Other Should I Become a Mathematician?

[mathwonk]

the time cannot be decreased. the point is to try to realize how much you are learning in a few pages of apostol. i.e. time spent on apostol expands. a few pages will last you a long time and take you a long way.

 

[Alpharup]

I don't know why Apostol like books can't be used for engineering mathematics..
I compared the topics covered in Engineering Mathematics Textbook(Erwin Kreyszig) and Apostol and found that they almost match in topics. Moreover, The engineering mathematics is not so rigorous in the approach. What I feel is that lack of rigour discourages mathematical learning. There should be continuity in ideas. I feel that Apostol gives the continuity of ideas. After reading a few pages, I got immersed in it and I didn't refer any other textbook. I think it is more self contained in concepts. On the contrary, when I read engineering mathematics, there is a need to refer some other book for results, proofs, etc..Many tough proofs are omitted and it irritates a lot. Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

 

[IGU]

Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

Obviously you can use Apostol, but for most engineering students the proofs are uninteresting and irrelevant, taking time away from practicing usage of the new mathematical tool. Apostol teaches math, not engineering. And he created and refined his books while teaching the material to Caltech freshmen and sophomores with no calculus background, who were much more about science than engineering (still applied math, but a little less so).

As I wrote earlier, you will benefit from learning calculus from Apostol. It will give you an advantage over your peers, engineering students who don't understand the math as well. So go for it! Just make sure you don't neglect practicing the application of what you learn to real-world problems.

-IGU-

 

[QuantumP7]

I became fully deaf about a year and a half ago. I've always had problems with my hearing and severe depression, so no degree yet. I've been studying finance so that I can try to make some money and get some cochlear implants (Medicaid in my state doesn't pay for it), and get off of SSI. I REALLY miss studying pure math, though. *sighs*

 

[mathwonk]

I have learned what I know of calculus by teaching it from several different books, learning something different from each one.

They include Spivak, Courant, Kitchen, Apostol, Thomas (an older edition), Cruse and Granberg, Edwards and Penney (several editions), Fleming, Loomis-Sternberg, Bers, Sylvanus P. Thompson, Stewart, Lang, ...

 

[nitro_gif]

Ok, got a few books on the go right now, in particular Lang's Basic Mathematics.

I like the content, but how can I retain and absorb more information? I feel like I read stuff but don't retain what I should, so I reread it again and still don't retain enough.

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

 

[epenguin]

Is it worth writing notes from the text as you are reading?

Maybe it is worth making notes after reading and then find out if you know what you have read.

(Do what I say not what I do. )

 

[dkotschessaa]

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

Yes, you pretty much have to. Except perhaps for some exceptional people, if you're not at least doing some pencil and paper work while reading, you're not really going to learn much.

Math textbooks are dense and leave a lot of stuff out, intentionally. Proofs in particular, with good reason, do not show all the "background" steps involved in getting from point A to point B. So you need to fill in those blanks, and you need to "convince yourself" that the things the books is saying are true.

If something is abstract, you may need to scratch out some concrete examples. For example, if you were reading an algebra text that tells you that a^xa^y=a^(x+y) then you'd want to plug some numbers in there to see that it "works."

-Dave K

 

[mathwonk]

one of my best math teachers, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

 

[dkotschessaa]

my best math teacher, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

Fantastic!

I've discovered the joy of the whiteboard now. I have a standard one, plus sticky-whiteboard sheets plastered all over my office wall. I am enjoying the hours of lively activity, working out examples, proving theorems, writing definitions until I know them from memory, ironing out all the details and just generally mathematically playing around. I've found it is better for someone as hyper as me, rather than trying to sit still, hunched over a desk. I'm learning quite a bit this way.

-Dave K

 

[nitro_gif]

Fantastic!

I've discovered the joy of the whiteboard now. I have a standard one, plus sticky-whiteboard sheets plastered all over my office wall. I am enjoying the hours of lively activity, working out examples, proving theorems, writing definitions until I know them from memory, ironing out all the details and just generally mathematically playing around. I've found it is better for someone as hyper as me, rather than trying to sit still, hunched over a desk. I'm learning quite a bit this way.

-Dave K

I have considered getting a white board. Sitting is no fun to me.

 

[Mandelbroth]

I suppose this is the place to ask this.

I'm just entering my junior year of high school. I like to consider myself a "mathematician," though I don't really do it professionally.

