimi wrote:In a few weeks, I'm going to attending a math team state competition. One of the tests I'll be taking is on the History of Prime Numbers. Does anyone have any good books, or internet sources, that I could read that covers prime numbers well?


The Prime Pages at the University of Tennessee at Martin.

http://primes.utm.edu/

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Re: Math Books

Postby greengiant » Thu Mar 24, 2011 7:17 pm UTC

Can't say I've personally read it, but Music of the Primes by Marcus Du Sautoy sounds like it would fit the bill.

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Re: Math Books

Postby vector » Thu Mar 24, 2011 9:43 pm UTC

greengiant wrote:Can't say I've personally read it, but Music of the Primes by Marcus Du Sautoy sounds like it would fit the bill.


I've read it, and yeah, it would be absolutely perfect.
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Re: Math Books

Postby Ddanndt » Fri Apr 08, 2011 8:19 am UTC

Does anyone know a good LaTex book covering some basic stuff like text formatting, maths formatting, figures and drawing with TikZ?
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Re: Math Books

Postby jestingrabbit » Fri Apr 08, 2011 8:23 am UTC

Most of those are in lshort.pdf, but not TikZ.
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Re: Math Books

Postby Cornish » Sun Apr 10, 2011 3:47 pm UTC

Can anyone give me any suggestions for a good problem based calculus book? I already have Calculus (3th Ed) by Michael Spivak; which I really like, but would love to have a few more problems to work with (in the same style) for pretty much every chapter. In other words... I guess I'm looking for a book much like the one I have, but with more or different problems.

Thanks in advance!

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Re: Math Books

Postby Metaphysician » Thu Apr 14, 2011 12:20 am UTC

Hello everybody, this is my first post here on xkcd. I've recently started to become very interested in mathematics. My experience with math is limited. I coasted through Algebra 2 and Trig in highschool and never pursued any further knowledge in math as I've always been more interested in philosophy, literature and language. I dropped out of college for financial reasons and never had the opportunity to take any of the mathematics electives I was interested in. As I've studied philosophy over the years I've begun to realize how much philosophy draws from the mathematical world. Thus, I have decided to construct a math curriculum for myself. I have no illusions of becoming an influential mathematician or even entering a profession requiring copious practical application of mathematics or pursuing mathematics as anything more than a hobby. I'm simply a 26 year old college dropout philosophy junkie that desires understanding for the sake of understanding. Now, I have read through most of this thread and compiled a tentative reading list to start out with and would like some input on my selections. I'm basically looking for an idea of what may be within my reach and what may be too advanced and obviously any suggestions if you feel I'm missing something... well honestly any advice would be great at this point as the world of math is massive and I have no idea where to start. I believe most of my selections are more about the philosophy of math rather than actual instructional texts so any input on instructional texts (preferably texts that focus on explaining the theory and underlying thought process behind the techniques rather than just saying "here is how you solve this or that type of problem") would be helpful. My tentative reading list is as follows.

A Mathematician’s Apology by G.H. Hardy
Mathematics: A Very Short Introduction by Timothy Gowers
Makers of Mathematics by Hollingdale
Fermat’s Last Theorem by Simon Singh
Does God Play Dice by I. Stewart
From Here to Infinity by Ian Stewart
Solving Mathematical Problems by Terence Tao
What is Mathematics? by R. Courant and H. Robbins
Beyond Numeracy by J.A. Paolos
Chaos by J. Gleik

EDIT: And after doing some research, I find that my Algebra is incredibly rusty after eight years of non-use.

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Re: Math Books

Postby doogly » Mon Apr 18, 2011 2:04 am UTC

You are going to have to soldier on through some frustrating math then. If your algebra is rusty and you want to deal with 'the theory behind' it, it won't work. Algebra, trig, these are things that require drilling. Pieces of paper with lots of repetitive problems, and you solve them all. I'm not sure there is any way around this.
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Re: Math Books

Postby Metaphysician » Mon Apr 18, 2011 7:06 am UTC

doogly wrote:You are going to have to soldier on through some frustrating math then. If your algebra is rusty and you want to deal with 'the theory behind' it, it won't work. Algebra, trig, these are things that require drilling. Pieces of paper with lots of repetitive problems, and you solve them all. I'm not sure there is any way around this.


Ah, well, in that case, any suggestions as to what resources would be best for brushing up then?
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Re: Math Books

Postby Mammon » Wed Apr 20, 2011 6:47 pm UTC

I got a copy of Introduction to LOGIC, 2nd Edition, by Irving M. Copi from a used book sale from my library. Is this book any good?

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Re: Math Books

Postby doogly » Wed Apr 20, 2011 7:46 pm UTC

Metaphysician wrote:
doogly wrote:You are going to have to soldier on through some frustrating math then. If your algebra is rusty and you want to deal with 'the theory behind' it, it won't work. Algebra, trig, these are things that require drilling. Pieces of paper with lots of repetitive problems, and you solve them all. I'm not sure there is any way around this.


Ah, well, in that case, any suggestions as to what resources would be best for brushing up then?

It's not stuff I've looked at in a while, but basically anything with a lot of problems should do the trick. You just need to be very solid on algebra and trig, and some essential function stuff. Plotting things. I don't have any specific books I know of for these things.

If you enjoy geometry, there are books like "Elementary Geometry from an Advanced Standpoint," by Moise. That might be a good one for you. It does cover everything you need, but it is a little more sophisticated about why everything is happening.

