Should I Become a Mathematician?

[mathwonk]

maybe artin's algebra would bring you up to speed in proofs.

 

[Darkiekurdo]

I just picked up Spivak's 'Calculus', third edition. :!) I'm so happy.

 

[mathwonk]

i learned one variable calc from spivak's calculus, and i learned several variable calc from his calculus on manifolds.

 

[symbolipoint]

i learned one variable calc from spivak's calculus, and i learned several variable calc from his calculus on manifolds.

What kind of book is that "manifolds" book of Spivak? Does it contain a section on several-variables Calculus, or IS IT a book on several-variables Calculus? Further, who applies the concept of manifolds (I really have no idea about manifolds)?

 

[quasar987]

mike artins book algebra is probably the best out there for those who are ready for it. i would not have written my book if i had been better acquainted with it, i would have just used his book in my class.

How is Dummit & Foote?

 

[MrJB]

as to how to become a mathematician, I myself after a checkered career, just read the books and taught the courses and went and sat for the rpelim exams at the university of washingtona nd outperformed the studebnts there, and got an offer from them.

That's rather cool. I guess it comes down to how well you know the math.

 

[ircdan]

mike artins book algebra is probably the best out there for those who are ready for it. i would not have written my book if i had been better acquainted with it, i would have just used his book in my class.

What do you like about Artin? I will be using it for a course and so I'm just curious.

Thanks.

 

[Darkiekurdo]

Wow, Spivak is really rigorous!

 

[mathwonk]

manifolds are like n- space except they can be smoothly curved. this means you lose the precise value of a derivative, but you can still tell if a derivative is zero or not, i.e. you can tell if two curves are tangent even in a curved space.

thus calc on manifolds is just a generalization of several variables calc. so spivak begins with a chapter on sev vbls diff calc, then one on sev vbls integ calc, then goes into the algebraic machinery for discussing differentials and integrals abstractly, (chains and differential forms), then defines manifolds and generalizes the sev vbls calc to the setting of manifolds.

everyone who studies spaces that can curve uses manifolds, like space scientists, relativity theorists...artin is a world famous algebraist and algebraic geometer, so his grasp of the subject goes deeper than that of people like me. This means the material is so clear to him he can make it seem easy to the reader as well. as a master he is not dependent on copying the same proofs from other books, he often makes his own simpler ones.

he also wrote this book for sophomores at MIT, so although the entry level is high in terms of ability it is low in terms of sophistication. This is the best possible situation, when a deep master of a subject undertakes to explain it to beginners. the result is often that the rest of us can understand it too.

this is a very successful book. the only thing it lacks for some of us is that it does not treat every graduate topic, being an undergraduate book, so he does not give us the benefit of his explanation of tensors.

 

[mathwonk]

dummitt and foote is a fine book, written in more detail than artin's book, for average students, and is very popular. i prefer artin myself, although DF have succeeded in stating some facts very very clearly. e.g. i like the section on products and semi direct products of groups.

they say very clearly how to recognize a group G which is a product of two subgroups H,K, namely the subgroups, should satisfy HK = G, HmeetK = {e}, and both H,K are normal.

analogously, if the same holds except only K is normal then G is still a semidirect product of H,K.

this is very clear and very useful.

there is also much more coverage of more topics in DF than in Artin. But in my opinion, the discussion in DF just does not have the ring of the master, as Artin's does. Still they are obviously expert algebraists, and researchers, and they know what they are doing, and even their problem sets have a huge amount of information. I havea copy of DF, I just do not read it much. (Later I read it more and earned a lot.)

I myself disagree with their way of presenting some material as I have said elsewhere here. E.g. they give proof of the decomp. thm for fg modules over pids that to me is useless in understanding or using the theorem. then later when they need to use the theorem themselves, they refer you to a different proof in an exercise. i do not like this way of doing things.

Why waste your time reading a proof that will not be used, and omit the approach that wil be used? it's just not my cup of tea. but it is a good book with a lot of clearly explained and userful material. in particular it is easy to read.

 

[Darkiekurdo]

I just finished the first chapter of Spivak's 'Calculus'. And although very rigorous, I like his style. It is unlike everything I have read before. In all other books i have read they took the properties of numbers for granted. Spivak does niet. He shows, for example, why 'a times 0' equals 0.

 

[mathwonk]

yes, he will also make the key "least upper bound" axiom for reals very very clear.

but be sure to do the exercises, or his smooth explanations may fool you into thinking you understand more deeply than you do.

 

[Darkiekurdo]

yes, he will also make the key "least upper bound" axiom for reals very very clear.

What version did you read? And what chapter is that?

but be sure to do the exercises, or his smooth explanations may fool you into thinking you understand more deeply than you do.