I like to consider myself as talented, though this is really a biased opinion. I'm really a pure math person, but I've been interested in application to medicine for a long time now. However, I've recently been reading some papers on applied math, and I'm having trouble dealing with the estimations and approximations. My pure math background is much stronger than my applied math background. I've tried treating $\approx$ like an equivalence relation, but I have issues with its transitivity. Even then, I see a lot of what I call "abuse of equality."

Do any mathematicians have advice for how to jump that hurdle? I want to go into applied math, but I have no idea how to get passed this. Or, if this continues to bother me as much as it does, should I even go into applied math?

I have considered getting a white board. Sitting is no fun to me.

I got a whiteboard.

Best. Christmas. Ever.

 

[Dens]

Whiteboards are great. You start to write and "create" things you would never do with pencil and paper. I have experienced the amount of creativity output with it.

 

[dkotschessaa]

I suppose this is the place to ask this.

I'm just entering my junior year of high school. I like to consider myself a "mathematician," though I don't really do it professionally.

I like to consider myself as talented, though this is really a biased opinion. I'm really a pure math person, but I've been interested in application to medicine for a long time now. However, I've recently been reading some papers on applied math, and I'm having trouble dealing with the estimations and approximations. My pure math background is much stronger than my applied math background. I've tried treating $\approx$ like an equivalence relation, but I have issues with its transitivity. Even then, I see a lot of what I call "abuse of equality."

Do any mathematicians have advice for how to jump that hurdle? I want to go into applied math, but I have no idea how to get passed this. Or, if this continues to bother me as much as it does, should I even go into applied math?

I think just a better understanding of what approximation means would be useful. You don't need to be deciding about pure vs. applied math yet.

Any application of mathematics (except perhaps in some areas of computer science) will require some approximation. To make yourself more comfortable with that, you should understand that different applications of mathematics will require different degrees of approximation. A good engineer or applied mathematician will now what degree of precision is needed for the task at hand. If you're measuring the radius of a planet it is ok to be off by a few feet (or maybe a hundred or a thousand for all I know). If you're putting a pond in your backyard then "3" is probably as good as "pi". There is just no such thing as an exact answer in the "real world."

Also keep in mind that you will deal with approximations in pure math as well. 3.14159 is an approximation of pi. 1.4142 is an approximation of the square root of 2. This is true no matter how many digits There's an entire field of mathematics called approximation theory which can still be considered pure mathematics.

-Dave K

 

[Mandelbroth]

I think just a better understanding of what approximation means would be useful. You don't need to be deciding about pure vs. applied math yet.

I agree with this. I like to mentally chew on ideas for long periods of time, though, so thinking about what to do long before is helpful for me.

Any application of mathematics (except perhaps in some areas of computer science) will require some approximation. To make yourself more comfortable with that, you should understand that different applications of mathematics will require different degrees of approximation. A good engineer or applied mathematician will now what degree of precision is needed for the task at hand. If you're measuring the radius of a planet it is ok to be off by a few feet (or maybe a hundred or a thousand for all I know). If you're putting a pond in your backyard then "3" is probably as good as "pi". There is just no such thing as an exact answer in the "real world."

I understand that numerical answers are important, but if you give me something like $\displaystyle \int\limits_{(-\infty,+\infty)}e^{-x^2}~dx=\sqrt{\pi}$, the LHS and RHS are both cool. However, the fact that they are equal interests me. I think equality is the most beautiful part of that expression, and indeed in most of mathematics. I feel like by approximating things like $n!\approx \sqrt{2\pi n}(\frac{n}{e})^n$, we lose a lot of that beauty, which we could have left more precisely with $\lim_{n\rightarrow +\infty}\frac{n!}{\sqrt{2\pi n}(\frac{n}{e})^n}=1$.

Also keep in mind that you will deal with approximations in pure math as well. 3.14159 is an approximation of pi. 1.4142 is an approximation of the square root of 2. This is true no matter how many digits There's an entire field of mathematics called approximation theory which can still be considered pure mathematics.

I always thought approximation theory was more to do with series expansions and not approximating constants, but I'm sure there's something in there.

Thank you for your response. I appreciate your input.

 

[WannabeNewton]

Taylor expand and drop all 2nd order and higher terms :)

 

[Dens]

Hey mathwonk, does it look bad if you take an art class for your final year even though you have maxed out all the art credits you need? Going into pure math, but only a stats class is available for the taking and I am not particularly interested in statistics.

Do you also think it is necessary to explain one W and a "bad" grade ("bad" = A-, I also think my transcript will show class averages.)? Particularly, if the course has some relevance to what you are doing?

