Other Should I Become a Mathematician?

[mathwonk]

congratulations!

Sheldon Katz is a good friend of mine there, and I have met William Haboush, also very nice.

 

[Gib Z]

Is it always best to read one subject at a time?

Not always. Sometimes two different subjects can aid each others learning. For example, many people recommend taking multivariable calculus and linear algebra together.

Its amazing! I got an acceptance letter today from U of ill in urbana. I didn't see that coming; this was the highest ranked school I applied to.

Congrats =]

 

[Helical]

mathwonk, my main interest is physics and that is what I've planned on majoring in. But I've found that I'm also quite interested in math and am thinking about doing a double major. Is this feasible in four years as well as having a life other than my studies? I think I'm more intelligent than average but I'm not a genius so maybe staying for 5 years is a better option. I'll be waiting to make this decision until my first year to make sure I enjoy calculus. Also I am a senior in high school and am currently only taking pre-calculus.

 

[mathwonk]

helical, i don't really know the answer to this, but i do think it is worth trying.

you can give it your best informed shot, discussing it with college advisors at your chosen school, then see how it goes.

the only way to find out is to try, intelligently, i.e. by first getting advice and planning as you are doing.

if you don't try, you'll never know.

 

[Darkiekurdo]

How can I get better at solving Olympiad-type problems?

 

[mathwonk]

practice. i presume there exist books of problems.i have never done this in college, but in high school we practiced for contests by working lots of them and did very well.

 

[G01]

How can I get better at solving Olympiad-type problems?

There is nothing better than practice. So as many problems as you can. As you keep doing them you'll develop personal patterns and algorithms to solve problems. You won't be tied down to formula sheets as much, etc. Practice, practice, practice!

 

[jhicks]

How can I get better at solving Olympiad-type problems?

My library of math books is pretty limited due to a few years of uh... languishing, but one book I enjoyed to the fullest is called "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics". All of the problems are very accessible to someone who has done up through precalculus, but they range from easy to extremely difficult and cover just about any type of elementary problem you could think of, from divisibility to word problems to limits. I worked on the 320 problems for maybe a year total out of the 5 years I've had the book, and I have solved maybe 100 of them.

Edit: It's 12 bucks on Amazon. https://www.amazon.com/dp/0486277097/?tag=pfamazon01-20

 

[PhysicalAnomaly]

You've provided plenty of good book titles. However, you haven't said what has what prerequisite. Could someone please list good popular books for undergrads (like Spivak, Apostol and Kreyzig) in sequential order? I would like to cover linear and abstract algebra, topology and real analysis on my own (the uni course is a bit slow; my foundation is uk A levels and the australian system is a bit sluggish) but don't know which order of books to use. Also I am taking a double degree with chem eng as one side and would like to know some maths that'll be useful for engineering (I'm guessing something like fourier, greens, pde's). Any good books for this purpose? Will Kreyzig's book cover enough for me to skip things like Spivak and Apostol? What do you think of Strang's linear algebra?

I'm thinking it would be quite safe to follow Cambridge's maths syllabus (it's on their maths department's site). Would it be overkill? What they cover in 2 years is probably what mine covers in 3!

 

[eastside00_99]

Could someone please list good popular books for undergrads (like Spivak, Apostol and Kreyzig) in sequential order? I would like to cover linear and abstract algebra, topology and real analysis on my own

You could probably learn any of those topics right now. I don't see one as a prereq for another depending on what you mean by topology. If you mean point-set topology, real analysis will give you intuition but is not really necessarily if you have a propensity for abstract thinking. If you mean algebraic topology then you will need abstract algebra, point-set topology, and possibly linear algebra for intuition in homology theory. I don't know of any "standard" textbooks for these things save Munkres "topology" for point-set topology. Of course, you have Lang's "algebra" and other similar books for all of those subjects but they are not necessarily the best books to read to first learn the subjects but rather after a second go around at the subjects. The most economical list that I can come up with is the following:

linear & abstract algebra ------ Artin's "Algebra"
point-set topology ------ Munkres' "topology"
Real Analysis ------ Marsden's "Elementary Classical Analysis"

That list is pretty damn difficult enough. In the end, you will have to decide what books are most accessible to you and which challenges you enough.

 

[PhysicalAnomaly]

Thanks. I seem to be in luck. My university's library has Artin and Munkres.

