Should I Become a Mathematician?

[adame903]

So I am a little bit at a fork in the road and I was wondering if anyone could give me some advice. I am currently an undergraduate physics and applied math double major. However lately I have been considering just doing an applied or pure math major. I love both math and physics but I feel that I have much more enjoyment learning and studying math. With Physics, even though I made an A+ in my first general physics with calculus course this past semester, I remember becoming very frustrated with it and not having as much fun like I originally thought i would. I just got done completing calculus 1 and 2 this summer and I loved every second of it. (im technically a junior and pretty behind on the curriculum because I had no idea what I wanted to do at first) I don't want to pass off physics just yet because I've only had the first class, but I can't help but feel that if I did, then I could more fully immerse myself into learning more advanced topics in math that I'm interested in.

Also, a quick question. I'm registered for calc 3, linear algebra, and ODE this fall, plus gen physics 2 w/ calc. Is this suicide, in the sense of being able to successfully make A's and understand the material from each class? At my university the calculus sequence is separated into 4 classes, so calc 3 from what I know of here is all the series stuff and polar coordinates.

Thanks in advance to anyone's help/advice.

 

[iomtt6076]

So I am a little bit at a fork in the road and I was wondering if anyone could give me some advice. I am currently an undergraduate physics and applied math double major. However lately I have been considering just doing an applied or pure math major. I love both math and physics but I feel that I have much more enjoyment learning and studying math. With Physics, even though I made an A+ in my first general physics with calculus course this past semester, I remember becoming very frustrated with it and not having as much fun like I originally thought i would. I just got done completing calculus 1 and 2 this summer and I loved every second of it. (im technically a junior and pretty behind on the curriculum because I had no idea what I wanted to do at first) I don't want to pass off physics just yet because I've only had the first class, but I can't help but feel that if I did, then I could more fully immerse myself into learning more advanced topics in math that I'm interested in.

Also, a quick question. I'm registered for calc 3, linear algebra, and ODE this fall, plus gen physics 2 w/ calc. Is this suicide, in the sense of being able to successfully make A's and understand the material from each class? At my university the calculus sequence is separated into 4 classes, so calc 3 from what I know of here is all the series stuff and polar coordinates.

Thanks in advance to anyone's help/advice.

Regarding your course schedule, I don't think it is academic suicide and you should manage fine. Although I would say if your linear algebra course is heavily proof-based, then be prepared to put in a lot of work. Even so, I think you should be okay. (I'm assuming that a huge portion of your time isn't taken up by a job or something like that.)

About choosing your major, it is difficult to say since you're still in the early stages of your math/physics coursework. If possible, I'd say wait until you've had more coursework or research experience to decide. If not, since you weren't overly thrilled by your general physics course, perhaps you should go with applied math. Later down the line, if you feel interested in doing physics, you can always join a physics research lab (perhaps in the summer) and pick up physics knowledge there. Or maybe if your schedule allows, do the applied math but take a couple more higher level physics courses (like E&M, quantum).

 

[rockitscience]

So I am a little bit at a fork in the road and I was wondering if anyone could give me some advice. I am currently an undergraduate physics and applied math double major. However lately I have been considering just doing an applied or pure math major. I love both math and physics but I feel that I have much more enjoyment learning and studying math. With Physics, even though I made an A+ in my first general physics with calculus course this past semester, I remember becoming very frustrated with it and not having as much fun like I originally thought i would. I just got done completing calculus 1 and 2 this summer and I loved every second of it. (im technically a junior and pretty behind on the curriculum because I had no idea what I wanted to do at first) I don't want to pass off physics just yet because I've only had the first class, but I can't help but feel that if I did, then I could more fully immerse myself into learning more advanced topics in math that I'm interested in.

Also, a quick question. I'm registered for calc 3, linear algebra, and ODE this fall, plus gen physics 2 w/ calc. Is this suicide, in the sense of being able to successfully make A's and understand the material from each class? At my university the calculus sequence is separated into 4 classes, so calc 3 from what I know of here is all the series stuff and polar coordinates.

Thanks in advance to anyone's help/advice.

