Other Should I Become a Mathematician?

[eastside00_99]

In all honesty, I have not read Lang or Dummit & Foote. I have read most of Artin's Algebra and Gallian (this was the book for my required undergraduate course in algebra). Artin isn't really that bad. I especially enjoy his discussions about the formal development of Group, Ring, Field theory et cetera. I do think it is strange though that computation and discussion take place over some important theorems. What I mean is often artin gives computational examples or long discussions within the section and then leave important theorems for proof by the reader. I don't see this as all that bad, but, for instance, the second and third isomorphism theorems of groups is left as an exercise in the section on multilinear algebra. To me it seems, you would want to at least prove it for groups in the group theory section and then allow for the reader to extend the results for rings and modules as the material progresses. But, that is only a minor quarrel I have and such proofs could be found in other books.

I actually tried to read Lang a while back but found it inaccessible at the time. I remember that in the exercises in the first chapter there was a question about abelian categories something like show that the category of abelian groups form an abelian category. At the time, I wouldn't have a chance of showing that just because of my immaturity. Now, the problem would probably be trivial. That highlights the fact that sometimes it is best to use the book that is not to far from one's level because a lot of the material in a book by Lang can be understood very easily if you have the intuition and practice that book like that of Artin's can provide.

One question I have about notation in group theory that a friend of mine brought up that I would like to ask you MathWonk is why do we refer to the order of a group by |G|. I understand it probably has its roots in the written work of Galois. But, it would seem better to write [(e):G] where e is the identity and (e) is the subgroup generated by the identity. The problem with this may be manyfold such as not extending to semi-groups and doesn't correspond to the way we write the order of an element, but still this gives a nice correspondence between the Tower theorem for fields and the formula

|G| =[H:G]|H| where H is a subgroup of G and which we can now write as

[(e):G]=[(e)][H:G].

Anyway, what is your advice for qualifying in algebra. Would you recommend working most of the problems in the reference books for the course? This would be a tall order at my school as about four books are used as reference books for the graduate course in algebra. Of course, I guess people should do as many problems as they can. But, what advice do you offer to your algebra students?

 

[k3N70n]

artin wrote his book for sophomores in college, so it is a high level beginning book, not a graduate book. but he is a MUCH better mathematician than most authors, apparently such as Dummit and Foote (or me), so his book has more expertise flowing through it than ours.

so the choice between those is a choice between an undergraduate book by a master and a graduate book by lesser mortals.

I myself think dummit and foote has a great deal of useful information, clearly explained. BUT i do not like the lack of insight in the discussions., I own one and i use it for some references, but i do NOT get much extra insight by listening to what they say.

dummit and foote is indeed the now standard text for most courses at most places, which means it is the current blandly written book that conbtains everything, and can be read by anyone. it does not mean it is the book that future professionals need.

i.e. you will not learn as much from it as if you read a book by a master like jacobson.

years ago hungerford was the current standard dumbed down algebra book (i.e. easier than lang to read, but not as deep). nowadays dummit and foote make hungerford look hard.

note the first part of dummit and foote is also an undergraduate book, but not as good a one as artin in my opinion.


but these discussions are pointless. get which ever one you can read. but be aware, you will NOT get the deepest understanding from a dumbed down book intended to be readable by every average grad student.

the classic best graduate books for experts are (older) van der waerden, and (more modern) lang and jacobson. but i recommend having hungerford and dummit and foote also for their problems and examples.

but if you are ready for a beginning grad book more advanced than artin, what do you think of my notes for math 8000, free in my website?

or better, the free notes and books of james milne on his?

but basically my attitude is that you learn more from reading a high school algebra book by a master, than a phd level book by an non master, so i recommend books by masters, like artin.

Thanks for the advice Mathwonk! I'll probably pick up Lang.

 

[mathwonk]

well let's see, this may lead you astray, but when i myself was a student, it seemed to me that the problems in herstein sufficed to pass a lot of quals!

hungerford was written explicitly to provide adequate quals preparation.

i recommend reading the guidelines for your uni on passing quals and looking at old ones. indeed in this very thread, there was a segment on passing quals, complete with sample exams from several universities. i know this thread is too long, but please search, and you may find my own exams.

try pages 10-13 of this thread.

 

[eastside00_99]

Thanks. I looked at those pages and will refer back to them.