More questions:

I've gathered from trawling the forum that Spivak would be easier than Apostol. That true?
Kreyzig or Stroud? (And would I need strang after that?)
If Stroud, what's the difference between engineering maths, further engineering maths and advanced engineering maths?
I'm partway through strang's linear algebra. Which book do you recommend after this for linear algebra?

What is point-set topology and would it be required to something else?

Is Marsden the standard complete text for real analysis? Will it be redundant if I read Spivak's calculus? Does Spivak's book on manifolds follow from his Calculus or is there overlap?

Thanks for your guidance.

PS If I study the books that go in depth for maths, does that mean I can do without engineering maths books?

 

[Gib Z]

PS If I study the books that go in depth for maths, does that mean I can do without engineering maths books?

Not always true. Just because you know the theory does not necessarily indicate you can also apply it effectively. Chances are your math book won't go into much detail about applications to engineering, so for that course you may need a good separate book that is filled with those types of problems.

 

[quasar987]

Is Marsden the standard complete text for real analysis? Will it be redundant if I read Spivak's calculus? Does Spivak's book on manifolds follow from his Calculus or is there overlap?

I wouldn't call Marsden standard, but it is in my opinion, the best analysis book for beginners. However, it is deficients in some ways (rudimentary treatment of power series) and overdoes it in others (it develops the Riemann theory of multiple integrals! this is useless)

Spivak's calculus is not a real analysis text. It's a calculus text, whatever that means. In either case, 'Calculus on Manifold' does not follow and even if it did, it is a bad introduction to calculus in higher dimensions because it is extremely dense with no examples.

 

[eastside00_99]

I've gathered from trawling the forum that Spivak would be easier than Apostol. That true?
I thought it was the other way around; I have read neither.

Kreyzig or Stroud? (And would I need strang after that?)
Kreyzing for what? "introductory functional analysis"?

I'm partway through strang's linear algebra. Which book do you recommend after this for linear algebra?

Any linear algebra book that discusses canonical forms.

What is point-set topology and would it be required to something else?

Point-set topology is a generalization of the concepts of space of R^n that are not associated with distance. It would be required for a myriad of things: algebraic topology, manifold theory, several complex variables, algebraic geometry, etc, etc



Anyway, take it slow. I mean unless you are just some freak of nature or study 18 hours a day, you are not going to be able to master these four subjects in a month. It wouldn't really be possible within a semester while you are also taking other classes. Next semester I would just recommend you taking a more advanced math course and working hard in it.

 

[jhicks]

Kreyzig or Stroud? (And would I need strang after that?

My personal opinion of Kreyzig was that it was an information overload (like a lot of engineering classes) with minimal theory discussion. To me it's a reference book only, but it does cover absolutely everything you're likely to see in engineering.

 

[PhysicalAnomaly]

I'm not that crazy. I intend to take a year or two to finish. But I have limited time so I have to make sure that I'm using the best books since I won't have time to go back and use an alternative. Australian universities don't usually have very advanced maths classes. Not until 3rd year anyway. Cambridge students cover almost our entire 3 year syllabus in 2 years and in more depth too!

 

[mathwonk]

you should read both spivak and apostol to see which you like better. there are no applications and no physics in spivak at all, and this is a limitation for many people.

one nice thing about apostol, is that although differentiation makes evaluating integrals easier, integral calculus is much older than differential calculus and does not depend on it at all, up to a certain point. it seems it was essentially invented by eudoxus and especially archimedes.

this is obscured by most books which do differential calc first. apostol is unique in thoroughly explaining integral calc first, which is historically sound.

 

[Hydrargyrum]

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.

Where can I find writing from Gauss, Newton, Euler, etc?

 

[quasar987]

Here's a book by Euler (translated of course): http://www.amazon.com/dp/0387985344/?tag=pfamazon01-20

Newton's PRINCIPIA MATHEMATICA is also easily found in english, but not easily read.

For people of the era ofGauss, Riemann, Weierstrass, Cauchy, etc. and onward, if most universities are like mine, the library should have books going by names like "Collected works of [...]"

 

[Hydrargyrum]

i was thinking about books that i could download . . . for free

 

[pivoxa15]

Mathwonk, what do you think about the idea of starting a Phd in a field in which a student have no idea in?

 

[paniurelis]

Let me introduce myself: I am 29, studying as a part timer computer science in eastern europe, Lithuania. Math is a hobby for me. I have been solving elementary math contest type problems for a year before enrolling into bachelour program in local university, basically for review of elementary math and for fun :-).

