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The Should I Become a Mathematician? Thread

by mathwonk
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mathwonk
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Jun5-06, 12:05 AM
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I am interested in starting this discussion in imitation of Zappers fine forum on becoming a physicist, although i have no such clean cut advice to offer on becoming a mathematician. All I can say is I am one.

My path here was that I love the topic, and never found another as compelling or fascinating. There are basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis.

There are several excellent books available in these areas: Courant, Apostol, Spivak, Kitchen, Rudin, and Dieudonne' for calculus/analysis; Shifrin, Hoffman/Kunze, Artin, Dummit/Foote, Jacobson, Zariski/Samuel for algebra/commutative algebra/linear algebra; and perhaps Kelley, Munkres, Wallace, Vick, Milnor, Bott/Tu, Guillemin/Pollack, Spanier on topology; Lang, Ahlfors, Hille, Cartan, Conway for complex analysis; and Joe Harris, Shafarevich, and Hirzebruch, for [algebraic] geometry and complex manifolds.

Also anything by V.I. Arnol'd.

But just reading these books will not make you a mathematician, [and I have not read them all].

The key thing to me is to want to understand and to do mathematics. When you have this goal, you should try to begin to solve as many problems as possible in all your books and courses, but also to find and make up new problems yourself. Then try to understand how proofs are made, what ideas are used over and over, and try to see how these ideas can be used further to solve new problems that you find yourself.

Math is about problems, problem finding and problem solving. Theory making is motivated by the desire to solve problems, and the two go hand in hand.

The best training is to read the greatest mathematicians you can read. Gauss is not hard to read, so far as I have gotten, and Euclid too is enlightening. Serre is very clear, Milnor too, and Bott is enjoyable. learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

If these and other top mathematicians are unreadable now, then go about reading standard books until you have learned enough to go back and try again to see what the originators were saying. At that point their insights will clarify what you have learned and simplify it to an amazing degree.


Your reactions? more later. By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.

Remark: Arnol'd, who is a MUCH better mathematcian than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's.

I intend to revise my linear algebra notes now to point this out.
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courtrigrad
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Jun5-06, 01:44 PM
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I probably want to become a mathematician. I am not sure whether to go into pure or applied math. I will probably opt for the latter, as I like being able to develop ideas useful for the world. Mathwonk, I am currently reading and doing problems from Apostol's vol. 1 Calculus. I realized in the past years, that I was very obsessive compulsive about doing every single problem. If I got stuck on one problem, I had to finish it. But now I just take the problems that really pertain to the material (i.e. not plug and chug problems), and if I get stuck, I just move along and post the problem here.

If I want to become an applied mathematician, is studying the book by Apostol ok? I want to really understand the subject (not some AP Calculus course where I just "memorized" formulas). Last year, I tried reading Courant's Differential and Integral Calculus, but it seemed too disjointed. I like Apostol's rigid, sequential approach to calculus.

Also, if I want to become an applied mathematician, should I, for example, major in math/economics? Here is my tentative plan of future study:

Apostol Vol. 1: Calculus
Apostol Vol. 2: Calculus (contains linear algebra)
Calculus, Shlomo Sternberg
Real Analysis
Complex Analysis
ODE's

What would you recommend an applied mathematician take? Also, would you recommend me to go back and reconsider the old Courant, as I remember you saying that his book contains more applications? Or am I fine with Apostol?

Thanks a lot
matt grime
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Jun5-06, 03:49 PM
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Quote Quote by courtrigrad
Also, if I want to become an applied mathematician, should I, for example, major in math/economics?

Hell no. Maths and economics majors know jack about maths either pure or applied. Economists struggle to add up, never mind do maths properly (including applied maths).

If you're going to be a good applied mathematician then you'll be able to do Apostol and the purer stuff: you might not see the utility of it a great deal at times, but you will be able to do it, and it might well come in useful later.

mathwonk
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Jun5-06, 05:41 PM
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The Should I Become a Mathematician? Thread

I am glad to see Matt is chipping in. Courtigrad, I think Apostol is outstanding and probably more than sufficient for training in any future direction, but at intervals I suggest going back and reconsidering Courant. I also did not like it as a student, but appreciate it more now.