I am quite worried about the qualifying exams. I don't think I am assured to pass the real or topology exam without the corresponding courses at CUNY. But, I think if I work with Lang's Algebra through the summer; I should be able to pass the algebra qual. I don't know. It really depends on the material covered in the corresponding courses and without taking the course there is no way to know exactly what that is. So, my plan now is to study algebra all summer, Pass the algebra qual in september, take real analysis, topology and advanced algebra, pass the quals in topology and real analysis at the end of the academic year. I think this is reasonable. Because I know the book used in the algebra course, I should be able to pass the qual. I sort of feel like I could do the topology qual since I took one of the CUNY ones for fun and did quite well, but I don't want to push it. I should consult someone at CUNY about this topic.

 

[mathwonk]

hungerford and dummitt foote seem to me more quals oriented, while lang seems more research oriented, but you should ask the locals.

 

[ircdan]

Thanks. I looked at those pages and will refer back to them.

I am quite worried about the qualifying exams. I don't think I am assured to pass the real or topology exam without the corresponding courses at CUNY. But, I think if I work with Lang's Algebra through the summer; I should be able to pass the algebra qual. I don't know. It really depends on the material covered in the corresponding courses and without taking the course there is no way to know exactly what that is. So, my plan now is to study algebra all summer, Pass the algebra qual in september, take real analysis, topology and advanced algebra, pass the quals in topology and real analysis at the end of the academic year. I think this is reasonable. Because I know the book used in the algebra course, I should be able to pass the qual. I sort of feel like I could do the topology qual since I took one of the CUNY ones for fun and did quite well, but I don't want to push it. I should consult someone at CUNY about this topic.

your priority should be on solving problems from past quals if the exams are available, don't neglect these! I think most schools try to keep their quals each year similar, so doing problems from old exams helps a ton. I would of course as already mentioned, ask at your school.

everyone worries about quals, it's a requirement, if you didn't worry you probably wouldn't study hard enough to pass now would you

goodluck!

 

[asdfggfdsa]

well lang is good but not sufficient, as it is all theory and no examples.

i recommend you add hungerford to it.

Thanks for your recommendation (and for all of the other information you've provided in this thread).

 

[The|M|onster]

Mr. mathwonk, sir, would you mind posting solutions to that practice Vector Calculus exam that you posted on the last page? It would be greatly appreciated (I'm actually studying for a Vector Calculus final that is coming up in a week or so).

 

[mathwonk]

Dear all,

Questions IA,B, were testing knowledge of the big theorems useful for computing integrals and recognizing gradients.

IA,a: f(q) -f(p).

b. the path integral of F dot dr = the surface integral of (del cross F) dot n dsigma, page 906.

c. the surface integral of F dot n dsigma over S, equals the volume integral of del dot F dV, over R.
IB a) True, the curl of a gradient is always zero, by the equality of mixed partials.

i.e. the entries are the differences of second partials of f taken in the opposite order.

i.e. curl f = (fyz - fzy, fzx - fxz, fxy - fyx) = (0,0,0).

b) False, curl(F) = 0 only guarantees that F is locally a gradient, as we saw an example "dtheta", of a field wioth zero curl, but only a gradient in regions that do not wind around the origin.

c) True, here the region U is simply connected so curl(G) = 0 does guarantee that G is a gradient in U, so all closed curve path integrals are zero.

d) True, stokes theorem equates the flux integral of curl(F) over a surface, with the path integral of F itself over the bloundary curve.

so if two surfaces have the same boundary ciurve, then stokes equates both flux integrals to the same path integral.

we had explicitly answered this question, a homework problem from the book. page 913, problem 11.

e) this is true, by the divergence theorem, since every sphere is the boundary surface of a ball, and the divergence theorem

says to get the surface integral, we can just integrate the divergence, which is zero, over the ball.IIa) This is a simple path integral we did several times, for the area of the region inside the path, namely an ellipse of semi - axes a,b, the area is pi ab,

which here is 6pi.

IIb) here is one way to see it gives area, since by greens theorem, it equals the double integral of dxdy over the interior of the ellipse,

i.e. area., see problem 21, page 885.

IIIa) del cross F here i.e. curl(F), is just ( y, x, 1).

By the true statement IB d), we can replace the hemisphere H by any other simpler surface with the same oriented boundary,

such as the disc of radius 2, in the x,y plane.

then the normal vector to the disc is just (0,0,1), so in the flux integral, the dot product of curl(F) with n is just 1,

and the surface flux integral becomes just dxdy over the disc,

i.e. the area of the disc, or 4pi.

the path integral is not too hard either, and during the test i even did the surface flux integral over the hemisphere,

using spherical coords, and it was not too bad either. it finally came out as the integral from phi = 0 to phi = pi/2,

of 8pi sin(phi) cos(phi) which is again 4pi.