Felix Klein, a 19th century super star, wrote "Elementary Mathematics from an Advanced Standpoint," with an algebra/analysis volume and a geometry volume. These also have the benefits of being cheap.
http://www.amazon.com/Elementary-Mathem ... pd_sim_b_1

And Gelfand wrote some elementary books. One of the greatest beasts of the 20th century will go over trig for you. This is crazy.
http://www.amazon.com/Trigonometry-I-M- ... pd_sim_b_6
http://www.amazon.com/Functions-Graphs- ... pd_sim_b_5
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Re: Math Books

Postby Metaphysician » Wed Apr 20, 2011 11:09 pm UTC

doogly wrote:
Metaphysician wrote:
doogly wrote:You are going to have to soldier on through some frustrating math then. If your algebra is rusty and you want to deal with 'the theory behind' it, it won't work. Algebra, trig, these are things that require drilling. Pieces of paper with lots of repetitive problems, and you solve them all. I'm not sure there is any way around this.


Ah, well, in that case, any suggestions as to what resources would be best for brushing up then?

It's not stuff I've looked at in a while, but basically anything with a lot of problems should do the trick. You just need to be very solid on algebra and trig, and some essential function stuff. Plotting things. I don't have any specific books I know of for these things.

If you enjoy geometry, there are books like "Elementary Geometry from an Advanced Standpoint," by Moise. That might be a good one for you. It does cover everything you need, but it is a little more sophisticated about why everything is happening.

Felix Klein, a 19th century super star, wrote "Elementary Mathematics from an Advanced Standpoint," with an algebra/analysis volume and a geometry volume. These also have the benefits of being cheap.
http://www.amazon.com/Elementary-Mathematics-Advanced-Standpoint-Arithmetic/dp/048643480X/ref=pd_sim_b_1

And Gelfand wrote some elementary books. One of the greatest beasts of the 20th century will go over trig for you. This is crazy.
http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144/ref=pd_sim_b_6
http://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/ref=pd_sim_b_5


These look pretty much perfect for me, I'll check them out. Hopefully I won't be embarrassingly ignorant on this subject for much longer. Thanks for helping a n00b out :)
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Re: Math Books

Postby Malarowski » Wed Apr 27, 2011 4:09 pm UTC

Hello,

I am looking for a Statistics and Probability book. Now, the problem is, I don't know how 'advanced' the course is I am taking. We covered basic data collection and stats in 2-3 lectures and went on to probability (permutations etc) through normal distributions, z/t scores and test, and ending now with Null Hypotheses and tests associated with them with paired tests being last. If anybody could recommend a good book for review (for my final), I would appreciate it. I have a decent understanding of most topics but would like to have another explanation for them to see if I missed anything. I suspect that a intro book should be enough, but I don't know for sure if those don't stay with Statistics too long and don't cover Probability enough.

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Re: Math Books

Postby doogly » Sat Apr 30, 2011 12:54 am UTC

Metaphysician wrote:These look pretty much perfect for me, I'll check them out. Hopefully I won't be embarrassingly ignorant on this subject for much longer. Thanks for helping a n00b out :)

They looked good to me, but I'm not intimately familiar with them, so if they work well for you let the thread know!
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tameree
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Re: Math Books

Postby tameree » Sun May 08, 2011 1:39 am UTC

Hi,

I am looking to develop my mathematical mind as much as possible. The "highest" classes I have taken are Calculus I & II as well as linear algebra. I'd be looking for either a bunch of books or some sort of guide (similar to the guide "How to become a good theoretical physicist" for physics) that ranges from algebra up to topology, complex analysis and partial differential equations.

Basically, I want to master as much of the mathematical field as I can. I'd also be interested in a book (if any exists) that teaches maths at the same time as physics, as I think that learning one reinforces the other. (Is that what's called "Mathematical physics"?)

Thank you.

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doogly
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Re: Math Books

Postby doogly » Tue May 10, 2011 2:11 pm UTC

Mathematical physics is a whole branch of research. It's what I do! It is physics where you prove things. Formal results about quantum field theory, statistical mechanics, dynamical systems, general relativity, etc. These are the people whose eyes light up at "exact solutions." That sort of thing.

You've got some more calc type things ahead of you. Some vector calc, most likely - Div, Grad, Curl and All That, sort of thing. If you get some Purcell, Griffiths or Marion and Heald for electromagnetism, that should help out a bunch with the vectors in physics. And also in Heald you'll see some Fourier analysis peeking in, which is another great place to go next. The awesomest of partial differential equations, totally. I don't know a good PDE textbook.

Korner's book looks great for Fourier, actually. Might be worth a look.

These sorts of things are better considered "mathematical methods," rather than "mathematical physics." Everyone in any branch of physics would need to know these sorts of things.
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Re: Math Books

Postby Dopefish » Wed May 11, 2011 1:32 am UTC

As far as I can tell, almost all the math I've ever found myself using over the majority of the physics half of my degree has been covered in Boa's "Mathematical methods in the Physical Sciences".

It's also made most of the math courses I was obligated to take redundant, as my course equivalent of mathematical boot camp was taught primarily out of that book. It's not always where I've directly learned from, as most of the courses have their own book (indeed, most of my vector calculus came in the context of griffiths electrodynamics), but it seems to have everything in there and then some, so if you want a whackload of physics practical math in one book, it's well suited. It's generally not been my favourite book, but it gets the job done.

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doogly
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Re: Math Books

Postby doogly » Wed May 11, 2011 1:39 am UTC

Yeah, I am usually unhappy with that sort of cookbook approach. I don't like to learn math and physics at the same time; I like math a la mathematicians, and physics a la physicists. Math as a pure tool just made it boring and confusing. But I am a minority among physics folk; most of them do not like abstractions as much as applications, and don't want to talk about math unless it "means something."
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