One thing a friend/student of mine said about his career in applied mathematics may be useful: he said it was primarily the difficulty of the pure mathematics he studied that readied him for applied mathematics work, not the specific knowledge. Having had to learn algebraic topology taught him how to learn something hard, and he had a big advantage over others in his field when he needed something new. He knew how to learn.
courtrigrad
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Jun5-06, 06:11 PM
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so I shouldnt major in math/economics if I want to become an applied mathematician? Just apply in pure math?
mathwonk
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Jun5-06, 07:43 PM
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It fits with my friend's experience, but you could ask some applied mathematicians. I agree with Matt though, a math degree is much more appropriate than an economics degree.
mathwonk
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Jun5-06, 07:44 PM
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Becoming a mathematician.

Being a mathematician means doing mathematics, but the activity is not the same as the job. Being a professional mathematician means being a professor, doing research and teaching and writing, or working in an industry using math tools to do things like design cars, or solve turbulence problems for aircraft, or to estimate the actual pollution in streams from samples. I only know about the professor side of it since I have been teaching and working in a university setting most of my life, but the behavior of learning and practicing mathematics is probably not too different for all intended lines of work. Ironically, a professor often has so many duties associated with teaching, grading, evaluating people, recruiting, etc,.. that he/she has to scrounge time to actually do math.

Getting started: Junior high.

Unless your parents will agree to send you to a special math school, and you live somewhere like North Carolina where these exist, or can get into and afford a prep school, you have little control over the training you get at this level. Your teacher may not know much math at all or even like it. But the good side is that school is usually pretty easy at this level so you should have free time to spend reading and learning on your own. A wonderful book to try is What is Mathematics? by Courant and Robbins. If you want a leisurely book for the public about a mathematical triumph that anyone can read, try the book by Simon Singh on Fermat’s last theorem as solved by Andrew Wiles.

math team, SAT’s
If possible join the math team at your school, and participate in math contests, practicing solving problems. This is not only fun, but good experience at test taking, a useful survival skill in the world of SAT’s. Being good at tests is not quite the same as being good at math, but it is the way talent is often measured and rewarded in young people, so it can help you earn scholarship money and admission to top schools. When I won the state math contest in high school in Tennessee I started getting job offers, and a high SAT score earned me a merit scholarship and fairly easy admission to Harvard. (Those were the old days.) Be aware though that the more important SAT test is the verbal one. It is harder, a better measure of your reasoning skill, and a better aid at distinguishing yourself. I am out of date here too, as I have heard they recently dropped the analogies portion, which of course was the most valuable part for detecting reasoning ability.

Caveat: This is a political world, and popular high stakes tests are not entirely intended to measure ability, but also to reward some political constituency, as witness the dishonest recent “recentering” of SAT scores, i.e. SAT grade inflation, making it harder for really good students to stand out. The ridiculous “no child left behind” rules have actually made it harder for some of my best students to get teaching jobs, because the criteria are so stupid that they work against really gifted people. For example some school districts require candidates to have taken certain mickey mouse math courses at the college level, whereas only a very weak student would not have taken them already in high schol or even junior high. But it is useless to protest, just learn to survive. Find out what the rules are for the goal you seek and make sure you qualify even if it seems silly and a waste of time. The books by the late Paul Torrance on gifted education, especially for the creatively gifted, are very helpful. They explain how to find resources for bright kids, and how to help them get credit for what they do.

Giftedprograms, TIP:
One thing that was a great experience for our young kids, is the TIP summer program at Duke University. They admitted bright junior high kids based on SAT scores taken when 12 years old, and offered a wonderful environment of talented students, excellent teachers, and fascinating introductions to topics like physics (delightful book: "Thinking physics" by David Epstein?) and number theory, that 12 year olds do not usually see in school. I presume it is still a good program. Of course before taking the SAT test, you should get hold of some practice SAT books and take a bunch of them to learn how they go. If you have an experienced professor or teacher around, or any parent who is test savvy, they can help you learn the difference between a true response and a correct response. This is hard to quantify, but my bright children often argued correctly that a certain multiple choice was actually true, but I knew from experience with tests that a different one was wanted by the tester.
Many states also have “governor’s honors” programs, and other special opportunities with various qualifications. Try to find out about them and get in on them. But do not despair if this is out of reach. Such things did not exist in my day and I never had any such special opportunities or training, just a mediocre classical high school math education, no calculus or anything advanced, just the added practice of the math team. I did have an excellent basic algebra course and Euclidean geometry, but that’s it. I knew the “root - factor theorem” for polynomials (r is a root if and only if (x-r) is a factor), and I had lots of practice trying to prove geometry facts. I didn’t even know trig.

