Other Should I Become a Mathematician?

[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.

 

[Zetetic]

sorry for the double post, I accidentally hit send before I finished.
Anyway, my question is: How much will my problems with details and my scatterbrained tendencies affect sucess in mathematics? Any advice on remedying my problem or assuaging my anxiety would be greatly appreciated.

 

[rudinreader]

... 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.

Did studying epsilon-delta make you forget how to add and subtract numbers? Drill problems alone will not get you to higher math. You probably just need to be patient, apostle is a good choice.

 

[adrian_durham]

No one seems to be interested in my post in the Calc/Anal forum...

I was wondering what would be the physics equivalent of the Courant Calculus text. Also, while we're at it, isn't Courant more rigorous than Apostol? I actually have the first volume of Apostol, and it's not bad at all. But, it still seems preoccupied with computational problems a lot of the time. I haven't really gone through it, myself, so maybe I am being too picky and not thorough enough. I took calculus out of a "normal" book and then only later on took real analysis, etc. out of more advanced books. At any rate, I have ordered some used Courant, wondering how different that may be. I am also interested in something along those lines for physics. You know -- like Halliday, Resnick and Walker is to Thomas and Finney as what is to Courant?

 

[mathwonk]

well there is a text by courant, (and hilbert) called methods of mathematical physics.

 

[adrian_durham]

well there is a text by courant, (and hilbert) called methods of mathematical physics.

Well, that probably isn't the equivalent, though, is it? It is more like a follow up to the Calculus book. And, does it really systematically hit classical mechanics, e&m, etc. like Halliday, Resnick and Walker -- perhaps the Thomas and Finney of physics -- would? I was also looking at the reviews in Amazon -- does it have exercises? (For some reason the "look inside" feature on Amazon stopped working for me.)

At any rate, I would expect that book to be the equivalent of books by similar names for like an upper level math sequences for scientists and engineers. I'm looking for the "calculus" of physics that all freshman take that hits all of the major areas of classical physics perhaps even with a little special relativity and quantum mechanics -- that kind of thing. Of course, most freshman would take Thomas and Finney and HRW. But, if you had a freshman taking Courant, then his physics text would be...? Suppose you did Courant in a 4 semester course sequence, what text would you use starting in the second semester, say, for a concurrent physics sequence like the way they do it with TF and HRW? (The real answer to that question might be that you just shouldn't do it that way -- you should do Courant and then skip up a level to better physics texts aimed at each specific area.)

 

[mathwonk]

well how about the berkeley physics course?

 

[adrian_durham]

well how about the berkeley physics course?

Alrighty I'll take a look at that, then. Thanks!

 

[rudinreader]

apostle is a good choice.

No one caught that? For the record my favorite fish is salmon..

well there is a text by courant, (and hilbert) called methods of mathematical physics.

Two book questions.

First, I have seen the book Differential Operators of Physics by Hellwig referenced a few places.. Is that good to plug?

Second (more important for me), The book (around 1972) Symmetry Groups and Their Applications by Miller (available online) comes across to me as very good for "serious reading", by looking at it's heavyweight bibliography. Yet, it's out of print and otherwise I never it mentioned on PF. The only critique I can give is not really criticism because I haven't read it - that he seems to write in a "low level language" (via use of the word "local") despite it seemingly being of "high level interest". This is not necessarily a drawback (but is it?). The other point is that finite representation theory, lie group theory, and mathematical physics don't seem to be presented in the same way as recent books. The only comment from a mathematician I have heard of the book is that it is "an invaluable reference for those interested in dynamics". So in conclusion, is this a fresh tomato that's been hiding, or otherwise is it not the best for the picking?

 

[KGZotU]

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.

I feel just the same way. The moment I knew what a great book it was was when he was giving the axioms for one of the number systems and he said something along the lines of:

Such and such, such that 1.
0 such and such.
..
Such and such, 0, such and such, 1, and this 0 and 1 are the same 0 and 1 referred to above

I got so excited that he would write that down, I ran downstairs and showed my wife.

I wanted to thank mathwonk for his inspiration. I realized that if I never spend any time at my desk over a book and a pad of paper that I'll die just as good at math as I am today. On that note, I wanted to ask a question of my peers. I use a stopwatch to time how long I'm at my desk, reading, working problems, or using LaTeX. I can get about 6 hours in a day before I stop picking stuff up. Am I wimping out? Can the brain do more? Can yours? I can add hours on by learning in other ways, like my course lectures, but that seems to be about it for learning at my desk.

Also as a note, I'm reading Ross's Elementary Analysis, and he is extremely easy to read. Great book for someone like me who is just getting into the underpinnings. Requires experience with proofs, though, which I'm taking this semester.