Honestly most physics professors don't even like teaching physics 101...and they might even skip some of the "boring" stuff. It's just a foundation, but it's not the bulk of the entire physics curriculum. Your first class is always going to be a bit tedious and the kinematics can be frustrating...but don't lose hope! My first class wasn't the most inspiring, my professor just wanted to weed out the numbskulls. Upperlevel physics is challenging but more meaningful, so don't let this one class discourage you (you're NOT calculating a block up a slope). Personally modern physics is what really first got my attention. So maybe it's best to wait awhile before deciding. Physics and math is not a bad pair, you can't go wrong with either one, but if you have it in you to do both - then go for it. It can definitely be beneficial to have knowledge in both when seeking research opportunities. A well rounded person is always nice to have on the job, as one knowledge compliments the other.

As for your schedule, I don't foresee a problem with the classes you chose. In an engineering program it is common to have several math and engineering courses all at once. You seem to be a serious student, so I think as long as you keep your work in check and that determined outlook, you will be fine! I hope this helps. Good luck!

 

[RJinkies]

adame903> I am currently an undergraduate physics and applied math double major. However lately I have been considering just doing an applied or pure math major.

adame903> I have much more enjoyment learning and studying math. With Physics, even though I made an A+ in my first general physics with calculus course this past semester, I remember becoming very frustrated with it and not having as much fun like I originally thought i would...I don't want to pass off physics just yet because I've only had the first class, but I can't help but feel that if I did, then I could more fully immerse myself into learning more advanced topics in math that I'm interested in.

adame903> Also, a quick question. I'm registered for calc 3, linear algebra, and ODE this fall, plus gen physics 2 w/ calc. Is this suicide, in the sense of being able to successfully make A's and understand the material from each class?

-----

it can always be dangerous tossing advice, but i think a lot of these questions, can't really be answered till you are at 'least' halfway into your second year of mathematics and physics both. Since you're getting A's and B's [or capabie of them without killing yourself] and not seeing the studying and passing of exams as that much of a 'chore' it's a good sign.

Once you slog through a text like Kleppner/Kolenkow or Symon [ideally both!] for intermediate mechanics and a book on EM like Purcell and peek 50 pages into Jackson's Electrodynamics Text... then you can say, i reallly want to get out of here and go into pure math, or keep up with math/physics both.

If you're considering applied math, there is a lot of physics there too. Things like on the level of Kolenkow's Addison-Wesley Text 'Mathmatical Physics' because applied math people would probably be touching on mathematical methods in physics a fair bit. And any applied math would use say, lots of differential equation stuff, and there is a lot of physical phenonmena there, which is like 65% of taking an actual 2nd/3rd year physics class.

On the other hand, i think it's healthier to see it as a 'mastery of mathematics' than saying i want to go purely into 'applied math' or purely into 'pure math'. I think if you want to really be king of the pure mathematicians, it's good to have grappled significantly with applied stuff as well as the pure.

A good set of questions:

- what do you like about math
- what don't you like about math so far
- what do you like about physics
- what don't you like about physics so far

Mechanics with or without calculus, can see dry and boring to people in high school/first year, and they can be dry for people who teach. Some people like the fields where there are a lot of unknowns and things to be discovered. But the tools of mechanics can play into a lot of phenomena, and people at all sorts of levels can have a love/hate for it. Same goes for people taking EM, if it's PSSC high school, Halliday/Resnick first year, or Jackson [third year/grad school], you can like it, or shudder, struggling with it, or breezing through it and going forth on.

it's good to see what the next courses up in your math and physics classes are, just do you know what 'directions' they lead into, so many people take first year math and don't peek at the second or third year textbooks, and the same goes for physics. It can be a sense of frustration and feeling lost, and there can be a sense of joy and wonder, usually both. But you get a sense of where you 'may' be going.

adame903> Also, a quick question. I'm registered for calc 3, linear algebra, and ODE this fall, plus gen physics 2 w/ calc. Is this suicide, in the sense of being able to successfully make A's and understand the material from each class?

Sometimes, it's something 'no one knows', people who went through it, or yourself. Depends if you want to 'pass' the course or get a A or B. Depends if you study well and can put the hours and effort into it.

A lot could be based on the textbooks used. It could be a demanding text, or it could be one without a lot of theory and proofs and abstraction. If you're using Swokowski's calculus text or Apostol's calculus text, one is going to be 4 times as much effort.