Here is my problem: I can't keep pace with a math lectures(Calculus I, II). There is lots of material, and the problems solved in the classes are usually very simple. The course is not proof based, and prof. does not demand a proofs of theorems during the exams, only the simple problems, some definitions and theorems are required. Don't get me wrong, all my grades are 10(A+), but I am not satisfied with the level of skill and knowledge gained during the course. I don't have any troubles with proofs, and I have TONS of good math books(mainly russian) with creative problems & solutions. So, at the beginning of first semester I decided to study proofs of main theorems and solve as many hard problems, especially NOT calculation based ones, in addition to the coursework. BUT, there are problems:
a) After some time I forget the proof of theorems learned earlier, also it takes time to learn the theory, and after a while, I need to review it more than once. I have a day job, usually study early in the morning, or at weekends. Reviewing and learning new material takes A LOT OF time, so I am constantly falling behind the course.
b) Creative problems require time, which I don't have, for example, while I am solving monotonic sequence limits problems for couple weeks, class is done with sequence limits, and is finishing function limits.

It is really depressing experience: I am constantly not satisfied with my progress. Has anyone of you had such problems ? Some suggestions ? Any opinions will be appreciated.
Maybe I worry to much ? Maybe my attitude is wrong: I don't think I know material enough if I see a problem which I am not able to solve ?
Thanks for advise in advance :-)

 

[mathwonk]

books by gauss, riemann, euler, etc, seem worth buying if anything is. do you spend money for cigarettes, or beer?

starting a phd is not sensible in a field you know nothing about, no. why would anyone think of this?

paniurelis, you seem to be struggling to find your niche in the world, a laudable and hard experience. i think you are to be congratulated.

 

[Ronnin]

I just started reading into Apostol's Calculus and I have never seen a book quite like it. I have taken 3 semesters of calculus and after starting this book I realized I never had a deep knowledge of the subject at all. I wish I would have been exposed to this book years ago when I first started. I like it a lot.

 

[rudinreader]

paniurelis I could go on and on about what you have mentioned.. but ...
The Cal 1-3 curriculum is completely determined by economics, to the point that:
math departments refer to engineering/other students with calc+ prerequisites as clients of the math department..

Sounds like you are in an economically difficult situation - not necessarily in terms of money -- but in terms of wanting to satisfy your math curiosity. But with the structure of intro calculus, not really knowing how to proceed, you are smart to throw a post out on PF.

If you create a post "how to prove calculus on your own" in Calculus/Analysis or General Math, I will try to help.. But only to sway you in the right direction.. hopefully others will help too. Proving it on your own means you can skip some things that may be inessential in terms of curiosity. A worthwhile project in this direction would differ drastically from the Calculus 1-3 assembly line factory course. If you like it, and if you think you are good at it, it could even be helpful if Computer Science is your thing.

 

[rudinreader]

paniurelis, I went ahead and created a thread for you here:

https://www.physicsforums.com/showthread.php?t=214916

just to guide in the right direction hopefully.. I hope it is of help..

 

[Hydrargyrum]

books by gauss, riemann, euler, etc, seem worth buying if anything is. do you spend money for cigarettes, or beer?

I guess you're right, but I don't even have money to buy food right now

 

[pivoxa15]

starting a phd is not sensible in a field you know nothing about, no. why would anyone think of this?

It's just that a student may not have had the opportunity to learn a field but realizes that it may be of interest. Maybe he/she could read up on it themselves prior to enrolment?

 

[Ender[x86]]

Does anyone know where I could find mathematical texts from Gauss, Euler, etc. that are freely available online and in English?

I know this question has been asked a lot, but I'm asking about places to look online in particular.

 

[Zetetic]

Hey, I am a freshman physics mathematics dual major and I have an interesting sort of predicament. I understand on a deep level the material covered in my classes and in fact nearly taught myself enough caclulus in 2 months to test out after nearly two years away from math ending at algebra 2, however, I put too much emphasis on really understanding the delta epsilon proofs for each rule of differentiation ect. and not enough directed towards the more topical approach and my knowledge of certain techniques (derivative of natural logs and inverse trig functions and population growth problems) was a bit deficient.
Anyway, I have continued my self study utilizing the first Apostol text and Gilbert Strang's book on linear algebra. I can handel the material just fine and am quite good at finding patterns and setting equations to them (though I can't allways do the proof), however, dispite my understanding of the subject, I am very prone to doing a medeocre job on tests as I am terrible at keeping track of details and eceedingly scatterbrianed when I need to put together the simplest set of techniques to solve a rudementary problem.