Thanks,
Joe

 

[rudinreader]

Also as a note, I'm reading Ross's Elementary Analysis, and he is extremely easy to read. Great book for someone like me who is just getting into the underpinnings. Requires experience with proofs, though, which I'm taking this semester.

Speak of the devil! I actually had a copy of that book and I mailed it to my brother who's serving in the army. To tell the truth, it was a difficult book to depart from - all of my good books are difficult to depart from. I'm not going to tell him that though! - better to give when you can!

 

[omgwtfbyobbq]

Does anyone have any recommendations for algebraic geometry texts? I've been bouncing back and forth between going back for a MS, and from there who knows, when I can (about three years from now) and since I liked algebra and algebraic geometry, I figure it's something to look into before I head back. I'm also planning on picking up baby Rudin, as well as his other real/functional/complex analysis, but aside from that I don't know what to look for. Any suggestion, thoughts, tomatoes? :)

 

[mathwonk]

for accessible introductions to algebraic geometry, there is miles reid's undergrad text, and william fulton's book on curves, and shafarevich's book basic alg geom, and phillip griffiths lectures on curves from china, and rick miranda's book on curves and riemann surfaces, and joe harris' book, ...just search on the topic on amazon... there are lots more.

the books by miles reid and shafarevich are algebro geometric, and the books of griffiths and miranda are more complex analytic.

it never hurts to just start with shafarevich, vol. 1, chapter 1. and work the exercises.

then there are more ambitious books by griffiths - harris, hartshorne, ueno, george kempf, mumford...

 

[qspeechc]

I am always amazed by how much maths you know mathwonk- it's quite incredile!

Apostol, Courant or Spivak? For Calc1 & 2? Or does it not matter (btw, we use Stewart, which I dislike for all its numerical stuff, and 'application to life sciences', and general lack of rigour, and so many just-so statements)

 

[symbolipoint]

I am always amazed by how much maths you know mathwonk- it's quite incredile!

Apostol, Courant or Spivak? For Calc1 & 2? Or does it not matter (btw, we use Stewart, which I dislike for all its numerical stuff, and 'application to life sciences', and general lack of rigour, and so many just-so statements)

Could anyone give their opinion of ranking of quality of these undergraduate Calculus books?

Thomas-single variable Calculus, Howard Anton Calculus-the thick old book with picture of some old man, published bout 20 years ago, Larson & Hostetler Calculus, Sallas & Hill Calculus...

Rank them any way you all think is best and give your feelings/reasonings. This may help some of us who may like to study on our own...

 

[torquerotates]

Salas and Hilles is pretty good. It fairly rigorous. And it also has numerous worked out examples. It has a wide range of problems. Ranging from easy to really hard. Its a book that's not a simple as Stewart but not as rigorous as Apostol.

 

[d_leet]

Salas and Hilles is pretty good. It fairly rigorous. And it also has numerous worked out examples. It has a wide range of problems. Ranging from easy to really hard. Its a book that's not a simple as Stewart but not as rigorous as Apostol.

I don't think I can possibly disagree more... My friends use this book for their calculus class, and it is without a doubt one of the worst math books I have ever seen, the organization of certain topics is very poor, in my opinion, and also some comments made at the start of chapters are completely worthless, stupid, and things no mathematician should ever say. Stewart's is not a great book, but it is pretty good for a first calculus course that does not intend to cover much theory.

 

[mathwonk]

for rigorous honors level books, spivak is the most fun, apostol may be the driest but very intellectually honest and excellent, courant has more physics and diff eq than spivak, but any of them is outstanding.

another superb honors level book on the same level is the one by joseph kitchen, but not easy to find.

I always heard salas - hille was a good honors level book, not on the level of the four just mentioned but better than average. most of the other books are all cookbooks, not theoretical. stewart is a well liked cookbook.

 

[torquerotates]

Well, according to amazon, stewart scored slightly lower

https://www.amazon.com/dp/0471316598/?tag=pfamazon01-20

https://www.amazon.com/dp/0534359493/?tag=pfamazon01-20

 

[torquerotates]

for rigorous honors level books, spivak is the most fun, apostol may be the driest but very intellectually honest and excellent, courant has more physics and diff eq than spivak, but any of them is outstanding.

@ mathwonk. I'm curious, is Apostol an analysis level book? I'm currently using it for self-study as a supplement to Rosse's elementary real analysis and it turns out that Apostol is on a whole different level! The problems in Rosses' book we're doable. With Apostol, I got stuck on the first problem.

Would you say that at most universities, Apostol is on the level of real analysis?

 

[d_leet]

@ mathwonk. I'm curious, is Apostol an analysis level book? I'm currently using it for self-study as a supplement to Rosse's elementary real analysis and it turns out that Apostol is on a whole different level! The problems in Rosses' book we're doable. With Apostol, I got stuck on the first problem.

Would you say that at most universities, Apostol is on the level of real analysis?