more later.
mathwonk
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Jun5-06, 11:07 PM
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2) Basic Preparation, High school:

In high school, it is usual nowadays to take AP calculus. More important for mathematical background, is to get a good course in polynomial algebra and Euclidean geometry, with thorough treatment of proofs. A course in logic would help as well if it is available. Again, one must make do with what is available, but be aware that courses like AP calculus are more designed to please parents and impress admissions officials than to train mathematicians. Most of the people making decisions about what to offer are completely ignorant of the needs of future scientists, and are only concerned with entrance to prestigious schools. Again one must play the game successfully, so even though these people have no idea what you need to become mathematician, they still are able to make decisions on who gets into top schools, so it is prudent to impress them, while also trying to actually learn something on the side.

So what I am saying is this: in order to succeed in college calculus, one absolutely MUST have a solid grasp of high school algebra and geometry, although most high schools shortcut these subjects to offer the more prestigious but less useful AP calculus. Thus it is wise to work through an old fashioned high school algebra book like Welchons and Krickenberger (my old book), or an even older one you may run across. A wonderful geometry book is the newer one by Millman and Parker, Geometry: a metric approach with models, designed for high school teacher candidates in college. If you can find them, the SMSG books from Yale University Press, published in the 1960’s are ideal high school preparation for mathematicians. These were produced by the movement to reform high school math in the early 1960’s but the movement foundered on the propensity to put profit before all else, the lack of trained teachers, and the unwillingness to pay for training them.

e.g. here is a copy of a precalculus book from that era:

MATHEMATICS FOR HIGH SCHOOL ELEMENTARY FUNCTIONS TEACHER'S COMMENTARY

Bookseller: Lexington Books Inc
(Garfield, WA, U.S.A.) Price: US$ 35.00
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Book Description: Yale University Press., 1961. Good+ with no dust jacket; Contents are tight and clean; Ex-Library. Binding is Softcover. Bookseller Inventory # 41816

and an algebra book:

Mathematics for High School First Course in Algebra Part I Student's Text
Bookseller: Bank of Books
(Ventura, CA, U.S.A.) Price: US$ 14.25
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Book Description: Yale University Press. Soft Cover. Book Condition: ACCEPTABLE. Dust Jacket Condition: ACCEPTABLE. USED " :-:Fair:-:Writing on first page, covers bent and creased, a little water damage, corners bumped, covers dirty, page edges dirty, spine torn.:-:" Is less than good. Bookseller Inventory # 19620

another algebra book:

CONCEPTS OF ALGEBRA
Clarkson, Donald R. Et. Bookseller: Becker's Books
(Houston, TX, U.S.A.) Price: US$ 15.00
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Book Description: Yale University, 1961. Book Condition: GOOD+. wraps School Mathematics Study Group Studies in Mathematics Volum V111. Bookseller Inventory # W040215

and one on linear algebra:

Introduction to Matrix Algebra. Student's Text. Unit 23.
School Mathematics Study Group Bookseller: Get Used Books
(Hyde Park, MA, U.S.A.) Price: US$ 25.00
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Book Description: Yale University Press. Paperback. Book Condition: VERY GOOD. USED 4to, yellow wraps. Slightly skewed; wraps sunned and a little worn at spine; text fine. Bookseller Inventory # 44146

Here are some books I use currently in teaching math ed majors, which would be bettter used in high school:

An Introduction to mathematical thinking, by William J. Gilbert and Scott A. Vanstone. paperback, ISBN 0-13-184868-2, Pearson and Prentice Hall.

also: (better) Courant and Robbins, What is Mathematics?

After mastering basic algebra and geometry, there is no harm in beginning to study calculus or (better) linear algebra, and probability. A good beginning calculus book is Calculus made easy, by Silvanus P. Thompson, (ISBN: 0312114109)
Bookseller: Great Buy Books
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Book Description: St. Martin's Press, 1970. Paperback. Book Condition: GOOD. USED Ships Within 24 Hours - Satisfaction Guaranteed!. Bookseller Inventory # 2397224 .