IV. div(F) here is just z.
using the divergence theorem, we are integrating z over the tetrahedron, so at each height z, if we integrate in the order z,x,y, we are

integrating z times the area of the triangular slice at height z, and that area is (1/2)(1-z)^2.

so we are integrating (1/2)z(1-z)^2 from z=0 to z=1, and get 1/24.

i also parametrized the faces of the tetrahedron and did the masochist's computation of the flux integral, and finally got the same thing.

there are three pieces to the surface integrand as usual, one each for dydz, dzdx, dxdy, and 4 faces for the tetrahedron, so potentially 12 parametrized area integrals to do, but 10 lf them are equal to zero,

because dzdx for instance is always zero in the x= 0 plane and z=0 plane, and ydzdx will be zero also in the y=0 plane.
and one of the two non zero integrals cancels part of the other one, for reasons of opposite orientation,
so we are left finally with an integral over the triangular base that also comes out 1/24.

Archimedes knew the value of this integral by the way because he knew the center of gravity of a tetrahedron is 1/4 of the way up from the base,
so at height 1/4, but the height of the center of gravity is the average z coordinate, which equals the integral ,of the z coordinates divided by the volume of the tetrahedron, as we know, (pages 817-818),
so the integral of z is the producto f the height of the centyer of gravity by the volume of the tetrahedron, i.e. (1/4) times 1/6 = 1/24.

recall the volume of a pyramid is 1/3 the product of the height by the area of the base.

actually archimedes computed centers of mass first and then deduced formulas for volume.best regards,

roy

 

[eastside00_99]

your priority should be on solving problems from past quals if the exams are available, don't neglect these! I think most schools try to keep their quals each year similar, so doing problems from old exams helps a ton. I would of course as already mentioned, ask at your school.

everyone worries about quals, it's a requirement, if you didn't worry you probably wouldn't study hard enough to pass now would you

goodluck!

thanks for the advice.

 

[RocketSurgery]

Hey mathwonk. I've heard that a lot of mathematicians think that is good to "learn from the masters and not their students".
How do you feel about this idea? Is there even a textbook (or even a complete set of papers) in every area of math that is written by a "Master" and how do you Define master?

I'm learning Calculus from Apostol but is Apostol a master? or even Rudin for that matter?

 

[mathwonk]

well i do think apostol is a master of calculus, and i do recommend learning from masters as soon as their writings are accessible to you.

once in grad school for the heck of it i went to the libs and tried to read the famous paper on the concept of a singular point, by the master of algebraic geometry oscar zariski.

i struggled for hours to get through even a few pages and felt discouraged. but the next day in class, when the prof brought up the idea of a regular local ring, and regular sequences, i knew the answer to every question he asked instantly, so much so that ultimately he told me to shut up as i obviously knew the subject thoroughly.

that was my best day ever in class, and the only day i was ahead of the lesson.

 

[RocketSurgery]

Oh very nice example Mathwonk. I was just curious like what people mean by master and what books would be good to read from the so called masters.

From your point of view it seems you consider Apostol a master of calculus bc of how well he knows Analysis/Calculus but not necessarily because of his own contributions to the field.

Like some might consider Newton a Master and say that to understand calculus you should learn from his writings. However I don't think anyone is probably going to be better off then they would learning from Apostol or Spivak then going back to Newton's work for some enlightenment.

 

[mathwonk]

it is excellent to read Newton. for example one could learn there, well before riemanns well known definition of integration, that all monotone functions are integrable, (which one can also learn in apostol).

i myself have the book on analysis by goursat, which is also recommended.
i do not know apostol's contributions but anyone as outstanding as he must have made some.
i do know spivaks contributions to differential topology, namely the concept of the spivak normal fibre bundle, a fundamental tool in the subject.
probably apostol has some work in analytic number theory. i will check it out.

 

[mathwonk]

well he got his phd at berkeley in 1948 and is so famous since then for his book I am having trouble finding older data. his research in the past 10 years or so has been handsomely funded for projects in education of high school students.

http://www.maa.org/reviews/earlyhist.html

oh yes i believe i have commented here on some recent research by apostol on figures in solid geometry with area and volume formulas similar to those of spheres.
i.e. certain solids have the ratio of volume to area equal to something like R/3, where R is a "radius",
such as a sphere and perhaps a "bicylinder" (intersection of two perpendicular cylinders) and many others.