I will do that. :)

 

[mathwonk]

well somewhere there is a chapter called "three hard theorems". the lub axiom is used there. i read the first edition of course in 1969 or so.

 

[Darkiekurdo]

Wow.

 

[mathwonk]

chapter 7 is three hard theorems, and chapter 8 is least upper bounds, where he actually proves those 3 theorems.

 

[Darkiekurdo]

I'm afraid I can't finish the whole book. It took me a lot of thinking to complete the first chapter. :(

 

[uiulic]

What a thread is this! I don't even have patience to read all through.I have been wanting to ask a question, and please allow it to be asked here.

For me, saying I want to be a mathematician is like saying I want to be King.

Looking back at how much time we (including you mathematicians) spent in mathematics,I then realized how difficult mathematics really is. At primary school, I were forced to recite 1*1=1 to 9*9=81 and from that time the language of maths was embedded in our brains and later on we are using what we remember in the way/style it is. I just feel lucky in passing those exams related to mathematics (or involved), which is a big release.

But now, there is such a situation for me (maybe also for many others). Whenever I read a bit more advanced mathematics books (say topology or advanced algebra, I never really started it), I find I am so disappointed with myself. In order to grasp an easy concept, I need recite it and then do a lot of exercises, which is what I did in primary school. I just fear this kind of experience, and I can't afford it because of my age.

How did you (those mathematicians and those who want to be) overcome such "fear"? Or share your experiences please if possible?

 

[mathwonk]

well it is just as hard for us to learn new things. the will to try to overcome our fear and struggle to learn is something we always have to work at.

it helps to realize it is natural, for everyone to feel uncomfortable in new situations, and everyone has to work to learn, and also remember the pleasure of succeeding at learning, and remember the added benefits of doing so.

just try one new thing today, and i will try one too! for me it will be something scary like trying to put my homework system online for the first time in my life. or even learn some deformation theory, which for me is challenging but more fun.

 

[mathwonk]

my webbooks are on my page whose address is visible in my public profile. just click on my name and the public profile link will appear.
pivoxa15, i now recommnend Hartshorne's book geometry: euclid and beyond as the best book on euclidean geometry, in connection with euclid's own work, available free online.

i am now using Hartshorne together with a book called geometry for the classroom by Clemens and Clemens. these are both books written by world renowned algebraic geometers for classes of young students at the undergrad or even middle school level, in Clemens' case. i love these books.

there is nothing like an elementary explanation from a high powered scientist.

 

[mathwonk]

any reaction to my new 100 page algebra book?

 

[quasar987]

I didn't know you had a new book, I'll go check it out immediately.

In the meantime, I was in the middle of applying to a university and it asks me if I want to apply for a

MA with thesis
MA w/o thesis
Ms.c with thesis
Ms.c w/o thesis

Would you be able to explain the difference btw the four, and why would one want to chose one instead of the other? Thx!

 

[mathwonk]

not fully, the best people to ask are the grad dvisors at the uni. but with thesis is for people who want to spend some time learning one topic rather well, and then practice writing by writing it up carefully. this is a good experience and will result in really knowing that one topic.

without thesis is for people who prefer to just take courses. this results in broader but less deep knowledge of more topics, and may take less time.

the 100 page algebra book has been up there for several weeks now, so may not be new to you, but is pretty new.

 

[mathwonk]

darkiekurdo, here's an idea, you do not need to read a math book linearly from front to back, in fact it is usualy a bad idea pedagogically.

so just turn right now to chapter 7 and read about the three big theorems.

of course you will realize you do need some stuff from earlier chapters, (namely the definition of continuity), but that will motivate you to go back and get it.

 

[Darkiekurdo]

I'll do that right now. Can I ask some questions while reading that chapter? Or you could refer me to parts where he explains that what I don't understand?

 

[mathwonk]

of course, good for you!.

you might think of it as a basketball game, if you get stopped driving down the side, you flip it out to the middle and go that way, if that doesn't work, you cross court. you cannot afford to let one blocked lane stop you from getting to the goal.

 

[ice109]

mathwonk is like the one man math subforum at pf

 

[Darkiekurdo]

I think I'll start at the previous chapter about continuity.

 

[Darkiekurdo]

I think my understanding of limits is not sufficient. So I'll go to the previous chapter about limits.

 

[Darkiekurdo]

I don't understand page 91; we have a function

x\ \sin\ \frac{1}{x}

and we want x to be within

\frac{1}{10}

of x, so,

-\frac{1}{10} < x\ \sin\ \frac{1}{x} < \frac{1}{10}

I understand it until that, but then he writes:

"|x\ \sin\ \frac{1}{x}| < \frac{1}{10}"

Now my question is probably very stupid, but why does he use the absolute value?