I love his motto: “what one fool can do, another can”

Do not laugh, this is a good book. And therefore his book on electricity and magnetism is probably also good (he was a fellow of the Royal Society of Engineers).

Elementary Lessons in Electricity & Magnetism. New Edition, Revised Throughout with Additions
Thompson, Silvanus P. Bookseller: Science Book Service
(St. Paul, MN, U.S.A.) Price: US$ 4.94
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Book Description: MacMillan Company, New York, NY, 1897. Hard Cover. GOOD PLUS/NO DUST JACKET. Red cloth covers are clean and bright with some wear at the tips and the head and foot of the spine; gilt lettering on spine is bright and easy to read; institutional lib book plate on inside front cover and lib stamp on copyright page; owner's signature inked on front flyleaf; binding cracked between front and rear endpapers and has been reinforced with clear tape; inside pages clean, bright and tight throuhgout. Overall, still a very useful, solid and clean working or reading copy. Bookseller Inventory # 008802.

Learn right now: the price of a book is unrelated to the value of the book as a learning tool, only to the scarcity of the book, and its popularity. [Notice how cheap these wonderful books are compared to the **&^%%$$!!! books that sell for $125. and up, that are required for college courses.]

Finally, if you are a very precocious high school student, and have learnt algebra and geometry, you may profitably study calculus. In fact, to play the game of college admissions, you may need to take AP calculus, even if the teacher is an idiot, just so the admissions officials will believe you have “challenged yourself”. There are many good calculus books, beyond the humorous (but valuable) Silvanus P. Thompson, although that may already suffice for an AP course. The delightful math book I had as a high school senior, was a combination of logic, algebra, set theory, analytic geometry, calculus, and probability, called Principles of Mathematics, by Carl Allendoerfer and Cletus Oakley. This was a wonderful book, and opened my eyes to what was possible after a long period of boring mathematics courses at the dull high school level.

here is a copy:

PRINCIPLES OF MATHEMATICS - SECOND EDITION
Allendoerfer, Carl B. & Cletus O. Oakley Bookseller: Adams & Adams - Booksellers
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Book Description: McGraw-Hill, N.Y., 1963. Hard Cover. Book Condition: Very Good. No Jacket. 8vo - over 73⁄4" - 93⁄4" tall. xii + 540pp. name on front endpaper. Bookseller Inventory # 014846.

I still have a copy of this book on my shelf.

A lovely calculus book, for beginners, with delightful motivation, is Lectures on Freshman Calculus, by Cruse and Granberg. They motivate integration by the “Buffon’s problem” of computing the likelihood of a needle dropped at random, falling across a crack in the floor. (I reviewed it in 1970, and criticized the flawed discussion of Descartes’ solution of the problem of tangents, but I wish now I hadn’t, as it might have survived longer.)

here is a copy:

Lectures on Freshman Calculus
Cruse, Allan B. & Granberg, Millianne Bookseller: Hammonds Antiques & Books
(St. Louis, MO, U.S.A.) Price: US$ 18.00
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Book Description: Addison Wesley 1971, 1971. Hardcover good condition with minor soiling, no dustjacket xlibrary with usual markings ISBN:none. Bookseller Inventory # LIB2958010770.

As before, participate in the math team, and practice your vocabulary, to pass high on the verbal SAT. And read lots of books. Mathematicians have to also describe what they do to literate folk, and of course also need to “woo women” (or your choice), as observed in dead poets society.

more later.
mathwonk
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Jun5-06, 11:14 PM
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Matt, feel free to jump in here and describe your possibly more helpful or normal path to becoming a math guy, or give any advice you want, or counter any goofy advice I have given. I kind of hung in there in spite of everything going hooey for a time in the 60's, and may not have as much to offer the average person. My motto was sort of "never give up" no matter what, and may not synch perfectly with the readers of this forum. Best, roy.
EbolaPox
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Jun5-06, 11:15 PM
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Similar to Courtrigrad, I'm reading Apostol Volume I. Currently, it's extremely entertaining. I've been working on proofs, and I think I'm getting okay at writing them (for my Linear Algebra book I'm reading.) I've been trying to get a list of books to begin studying. So far, my plan is

Study:

Calculus

Apostol Volume I
Apostol Volume II (Unfortunately, very expensive...)