 

[mathwonk]

well ok, apostol is not a creator of calculus as are Newton and riemann. i recommend reading them too, for what you can get, but you will learn a lot from apostol.

there are two types of masters of a subject, those who first created it, and those later brilliant people who have indeed mastered it, and show that by the depth of their writings.

galois created galois theory, but emil artin made it accessible to modern generations, and others such as his son mike, and other modern masters like jacobson and van der waerden, and lang have given expositions some of us find useful.

it might still be useful to consult dirichlet, gauss, and legendre, for related work, but i have not much done so.

to be specific, you are invited to read my notes on my webpage, but having done so, if they are found useful, they can at best serve as an introduction to those small parts of the subject i myself understand. afterwards move on up to reading better works by more qualified persons.

e.g. even though i have criticized details in their book, dummit and foote are more accomplished algebra experts than I, as one can see from perusing their research vitae, and their book contains more than my notes.

still some features of their book cause me to feel that they are either consciously writing down to their audience, which i find troubling, or for some reason do not convey the depth one senses in artin and jacobson.

 

[RocketSurgery]

Very good point. So I'll try to make a habit of learning a subject from a good modern textbook but also look into what the creators have written afterwards to get a deeper understanding. That way I can see the point of view of the two types of masters.

 

[The|M|onster]

Thank you very much for the solutions, mathwonk. Also, would you mind recommending me some good books on Number Theory?

 

[Vid]

I really like reading Hardy's Introduction to the Theory of Numbers, and he's definitely a master.

 

[RocketSurgery]

I really like reading Hardy's Introduction to the Theory of Numbers, and he's definitely a master.

Agreed but don't forget about his coauthor, Wright.

 

[mathwonk]

forgive me, i have temporarily forgot the names of the number theory experts here, greathouse? robert ihnot? ...

lets ask them. i agree with niven and hardy by the way, but you might also check out andre weil, basic number theory (misleading title).

also borevich and shafarevich, and ...

 

[RocketSurgery]

Haha yup can't leave anyone out.

 

[mathwonk]

heres my real fave: trygve nagell. check it out.

its $120 on amazon, but here is a used one:

Introduction to Number Theory.
Nagell, Trygve.
Bookseller: Monkey See, Monkey Read LLC
(Northfield, MN, U.S.A.)
Bookseller Rating:
Price: US$ 20.00
[Convert Currency]
Quantity: 1 Shipping within U.S.A.:
US$ 3.99
[Rates & Speeds]
Book Description: John Wiley, 1951., New York:, 1951. Hardcover. ex-library with usual markings, no jacket, sound copy, text is clean. Bookseller Inventory # 4463

 

[RocketSurgery]

Seems interesting. The amazon review says Nagell is similar to Hardy/Wright. I'll see if my library has it.

 

[mathwonk]

all books have the elementary result of fermat on which integers are sums of two squares, but nagell explains which integers are sums of three squares.

stuff like that. and it is well written. i however have not read hardy and wright so it might be even better.

like i said i am a rookie at number theory. there are several people here much more expert.

 

[The|M|onster]

Cool. I am going to head to the library right after I get off of work.

 

[RocketSurgery]

Hey guys I was wondering if anyone could recommend me any books on Game Theory that I would be able to understand.

I have a good grounding in Proof based math (Set Theory, Logic, Apostol Calculus)
and I've taken an elementary Probability class.

I'd like to read Neumann's book but I don't know if it will go over my head or not.

 

[Mokae]

Usually on site like this, people rarely introduce others any books because, as you might guess, users log in with different usernames, and even the writers of the books. People care to recommend their own written books, right ? So I doubt if anyone around introducing any book to you is not the writer himself

Why don't you look up in your school library or just go straight to your school teachers to make some questions on the same problem ? I am sure they are not that selfish to not even given their students a title of an interesting book they just read or so...

 

[RocketSurgery]

Luckily I do not share your cynicism. There are many mathematicians and mathlovers on this board who recommend books all the time including Mathwonk. Unless Mathwonk is secretly Tom Apostol then I don't think we have much to worry about.

Being as you just joined this forum recently you will realize that a lot of the regulars here are very helpful people and not businessmen just trying to make a buck.

Welcome to the forums though!

 

[Shukie]

Math at university is a little too abstract for me, I like more hands-on math like in physics.