Knowing what the textbooks are for those classes, others can toss a bit more help your way.

If you do well with high School or non calculus EM, and you sailed through calculus mechanics without a snag, the second part of your first year physics should be not too much trouble, but it will still be considerable work.

Knowing EM well enough with algebra is a big part of the struggle, and slowly and surely being confident with calculus word problems or applying the new math tools to the physics can be a snag for some.

Linear with a typical text should be pretty easy, though unusual. Depends a lot on how much theory there is, or how heavy or light the discussions on what a linear transformation is for the abstract stuff. Some classes are into just matrix stuff, and some do get difficult with excessive applications of it [the interesting but tricky economic/efficient paths/circuit path word problems]. Some texts or teachers could kill you with theory/proof/abstraction.

Taking calculus 3 slowly but surely is the best way to not have any trouble. And knowing your High School EM solid helps a lot for finishing first year physics with calculus.

first year EM, Calc III, linear, should be dense but doable, if you can study well and got the stamina and skill to do most all the textbook problems. Some might find a Diff Equations class on top of all that manageable, some might find it impossible.

A lot of this depends on the texts you're going to use, and how good you are at studying and mastering the material previously.

Some people prefer to do Diff equations during or after Calculus 3, but you can probably be comfy with just Cal I and II. Think of a new class as knowing new tools and new concepts, and for some it they can take to it smoothly, for others, it can be slightly choppy waters.

Seems like the worries about grades, studying are minimal here, so count yourself really fortunate, really. It's just will differential equations be too much, and do i like physics as much as i thought?

It's one thing to be frustrated and failing, and frustrated and getting an A. Tell us your likes and dislikes with the math and physics, and tell us the texts you used, and will be using.

I would find the calculus first year EM and third calculus course the most stressful, one or the other, or both at the same time.

Differential Equations might seen a 'strange new world' and the right or wrong text could colour things a lot. Sometimes the first chapter with some texts can be the hardest, depending on what your 'toolbox' is starting out.


Not sure if it's a great help, but i figured one more opinion can't hurt.

 

[Jimmy84]

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

Hi I am new to this talk I was wondering what is ODE's?

and is Apostol Vol. 2 good enough as a linear algebra text?

Greetings.


Im considering to do

 

[George Jones]

Hi I am new to this talk I was wondering what is ODE's?

Ordinary differential equations.

 

[thrill3rnit3]

Hi I am new to this talk I was wondering what is ODE's?

ODEs stand for Ordinary Differential Equations

and is Apostol Vol. 2 good enough as a linear algebra text?

Greetings.


Im considering to do

I haven't personally worked through the text, but looking at the table of contents, it looks more than adequate. Caltech uses this text for their Linear Algebra class, so I'd say it should be good.

Apostol's texts are generally hailed by the community as one of the bests.

 

[PieceOfPi]

Hi,

I am a junior at a state university, and my math classes actually start tomorrow (YES, we start that late!). While I haven't decided which math courses to take, I am thinking of taking two out of analysis, algebra, and topology.

I want to take analysis because I took it last year, and I did poorly that I ended up taking it pass/no pass. However, I feel more confident about my mathematical maturity that I want to give it a try again. The text is, of course, Baby Rudin.

I want to take algebra because that sounds like an interesting subject, and I feel like I need to learn algebra as early as possible if I want to become a mathematician (correct me if I'm wrong). The text is Beachy/Blair's Abstract Algebra.

I want to take topology because I heard that the professor is really amazing, and I'm also interested in learning this as well. The text is Munkres' Topology.

I will attend these courses to see which ones I want to stay, and I'll let you know how I felt about these courses tomorrow. I don't exactly know what I want to do with my life yet, but I'm kind of leaning toward going to a grad school in math. Plus, I also want to do REU in summer 2010, but this seems really competitive to get in, so it's probably a good idea to take a hardcore math course like one of these.

The ultimate option is, of course, I could take all of those three courses if I wanted to. But I'm not so sure if I have the enough mathematical maturity to do so.

Please let me know if you have any advice. Thanks.

 

[mrb]

If you want to do grad school, you should certainly take all 3 of those before you apply. In fact, generally schools offer a 2-course sequence in analysis and one in algebra also, and you should take the whole thing. At my school, topology was only offered every other year; if something like that is the case for you, this is your last chance to take it.