Differential Equations

Zill- This is the book I'm using now in my DE class. Unfortunately, my class doesn't focus on very much theory at all....Bunch of applications...boring.

Linear Algebra

Kolman (1970's, old, but easy to read and I like the Theorem-Proof-Example layout. I like to read the theorem, and try to prove it before I read there proof. So far, that's been going well.)
Friedberg (2nd Edition).

Analysis
Shilov- Elementary Real and Complex Analysis
Rudin- Principles of Mathematical Analysis (Also expensive...)

Modern/Abstract Algebra
No clue. Any suggestions?

Since I just graduated High School, I've been trying to spend my time productively studying Linear Algebra and Apostol. I really love Linear Algebra in the chapters on Vector spaces, subspaces, Linear Transformations, Isomorphisms, etc. However, manipulation of matrices (solving systems using boring matrix algebra) is a tedious process that doesn't interest me as much. I thoroughly enjoy proving the theorems the book provides.

I hope to be a mathematician and teach as a professor. Any recommendations for textbooks? Also, after thoroughly studying Linear Algebra, would it be wise for me to begin reading a text on Abstract Algebra? Or is there more mathematical preparation required?

Thank you, and thanks for making a topic about becoming a mathematician!
Hurkyl
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Jun5-06, 11:21 PM
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However, manipulation of matrices (solving systems using boring matrix algebra) is a tedious process that doesn't interest me too much. I thoroughly enjoy proving the theorems the book provides.
This is actually a very interesting problem... provided you don't actually have to execute the tedious details by hand.

On the one hand, there is all sorts of challenging work in trying to figure out how to compute these things efficiently, if you like that sort of thing.

On the other hand, manipulation of matrices is useful for solving all sorts of problems, and it's interesting to figure out how to set up the problem correctly, and the right algorithm to get the information you need!


For example, suppose that you're working in 16-dimensional space. You have two 12-dimensional planes which are denoted with the following data:

A point in the plane.
A basis for the vector space of displacements from that base point.

Or, equivalently, your planes are the images of maps of the form:

T : R^12 --> R^16 : x --> Ax + b

for some rank-12 matrix A and vector b.

Your challenge: figure out how one would compute the intersection of those two planes!

(Warmup problem: suppose that your planes are actually vector spaces: that is, your map is an actual linear transformation)
mathwonk
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Jun6-06, 12:00 AM
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thanks for pitching in, hurkyl! i suspect you are a physicist by training (?), but you are obviously a very strong mathematician by inclination and talent.
mathwonk
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Jun6-06, 12:01 AM
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as to those planning a career in math, here is a relevant joke i got from a site provided by astronuc:

Q: What is the difference between a Ph.D. in mathematics and a large pizza?
A: A large pizza can feed a family of four...
mathwonk
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Jun6-06, 12:08 AM
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here are the ode books i used in my spring 2006 course:

1. title: An Introduction to Ordinary Differential Equations
by Earl A. Coddington
ISBN: 0486659429
Dover Publications

2. title: A Second Course in Elementary Differential Equations
author: Paul Waltman
ISBN: 0486434788
Dover Publications

3) Differential Equations and Their Applications: An Introduction to Applied Mathematics
Martin Braun, Martin Golubitsky?,
Jerrold E. Marsden (Editor), Lawrence Sirovich (Editor), W. Jager (Editor)

4) Ordinary Differential Equations, by V.I. Arnold.
Paperback: 270 pages, Publisher: The MIT Press (July 15, 1978)
ISBN: 0262510189.