Thanks

 

[mathwonk]

don't sweat it. math is art, right?

 

[Dens]

Even if it is a language class? How does the committee even view stat classes? Especially when my school is lacking pure math classes?

 

[mathwonk]

I never worry about this sort of thing. Being able to read a foreign language is very useful for a mathematician. And it is a lot easier to earn a living in stat than in math. I am not a good person to answer these sorts of questions. I care about the subject, not the perception of it by committees, and I believe committees also are best approached just by being well qualified and not worrying about how your record "looks". Can you hold an intelligent conversation about math?

 

[dkotschessaa]

Even if it is a language class?

The community of mathematicians is small (compared to other fields) and very internationally diverse. I think having language skills endears you to this community in a very positive way.

 

[Seydlitz]

I was reading this thread from page 170 when I noticed intelligence trick involving integrating $\ln x$ without integrating by parts. Anyone recall the fact? I have already closed the page before noting it.

 

[muzak]

Hello,

I will be applying to graduate school soon and have no real idea of where to apply. I was wondering if any of you know of any schools geared towards the pure end of mathematics, primarily real analysis and functional analysis and/or variations of the two, etc. I've looked into a few, but I was hoping to get a more general sort of list due to many of you guys that are probably more aware of groups involved in these fields. I'm not looking for top 20 or anything, just somewhere that is relaxed enough taking in an average student with no math research in pure or otherwise.

I've taken Introductory Real Analysis I and II and Topology, enjoyed the former more than the latter (due to the fast pace and algebraic part, was a bit too advanced for me at the time) and have been exposed to oh so rudimentary levels of functional analysis. I just really enjoyed the "building up from foundations" aspect of analysis and the elegant proofs that I understood and was able to follow. I'd appreciate any ideas you guys might have.

 

[Shivam3013]

Hello; if you don't mind, can you (mathwonk) please message me your email?

 

[mathwonk]

please post a specific question here, for best results

 

[NATURE.M]

I was kinda wondering if I completed my undergrad in physics, could I still possibly go to graduate school in mathematics. Note, I will be taking all the fundamental math courses (real analysis, topology, complex variables, etc.).

 

[mathwonk]

yes. just read the requirements for admission to a grad school in math. i suspect you will never find a requirement that your undergrad degree is in math. i believe Ed Witten majored in history at brandeis. he then apparently enrolled in grad school in first economics, then applied math, then graduated in physics. then he received a fields medal in pure math essentially.

 

[Crake]

yes. just read the requirements for admission to a grad school in math. i suspect you will never fins the requirement that your undergrad degree is in math. i believe Ed Witten majored in history at brandeis. he then apparently enrolled in grad school in first economics, then applied math, then graduated in physics. then he received a fields medal in pure math essentially.

I'll never understand how a person like Ed Witten majored in history.

 

[dkotschessaa]

I'll never understand how a person like Ed Witten majored in history.

I do. I think the sooner we realize there is no formula for greatness the sooner we can stop questioning whether we are doing the right thing and just get on with it.


-Dave K

 

[epenguin]

I'll never understand how a person like Ed Witten majored in history.

He swotted hard enough to scrape through the exams.

 

[Alpharup]

can learning topology help me to design electric and electronic circuits better? I have not finished analysis... But if topology helps me in some way to design efficient systems, then I could self study both analysis and topology in these four years of my electronics engineering...

 

[Cod]

I'm having trouble understanding how to apply specific topics to specific events. For example, I enjoy solving systems of linear equations, matrix operations, and the like; however, I have no idea how this knowledge can translate to a research topic, job, etc.. Basically, I understand the application portion when I'm looking at textbook examples, but cannot seem to come up with my own applications.

Bottom line is, I really enjoy linear algebra and numerical analysis, but have little idea how to use these outside of the popular applications (cryptography, computational fluid dynamics, etc.).

Any thoughts are greatly appreciated.

 

[Mandelbroth]

I'm having trouble understanding how to apply specific topics to specific events. For example, I enjoy solving systems of linear equations, matrix operations, and the like; however, I have no idea how this knowledge can translate to a research topic, job, etc.. Basically, I understand the application portion when I'm looking at textbook examples, but cannot seem to come up with my own applications.

Bottom line is, I really enjoy linear algebra and numerical analysis, but have little idea how to use these outside of the popular applications (cryptography, computational fluid dynamics, etc.).

Any thoughts are greatly appreciated.

Does mathematics need to have applications?