I recommend you do all 3 unless you're really sure you will be crushed. From a learning perspective, you're a junior, and it's time you focus on what you want to do. If that's math, why clutter up your schedule with other crap? From a grad school application perspective, you need to get a ton of upper level classes on your transcript (with As!), and ideally be in grad classes next year, so it's not the time to be shy about taking math classes.

Forget about "mathematical maturity." If you love math and you are willing to study... and study... and study... and study... then go for it.

I actually took Analysis, Algebra, Topology, Calc 4, and an independent programming project in a lab in one semester. It went fine.

 

[PieceOfPi]

Thanks for your reply, mrb.

I went to all three of analysis, algebra, and topology today, and honestly, I would regret dropping anyone of those. I feel like I can understand analysis this term; algebra seemed like another good fundamental of mathematics that I should know ASAP, and topology seemed very abstract and interesting too. I think all of these instructors should be good as well.

I recommend you do all 3 unless you're really sure you will be crushed. From a learning perspective, you're a junior, and it's time you focus on what you want to do. If that's math, why clutter up your schedule with other crap? From a grad school application perspective, you need to get a ton of upper level classes on your transcript (with As!), and ideally be in grad classes next year, so it's not the time to be shy about taking math classes.

The other two courses that I'm considering taking are both computer science courses, but one of them are optional (meaning, I can take it later). I still want to take the other one, since this completes the intro sequence. So if I decided to drop the optional one, I can certainly take all three of those this term.

Forget about "mathematical maturity." If you love math and you are willing to study... and study... and study... and study... then go for it.

I actually took Analysis, Algebra, Topology, Calc 4, and an independent programming project in a lab in one semester. It went fine.

Thanks. At least a lot of people I know are doing at least 2 of those (each one with different grouping), so I think I can find study-group pretty easily. The getting A part might be pretty challenging though. Maybe I need a bit more confidence in my ability.

More suggestions/comments are always welcome.

 

[qspeechc]

Hi. I am also a mathematics junior :).

A lot of the motivation and background for topology comes from analysis, so I would say put off taking topology until after analysis (you'll be more "mature" as well.). You can do topology concurrently with analysis, but 95% of the time it's not done this way for a good reason.
That leaves you with algebra and analysis. Take them both, topology later, that's my suggestion.

I agree baby Rudin is not a great text. The book by Pugh "Real Mathematical Analysis" is at the same level as baby Rudin and covers pretty much the same material, but is a far better book imho. It is more modern, has better selection of problems (and more problems), and provides some intuition and geometrical insight into analysis; overall a much better book I think. It's also cheaper. Try picking up this book and using it in your analysis course alongside baby Rudin.

I don't know about your algebra text, but algebra is a very important course to have. You might find it interests you more than analysis or topology. It is a basic subject you pretty much have to know.

If you think you might go into physics or applied math grad rather than math, you'll find analysis very helpful (at least I think it is: differential geometry, differential equations, functional analysis, etc. all require a good understanding of classical analysis), topology is helpful too (I think), algebra less helpful (but still important to get "maturity", and important if you want to do math).

Hope I've helped.

 

[lurflurf]

^^
[Humor]In mathematics we do not care about motivation or background. [/Humor]95% of what now? Do you realize many schools calculus course (it would pain me to call a course out of baby Rudin "analysis") have a topology prereq? If you want to go swimming, be prepapared to get wet.

 

[qspeechc]

Obviously I am out of my league here. You're talking about universities like Princeton right? Where they expect freshman to have studied Spivak already right?
For the rest of us mere mortals, it's more usual to go:
Calculus(Stewart or Spivak) --> Analysis (baby Rudin) --> Topology (Munkres)
The only "topology" needed for baby Rudin is metric spaces. Even the Princeton handbook calls it Analysis in one variable or Analysis in Several Variables, and they're talking about baby Rudin. Even Spivak's Calculus has been called analysis, which strictly speaking, it is.
The 95% I was referring to was that it is the most common course to take real analysis before topology, and for good reasons; like I said the motivation and background for topology out of a book like Munkres is from real analysis, real analysis also gives you maturity.

As for your statement that mathematicians don't need motivation or background, I suggest you read the preface to Needham's Visual Complex analysis, even Munkres' preface talks of the need for motivation. All mathematicians need intuition, motivation and background, they're bluffing if they say they don't.