The one with the most to offer a beginner is Braun. The one I liked best with the most to offer me, i.e. the most sophisticated (try it if you want to see what I mean) was Arnol'd. The easiest one, that I had in college at harvard, was Coddington.

a great ode book that i did not appreciate until recently was by hurewicz. here is a copy:

Lectures on Ordinary Differential Equations.
Hurewicz, Witold.
Bookseller: Significant Books
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mathwonk
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Jun6-06, 12:17 AM
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here are cheap copies of rudin:

24. Principles of Mathematical Analysis 1ST Edition*(ISBN: 1114135615)
Rudin, Walter
Bookseller: Powell's Books
(Portland, OR, U.S.A.) Price: US$*15.00
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Book Description: MCGRAW HILL PUBLISHING COMPANY. HARDCOVER Mathematics-Real Analysis. USED, Less Than Standard. Bookseller Inventory # 04111413561502



45. PRINCIPLES OF MATHEMATICAL ANALYSIS.
RUDIN, Walter.
Bookseller: Robert Campbell Bookseller
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Book Description: New York: McGraw-Hill, 1964., 1964. Second edition. Hardcover. Very good in very good dust jacket. 270pp. Bookseller Inventory # 26517


i recommend this book as easier to read than rudin:

42. Introduction to Topology and Modern Analysis (International Series in Pure and Applied Mathematics)
Simmons, George F.
Bookseller: Chamblin Bookmine
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Book Description: McGraw-Hill Book Company, Inc., New York, New York, 1963. Hard Cover. Book Condition: Very Good. No Jacket. First Edition. 8vo - over 7¾" - 9¾" tall. Dark blue boards. 372 pages. Previous owner's name on inside front board. Bookseller Inventory # 12290.
cscott
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Jun6-06, 06:17 AM
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Quote Quote by mathwonk
thanks for pitching in, hurkyl! i suspect you are a physicist by training (?), but you are obviously a very strong mathematician by inclination and talent.
Im sure Hurkyl will respond, but I thought I remembered seeing him state in some thread that he was a mathematician pursuing some physics (?)
J77
#17
Jun6-06, 06:23 AM
P: 1,157
My name's J77, and I'm probably a mathematician...

Well, I have a degree in mathematics, a masters in mathematics and a PhD which was heavily maths related.

I'm probably even an applied mathematician!

If I were to narrow it down a bit, I do nonlinear dynamics. However, to quote a recent survey by Philip Holmes
Nonlinear dynamics, more grandly called "nonlinear science" or "chaos theory", is a rapidly-growing but still ill-defined field...
I publish in all many of journals from linear algebra, through physics, into aerospace engineering.

To become a mathematician... That's a hard one.

I was always good at maths tests from an early age - I think some people have a natural ability at maths. I certainly don't think that tests are a good indication of what makes a good mathematician though - and it sometimes saddens me to see so many threads in this forum which are just about obtaining so many points in this or that test. I'd say that a good mathematician should just have the ability to think laterally. Be able to throw a bit of imagination into the mix. Anyone can learn procedures/algorithms for an exam. However, the true test is seeing a new problem and using past experience, or developing new techniques, to solve that problem.

Also, as far as starting out goes, do a maths degree. Only after you start a university course will you see what parts of maths you like and what parts you dislike. Then, of course, be more specific with the masters and finally the PhD.

However, don't set out a course to follow from when you're 18 - just go with the flow

I hope this thread keeps going with good advice and will add more...

For now, advanced books on nonlinear dynamics/bifurcation theory:

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.

Y. Kuznetsov, Elements of Applied Bifurcation Theory.

And to keep with some of the previous posts:

V.I. Arnol'd. Catastrophe Theory.
jbusc
#18
Jun6-06, 07:19 AM
P: 212
Perhaps someone could give some advice...As a followup to what everyone has posted, I have been on the lookout for math resources for what I like to think of as "the theoretical engineer". I consider it to be that gray area between engineering, physics, and applied math that has emerged recently, where linear algebra, combinatorics, and discrete mathematics have found ample use.

I have taken several such courses, but the resources for self study seem to be slim. The courses themselves were primarily tought out of lecture notes supplemented by texts by Johnsonbaugh and Strang, which while excellent, didn't address most of the finer details of the courses.

In particular, I have been searching for resources on topics like measure theory, and differential geometry and manifolds, and I haven't found much that is written toward or even accessable to a non-mathematician, mostly due to the prerequisite vocabulary knowledge assumed by the reader.

For example, I recently found an excellent book on introduction to topology by Mendelson, which was very easily readable and understandable by anyone with some basic set theory background; unfortunately I have been unable to find books on other topics written at a similar level. I would certainly appreciate some pointers, thanks.


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