If I may give my opinion amongst the more experienced-backed opinions of the others who are probably better to answer this, you're fine. There is a difference in severity of the problem (see the following examples), but if I'm understanding you correctly, you should be alright.

There are two extremes for this kind of situation. If we have a problem like...

Solve the following system of equations: \begin{matrix}x+y=2 \\ x-y=4\end{matrix}​

...and you have trouble applying methods of linear algebra (or elementary algebra, for that matter) to that, you're probably in trouble.

However, I gather that you might be somewhere near the other extreme. If you look inside a physics book containing advanced topics such as relativistic necromancy (note: not an actual physics topic) and don't automatically think "I can apply eigendecomposition to this matrix and create a whole new subfield of relativistic necromancy!", you're probably okay.

 

[dkotschessaa]

Mathwonk and others,

This is a bit personal and I was going to journal it for myself, but I'm putting it out there despite the exposure.

I realized this morning (while meditating actually) that I still have a lot more anxiety about mathematics than I realized. I am in my senior year now and considering graduate school (at least a masters).

I realize that there are people who suffer from "math anxiety" and typically because of this they do not like mathematics and avoid it. But what about someone who does like mathematics? I realize that even though I've done well in most of my previous classes (though not extraordinary) I still worry about failing the next one. "I did ok in the last class, but this next one is more difficult!"

I know we're not psychologists here but I was wondering what your read on this is and how you may have experienced it. Some beliefs that are hanging me up are "I'm too old (37) to be doing this." "I am not naturally talented." and so forth. These are deep seated and I know consciously they are not a fact. I want to relax into it and enjoy it more.-Dave K

 

[Crake]

Mathwonk and others,

This is a bit personal and I was going to journal it for myself, but I'm putting it out there despite the exposure.

I realized this morning (while meditating actually) that I still have a lot more anxiety about mathematics than I realized. I am in my senior year now and considering graduate school (at least a masters).

I realize that there are people who suffer from "math anxiety" and typically because of this they do not like mathematics and avoid it. But what about someone who does like mathematics? I realize that even though I've done well in most of my previous classes (though not extraordinary) I still worry about failing the next one. "I did ok in the last class, but this next one is more difficult!"

I know we're not psychologists here but I was wondering what your read on this is and how you may have experienced it. Some beliefs that are hanging me up are "I'm too old (37) to be doing this." "I am not naturally talented." and so forth. These are deep seated and I know consciously they are not a fact. I want to relax into it and enjoy it more.


-Dave K

Hey, sorry that I'm not going to address your issue. (I don't have anything to offer you, honestly. I believe it's best to wait for the "pros").

I'd like to know more about your meditation habits. I'm thinking of starting to meditate, but I'm not sure if it's going to change anything tbh. Would you say meditation helped you? How so?

 

[mathwonk]

well i still have math anxiety, e.g. before posting on mathoverflow. once e.g. i asked a question about what some fancy theorem in algebraic geometry means. the first comment was from someone who was astonished that I didn't already know, because i am supposed to be an algebraic geometer. If i couldn't handle looking dumb like that, I would never get my questions answered.

The point is we are all ignorant but we are in there striving because we are interested in learning. I have occasionally also explained a few things to some really smart people who just didn't happen to know that one thing.

We are often afraid we will look dumb by asking a question, but actually one of the best ways to learn from someone is to let them look smart, by explaining what they know to us. people love to answer questions when they feel smart by answering them. they appreciate our giving them the chance to enlighten us, provided we allow them to enjoy the spotlight.

Why do you suppose so many people come on here and answer questions for free for years and years?

And I think meditation can be helpful in achieving balance and calm.

 

[Cod]

However, I gather that you might be somewhere near the other extreme. If you look inside a physics book containing advanced topics such as relativistic necromancy (note: not an actual physics topic) and don't automatically think "I can apply eigendecomposition to this matrix and create a whole new subfield of relativistic necromancy!", you're probably okay.

This part. I can work through problems (regular and applied) if the text "gives me the information". I just can't take linear algebra and apply it to something on my own, like the example you provided.

What are things I can do to help myself? Or do I just keep chugging at different topics I like and let it "come to me" eventually?

 

[Mandelbroth]

This part. I can work through problems (regular and applied) if the text "gives me the information". I just can't take linear algebra and apply it to something on my own, like the example you provided.

What are things I can do to help myself? Or do I just keep chugging at different topics I like and let it "come to me" eventually?