 

[mrb]

In mathematics we do not care about motivation or background.

"It is impossible to understand an unmotivated definition."
- VI Arnold

Who should we believe, lurflurf from an online forum, or VI Arnold? Somebody did not write down the definition of topological space out of the blue one day and start proving theorems. Instead, the definition was developed and refined over years with the specific purpose of coming up with a good generalization of concepts from analysis. If there weren't this connection, nobody would ever have been interested in topological spaces... except, apparently, lurflurf. The sad thing is that it seems people adopt this attitude so they can sound smart and condescending, but of course they just look foolish. (And nobody anywhere has ever learned calculus out of baby Rudin... learning Calculus BEFORE college, then taking a baby Rudin course early on is a very different thing.)

 

[PieceOfPi]

Thanks for your comments again

I agree baby Rudin is not a great text.

The good news is, I'm starting to like this book. When I read this book last year, I thought it was really difficult to read. I thought that there were so many theorems and definitions that I felt like I could never memorize. However, after taking a few more math classes, I finally realized you don't memorize definitions and theorems... rather, you try to understand why they are important. And it turned out that they are actually important in proving the big theorem at the end. For example, Rudin presents many definitions/theorems so that it gives me the important results like Heine-Borel and Weierstrass.

And mrb, I got to agree with VI Arnold for this one.

I will go to the lectures one more time tomorrow. So far, I'm leaning toward taking analysis and algebra, and take topology in my senior year. But then again, topology sure does sound interesting as well...

 

[PieceOfPi]

The another alternative is to take two of those courses this term, and start taking complex analysis (or functions of complex variables) that is offered in winter-spring quarters.

 

[lurflurf]

This should be much better...

^^
[Humor]In mathematics we do not care about motivation or background. [/Humor]

I can see how a lover of fallicies would hate mathematics.
Wow lots of fallicies in there...

"It is impossible to understand an unmotivated definition."
- VI Arnold

Who should we believe, lurflurf from an online forum, or VI Arnold? Somebody did not write down the definition of topological space out of the blue one day and start proving theorems. Instead, the definition was developed and refined over years with the specific purpose of coming up with a good generalization of concepts from analysis. If there weren't this connection, nobody would ever have been interested in topological spaces... except, apparently, lurflurf. The sad thing is that it seems people adopt this attitude so they can sound smart and condescending, but of course they just look foolish. (And nobody anywhere has ever learned calculus out of baby Rudin... learning Calculus BEFORE college, then taking a baby Rudin course early on is a very different thing.)

So we hare argumentum ad verecundiam, an argument stands on its own. A faulty argument by Andrew Wiles is still faulty. That Arnold quote is very silly, I will assume that is because it has been removed from its context, ironic.
Argumentum ad populum, popularity of a belief does not make it valid.
The part about you trying to sound smart, but looking foolish is spot on.
Multiple fallacies of Relevance and straw man. If people are not reading baby Ruding to learn calculus why are they reading it? Many people have used it with success as a primary source, though no one here suggested that, if such a person had difficulties, the causes would be having one source and that one source being poorly written. What you were trying to say with that bit I have no clue. My point being Munkres and Rudin could be read in either order or at the same time. Symbolically 0<[Munkres,Rudin]<epsilon if you like. Though one wanting to learn what those cover could choose better sources, they were presented as so called course books. Which one who enjoys motivation or background should agree with, Rudin in my view motivates the topology he introduces very poorly.

 

[mrb]

Yes, because every point made in an informal discussion in an online forum must be a rigorous proof. I completely forgot about that. If your earlier post was supposed to be humorous, then so be it, but I certainly didn't perceive it that way.

I tend to agree with you that the questioner probably has sufficient background now to take topology, if that's your point.

 

[lurflurf]

For the rest of us mere mortals, it's more usual to go:
Calculus(Stewart or Spivak) --> Analysis (baby Rudin) --> Topology (Munkres)
The only "topology" needed for baby Rudin is metric spaces. The 95% I was referring to was that it is the most common course to take real analysis before topology, and for good reasons; like I said the motivation and background for topology out of a book like Munkres is from real analysis, real analysis also gives you maturity.