No, Apostol is about at the level of Spivak, which is quite a bit more advanced than most calculus books, but not quite a real analysis book.

 

[d_leet]

Well, according to amazon, stewart scored slightly lower

https://www.amazon.com/dp/0471316598/?tag=pfamazon01-20

https://www.amazon.com/dp/0534359493/?tag=pfamazon01-20

I could be thinking of a different book, but I'm pretty sure the Salas book is one I am referring to, but a different edition than the one at that link.

 

[JasonRox]

Hey everyone!

I'm off to graduate school in September. I was originally wasn't going to go and I didn't even apply.

The day I was going to starting applying in a coffee shop in town I saw my professor walk in. Of course, I greet him and start talking. Then it came on the topic on where I was going for graduate school because he was assuming I was going somewhere. I told him how I don't want to go and that jazz. He insisted that I go and offered me a spot with him with a good offer. I couldn't let the opportunity pass up, so now I'm going to graduate school!

Let me say that I'm really excited. I'm still waiting for my acceptance letter though to make it official. I'm in though!

Anyways, I'm excited!

 

[omgwtfbyobbq]

Thanks mathwonk!

 

[morphism]

Nice Jason! Funny how that turned out.

If I recall, you're in Canada, right? Which school? And what will you be doing?

 

[Helical]

Mathwonk, when you say a book is a cookbook do you mean it is bad for learning out of period or it is bad for learning out of as a math major who wants a good theoretical understanding. I ask because I'm planning on majoring in physics and I believe my Calculus class will use Stewart.

 

[qspeechc]

From my personal experience using Stewart, I think it will be pretty good for physics majors, as there is a bit dedicated to that, and physics comes up quite a bit in the challenge problems.

 

[torquerotates]

From my personal experience using Stewart, I think it will be pretty good for physics majors, as there is a bit dedicated to that, and physics comes up quite a bit in the challenge problems.

The thing is that the hard problems in most physics texts are much more harder then the hard problems in Stewart. And they include calculus too. Making it redundant for physics majors to use Stewart.

 

[qspeechc]

I suppose your correct- I really don't like Stewart (I'm a maths major though...)

 

[JasonRox]

Nice Jason! Funny how that turned out.

If I recall, you're in Canada, right? Which school? And what will you be doing?

Hey!

I'm going to Brock University (in Canada).

My focus will be in Number Theory with emphasis on Algebra. I'll get reading material as soon as the term is over. Plus, he's getting another student which I haven't met, but she (a girl) sounds top notch. I met some of the other graduate students too. Should be good times.

 

[Ephratah7]

^_^... I want to be a mathematician. I'm not that good in math but i can understand math easily...i think...XD.. ^^.. I am just a 15-year-old girl dreaming of becoming a mathematician. BUT! I'm very interested in Math. It's my favorite subject.

 

[quasar987]

Pretty exciting news Jason! I'm happy for you.

morphism, you're in Canada too? Where?

I'm in montreal at Université de Montréal.

 

[mathwonk]

a cook book has recipes for doing things, not so good for understand why they work.

stewart is a fairly good cookbook, but a cookbook.

although the term cookbook has a sort of sneering tone, for many of us it is the place to begin.

i.e. before understanding why things work, it is useful to be able to actually perform the operations skillfully.

always choose whatever book seems enjoyable and useful at first. then try to progress upwards in level.

 

[JasonRox]

Pretty exciting news Jason! I'm happy for you.

morphism, you're in Canada too? Where?

I'm in montreal at Université de Montréal.

Nice! I'm hoping to land a summer job in Quebec City. I'd like to spend the summer there.

 

[uman]

Hi all,

I'm a junior in high school, very interested in math. I'm currently learning calculus out of Apostol's excellent book while concurrently taking AP calculus at my school (a waste of time, I must say...)

However it's clear to me that my background in geometry is deficient, having paid little attention when I learned it in 8th grade. I do not even remember what my teacher looked like, so it's obvious that I know precious little facts about geometry and have even fewer skills in geometric proof and problem solving.

I'd like to rectify this. Does anyone have any recommendations for a good, rigorous geometry textbook? Something difficult and thought-provoking but approachable to a (if I may be permitted to flatter myself) gifted high school student?

 

[mathwonk]

hartshorne, geometry: euclid and beyond.

 

[uman]

Thanks. I'll look into that.

 

[qspeechc]

You're in Junior High and reading Apostol? What, are you some sort of child genius?

 

[d_leet]

You're in Junior High and reading Apostol? What, are you some sort of child genius?

He's a junior in high school, not in junior high...

 

[qspeechc]

Ok, I'm not American, what grade is that?

 

[uman]

Normally 16 or 17 years old.

I'm 18 years old because I spent a year abroad.

So yeah, not a prodigy ;-)