Most of the time, you just have to think about it long enough. A good example comes from my economics class.

The other day, we discussed elasticity of demand and the formula for revenue. I noticed that, if the elasticity was equal to 1, the revenue did not stay the same (by the formula we were given), dispite what we were told. I thought about it a little, and then I noticed that, if we took the limit of part of the equation for elasticity, we got a formula $\varepsilon_D=-\frac{P}{Q(P)}Q'(P)$, which rather obviously implied the statement about if the elasticity was 1.

It just takes some extra pondering, I think.

 

[Mpopyft]

Hey guys; can anyone recommend some tough textbooks for math and science high school and calculus level. Not the 100$ new ones but some old ones such as some listed in this thread already?

 

[mathwonk]

i like this old edition of thomas calculus, 3d edition, 1965. for under $5.

http://www.abebooks.com/servlet/Boo...=an=george+thomas&bsi=120&kn=calculus

 

[theoristo]

Hi,I would like to ask if anyone had seen this book
Gems of Geometry John Barnes https://www.amazon.com/dp/3642309631/?tag=pfamazon01-20 which seems to be a geometry fun textbook or is it?Geometry is a beautiful subject and my friend claim this book make anyone fall in love with it.

 

[Cod]

Most of the time, you just have to think about it long enough. A good example comes from my economics class.

The other day, we discussed elasticity of demand and the formula for revenue. I noticed that, if the elasticity was equal to 1, the revenue did not stay the same (by the formula we were given), dispite what we were told. I thought about it a little, and then I noticed that, if we took the limit of part of the equation for elasticity, we got a formula $\varepsilon_D=-\frac{P}{Q(P)}Q'(P)$, which rather obviously implied the statement about if the elasticity was 1.

It just takes some extra pondering, I think.

Nice work. From now on, when I go through specific subjects, I'll try to apply it to something on my own once I have a solid grasp of the information. Thanks for the advice.

 

[Mandelbroth]

I'll never understand how a person like Ed Witten majored in history.

"Let $n$, the number of presidents, be an integer..."

can learning topology help me to design electric and electronic circuits better? I have not finished analysis... But if topology helps me in some way to design efficient systems, then I could self study both analysis and topology in these four years of my electronics engineering...

I can't see how it wouldn't.

Nice work. From now on, when I go through specific subjects, I'll try to apply it to something on my own once I have a solid grasp of the information. Thanks for the advice.

You're welcome.

 

[modnarandom]

I think someone might have mentioned it earlier, but what did people who did Part III in Cambridge think about it? Why did you go there? Who would benefit from it?

 

[Prince1]

I am a high school student and I want to get the most rigorous math education available in algebra and geometry. I was thinking the SMSG books from yale univ, but that may be outdated (they use stuff like "truth sets"). How about this plan:
Starting of with basic math by lang
Algebra by gelfand
Lang's geometry/kiselev geometry
gelfand trigonometry
Gelfand and sullivan's precalculus/"graphs and functions"
Is this enough to give me the strongest, most rigorous background in algebra and geometry? Or should I consider the yale univ SMSG books as well? Thanks.

 

[dustbin]

I've had instructors who say "truth sets" so I don't think that aspect is necessarily outdated... I'm sure they would still make fine books (I've only read a little), but that's if you can stand the typewriter typesetting.

Here are some other "lists" for you:

https://www.physicsforums.com/showthread.php?t=307797&highlight=pure+math+high school
http://www.artofproblemsolving.com/Store/curriculum.php? (They have classes, too)

 

[Prince1]

Thanks a lot. The writing style isn't an issue. I have gone through AoPS, but it isn't too rigorous. So should I go with SMSG or my other list (lang, kiselev, gelfand etc)? Or a combination of both?

 

[dustbin]

I'd personally do the Lang/Kiselev list. Be sure to check out some of the other "theory" books on the first link a gave you. Particularly the ones on inequalities.

 

[mathwonk]

In my opinion, the best geometry book is euclid, and the best guide to it is hartshorne: geometry: euclid and beyond,.

 

[Tobias Funke]

the best geometry book is euclid, and the best guide to it is hartshorne: geometry: euclid and beyond,.

I agree. Also, the Dover edition has its own commentary with plenty of good stuff to go along with Hartshorne, which is a great book but not absolutely necessary (but if you don't have the Dover edition of Elements with the commentary, it might be necessary!). Whatever coordinate geometry you need, which obviously isn't in Euclid, is probably in Gelfand, although I haven't seen his books for a while.