As for your statement that mathematicians don't need motivation or background, I suggest you read the preface to Needham's Visual Complex analysis, even Munkres' preface talks of the need for motivation. All mathematicians need intuition, motivation and background, they're bluffing if they say they don't.

The motivation this was a fuuny joke. The reader should bring some motivation of their own though. The topology books with 150 pages of streched out deformed giraffe show how easy it is to overdo that sort of thing.

You are almost making my point for me. Baby blue Rudin has about twenty pages of topology, reading say a hundred pages about topology (while not stricktly necessary) would provide background and motivation. Do not try make topology a slum of analysis, topology is a slum of combinatorics.[another joke] Courses in knots, combinatorics, differential geometry, or algebra would be at least as useful as preludes to topology as analysis "light". Even if your 95% is close it says nothing about which group (5% or 95%) is better off. One might say the 5% shows that topology first is a valid option. There are many courses that tend to procede others for no good reason.
Why take calculus before linear algebra?
The goals motivation and background are served by learning things as they are needed, not by learning lots of random things with the hope that they will become helpful in the future.

 

[economist13]

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 really think that is the case I suggest you look at modern economics again...in particular I might suggest Microeconomic Theory by Mas-Colell...or maybe
Recursive Methods in Economic Dynamics by Stokey, Lucas, Prescott

both standard PhD Micro/Macro books...

 

[n!kofeyn]

What are good mathematics publications/magazines? I guess something that a high school student can appreciate...

http://plus.maths.org/" article.

 

[Jimmy84]

I was wondering how good is the book Real and Complex Analysis by Rudin?

It has 424 pages it seems tempting to learn both real and complex analysis in such a short amount of pages. I was wondering how rigurous the book might be? Is the book a good preparation to start with differential geometry?

Im considering to do

Calculus, Apostol
Advanced Calculus, Loomis Sternberg
Real /Complex Analysis, Rudin (complementing with some other books on the subject)


Also searching on the net for Differential geometry books I found:

Differential Geometry, Analysis and Physics by Jeffrey M. Lee . I was wondering if someone knows about it and could recommend it?

The index is amazing, it seems to cover everything on the subject.

 

[martin_blckrs]

I was wondering how good is the book Real and Complex Analysis by Rudin?

It has 424 pages it seems tempting to learn both real and complex analysis in such a short amount of pages. I was wondering how rigurous the book might be? Is the book a good preparation to start with differential geometry?

Im considering to do

Calculus, Apostol
Advanced Calculus, Loomis Sternberg
Real /Complex Analysis, Rudin (complementing with some other books on the subject)Also searching on the net for Differential geometry books I found:

Differential Geometry, Analysis and Physics by Jeffrey M. Lee . I was wondering if someone knows about it and could recommend it?

The index is amazing, it seems to cover everything on the subject.

So what exactly do you want to study? Are you just starting with calculus and want to prepare for Differential Geometry?

Rudin's Real and Complex Analysis is an advanced book treating subjects like Measure Theory, Integration, some basics of Functional Analysis and quite a deal of Complex Analysis. If you just started with Calculus this is NOT the book you want to consider. This book will also tell you little of what you can use in Differential Geometry later on.

The book by Rudin is of course very rigorous (actually I think Rudin is a synonym for "rigorous" :-)) and you would generally consider the book, if you've already had a decent course on analysis (like Rudin's "Principles of Mathematical Analysis") and are considering going further in the field of Analysis.

If you've just started with calculus and want to prepare for DG, then Apostol and Loomis&Sternberg are a good preparation. You might also consider Spivak's "Calculus" and then also his "Calculus on Manifolds". Also Rudin's "Principles of Mathematical Analysis" is a great text as well as Munkres "Analysis on Manifolds".

For DG, I think there's no cannonical text, but there are some good books. A good introductory text is John M. Lee "Introduction to Smooth Manifolds". It's not really my taste (mainly because of lengthy and not so elegant proofs), but it covers a lot of topics and explains everything in detail (which becomes sometimes also its disadventage). Another good text is Warner's "Foundations of Differentiable Manifolds and Lie Groups" (less topics, more advanced). For more intuitive treatment and exercises there's a book by Fecko "Differential Geometry and Lie Groups for Physicists" ("for Physicists" says everything :-D).

 

[Jimmy84]

So what exactly do you want to study? Are you just starting with calculus and want to prepare for Differential Geometry?

Rudin's Real and Complex Analysis is an advanced book treating subjects like Measure Theory, Integration, some basics of Functional Analysis and quite a deal of Complex Analysis. If you just started with Calculus this is NOT the book you want to consider. This book will also tell you little of what you can use in Differential Geometry later on.

The book by Rudin is of course very rigorous (actually I think Rudin is a synonym for "rigorous" :-)) and you would generally consider the book, if you've already had a decent course on analysis (like Rudin's "Principles of Mathematical Analysis") and are considering going further in the field of Analysis.

If you've just started with calculus and want to prepare for DG, then Apostol and Loomis&Sternberg are a good preparation. You might also consider Spivak's "Calculus" and then also his "Calculus on Manifolds". Also Rudin's "Principles of Mathematical Analysis" is a great text as well as Munkres "Analysis on Manifolds".

For DG, I think there's no cannonical text, but there are some good books. A good introductory text is John M. Lee "Introduction to Smooth Manifolds". It's not really my taste (mainly because of lengthy and not so elegant proofs), but it covers a lot of topics and explains everything in detail (which becomes sometimes also its disadventage). Another good text is Warner's "Foundations of Differentiable Manifolds and Lie Groups" (less topics, more advanced). For more intuitive treatment and exercises there's a book by Fecko "Differential Geometry and Lie Groups for Physicists" ("for Physicists" says everything :-D).

Yea I finished my calculus high school book now I am reading Apostol, and I would like to prepare for Differential Geometry. I am looking forward to head into that direction though and perhaps into applied math. I am still not sure in what I am going to major though either math or physics. But for now I am having some spare time and I am studying on my own.

Im going to check Rudin's "Principles of Mathematical Analysis" Does it has a good complex analysis content?

thanks a lot for the recommendations.

 

[qspeechc]

Here's an article written by U. Dudley on calculus books. I thought some people might find it interesting. He talks about, among other things, how calculus books are too long, have silly apllications, not enough geometry and so on. I agree with most of what he says. He read 85 (!) calculus textbooks before making this review!

http://www.jstor.org/stable/2322923

 

[qspeechc]

Btw, I came across that article on this cool website:
http://mathdl.maa.org/

 

[n!kofeyn]

We have a politics forum to cater to those times when people want to talk about politics; the academic guidance forum is not the place for it. (And mathwonk's comment completely derailed the thread before it could even get started)

I don't doubt this. It's never good for Physics Forums to lose a member this way, but I personally found mathwonk's posts (not in the thread in question though) often very distracting. For instance this https://www.physicsforums.com/showthread.php?t=67268". Tom Mattson had found a version of David Bachman's book, A Geometric Approach to Differential Forms, on arxiv. Tom wanted to get a group discussion going where they would work through the book, but mathwonk almost immediately took over. In my opinion, he wasn't even participating in the discussion (and certainly not in the way Tom had hope for) and just rambled with very large posts, one after the other.

Tom even invited David Bachman, the author of the textbook and professional mathematician, to the thread, to which he accepted and started posting. Although, it wasn't long before mathwonk was basically insulting the author by constantly providing corrections or ways the material should have been presented, even in the face of statements by the author and Tom that the text was for undergraduates and that rigor was intentionally sacrificed for readability.

On top of that, mathwonk's self-indulging comments took over the thread and basically made it impossible for it to operate, which was very rude given Tom Mattson's original plan for the thread. In the end, mathwonk definitely seemed to irritate Bachman as seen in post 82, and you can easily see mathwonk's arrogance and complete disregard for the original purpose of the thread in https://www.physicsforums.com/showthread.php?t=67268&page=5#83". Just take a look at the thread, and you'll see near entire pages of the thread were just mathwonk posts.

I found this thread when I became interested in differential forms and found it completely useless due to mathwonk's meddling. I remember this frustrating me highly and even considered to quit coming here, even though I had basically just joined. mathwonk cost Physics Forums a possible member who is a professional mathematician and basically ran him off, as Bachman doesn't participate in the thread after the above mentioned posts.

All this is to say, mathwonk probably needed an infraction before this incident, and I find it a little frustrating he wasn't. I've seen other threads where this behavior of his took place as well. This has been bothering me because I've seen interesting threads shut completely down because they violated rules, in the case I'm referring to the post was deemed fringe science and not welcome. This is after just ONE post and a legitimate question in my opinion. The other point is that mathwonk's pinky up approach and condescending tone (see his winetasting https://www.amazon.com/review/R3RD2ULNTR37EU/ref=cm_cr_rdp_perm"&tag=pfamazon01-20 on Amazon :) is replicated somewhat by other PF members as well, which I think takes away from PF's ability to attract worthwhile members.

 

[tauon]

I'll be blunt and short: I'm a first year student reading mathematics and I was wondering if anyone here can recommend me some good textbooks...

I'm taking algebra, mathematical analysis, geometry and mathematical logic courses (which are mandatory) as well as an optional course in topology.

help? :P

 

[n!kofeyn]

I'll be blunt and short: I'm a first year student reading mathematics and I was wondering if anyone here can recommend me some good textbooks...

I'm taking algebra, mathematical analysis, geometry and mathematical logic courses (which are mandatory) as well as an optional course in topology.

help? :P

You should browse this https://www.physicsforums.com/forumdisplay.php?f=21".

 

[tauon]

You should browse this https://www.physicsforums.com/forumdisplay.php?f=21".

oh, I will. thanks for the tip. I don't know how I missed it. :)

 

[Bourbaki1123]

What is the probability of becoming a professor at some point after your PHD in mathematics? Also, to what extent does area of expertise affect this likelihood?

E.g. Suppose candidate X wrote his thesis on something in Automatic Theorem Proving candidate V wrote Something in Topos Theory, Candidate Y wrote his on something in Algebraic Geometry and candidate Z wrote his in some area of Analysis. Do these specializations affect qualification for an assistant professorship? I ask this because I wonder if being in a less popular area means less funding for research or if being in a more popular area means more competition or (more likely) some combination of both.

I'm talking about overall chances, so don't assume flagship school or state U, include southeastern state college X also.

 

[Chewy0087]

What is the probability of becoming a professor at some point after your PHD in mathematics? Also, to what extent does area of expertise affect this likelihood?

E.g. Suppose candidate X wrote his thesis on something in Automatic Theorem Proving candidate V wrote Something in Topos Theory, Candidate Y wrote his on something in Algebraic Geometry and candidate Z wrote his in some area of Analysis. Do these specializations affect qualification for an assistant professorship? I ask this because I wonder if being in a less popular area means less funding for research or if being in a more popular area means more competition or (more likely) some combination of both.

I'm talking about overall chances, so don't assume flagship school or state U, include southeastern state college X also.

this is the wrong section, you need to post this in the homework support - maths section, i'd consider a binomial approximated to a normal distribution.

 

[huenikad]

Hello,

I am a Gr. 12 Canadian student and I am deciding between math or engineering now for university. I was wanting to look into some math work to get a better idea of what I want to do. I've always found math at school to be ridiculously easy and have always enjoyed it but just get bored of the repetitivity. I have done math contests etc. over the years but haven't done too much further research into math yet. Sort of realizing how much I actually enjoy it now.

I am planning on looking at Courant and Robbins "What is mathematics", as well as Principles of Mathematics, by Carl Allendoerfer and Cletus Oakley. I was wondering if I should take a look at a specific calculus book or look for some more linear algebra type of stuff.

Any other books that I should take a look at that may pique the interest of a future mathematician?
Any books focused particularly on proofs would also be helpful.

 

[Landau]

I think Spivak's Calculus will certainly be of interest to you. It's a pleasure to read, but has also very challenging exercises. Take a look for yourself: click.

Of course, this is a 'serious' mathematics book. If you want to read a book about mathematics (instead of a mathematics book), I think Courant and Robbins may be a good choice.

 

[Bourbaki1123]

this is the wrong section, you need to post this in the homework support - maths section, i'd consider a binomial approximated to a normal distribution.

I guess that's an attempt at humor? Seriously though, if anyone has any actual insight into the process of becoming a professor (in mathematics) and what factors play into it and to what degree, I would appreciate it. I'm aware that it's highly competitive as far as getting a position and I want to know how to raise my chances aside from the obvious: pumping out tons good of research.