Should I Become a Mathematician?

[PowerIso]

Keep in mind I am speaking from my experience as an applied math guy. It's important to realize that there is a reason why it's called applied mathematics. The goal of applied mathematics is NOT to make tools for engineers or physicists, but rather to study interesting mathematical problems that may be applied but doesn't have to be applicable.

Just look at Combinatorial analysis. It can be applied to computer science, finical analysis, stats, and many other fields, however, much of the research that goes on within the field are purely mathematical questions.

Don't get me wrong though, there are a good number of applied people who do actively solve problems that can by used by engineers and physicist. If that is what you are interested in, then when looking for a graduate school in applied mathematics, try to find one that has a research group that is more about that than the what I presented earlier.

 

[mathwonk]

I am reminded that all research is the free flow of creativity and problem solving from the individual researcher, often without any focused regard for its use.

I have often made the error of assuming that research in math education was directed towards improving classroom instruction. while some is, much is just exploration of problems and concepts about learning.

I once asked a new friend who was doing research in learning psychology when his work would find its way into the classroom, and he replied he had no interest in that, but was merely engaged in "bringing order out of chaos".

 

[matqkks]

Paul Erdos said a mathematics is like a machine which coverts coffee into theorems and proof.
Marcus in his book "Finding Moonshine" says mathematician is a pattern searcher.
Lord Kelvin asked the question, whom do you call a mathematician?
He answered a mathematician is a person who finds the integral of e^(-x^2) from infinity to minus infinity as easy as you find 2x2=4.

 

[asdfggfdsa]

asdfggfdsa, classic texts in algebra are listed earlier in this thread, e.g. artin's algebra, and jacobson's basic algebra.

Thanks for the book recommendations - I have picked up Algebra, by Lang, from the uni library.

 

[mathwonk]

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

i recommend you add hungerford to it.

 

[tgt]

If there were two lecturers. One needs to refer to his notes every now and again and sometimes although rarely copies straight off his notes. The other dosen't use notes at all. Assuming talking on similar difficulty matieral and on stuff that is close to their research. Does it mean the one who dosen't use any notes at all knows the stuff much more? Is it also an indication that he who doesn't use any notes is more likely smarter and more capable?

 

[RocketSurgery]

It depends on how dependent the Professor is on them in my opinion. For instance my professor does follow a set of lecture notes he created however you can tell by his enthusiasm that he isn't simply reading his notes word for word but instead using them as a road guide.

Theres a difference between driving the car and reading the map after all
All though the driving is made easier by the directions one shouldn't be completely lost without them either if he knows all the roads. If the professor is lost without the notes then one could conclude he isn't very knowledgeable on the material he is teaching.

 

[mathwonk]

I have seen fields medalists refer to notes, so there is no easy rule on this.

also sometimes you are distracted by students or busy work just before class, and then it helps to have a some brief notes to look at to bring your mind back to the topic at hand, and remind yourself of the order you wish to say things in.

so lack of notes is a sign of recent preparation usually.

although professor john tate, or raoul bott, never used them on any occasion that i can recall, at least when teaching things elementary to them.

i also can recall bott not getting the details of some tiny calculation quite right, but I was not there to see tiny details from bott, but to get deep insights.

 

[mathwonk]

calc 3 test

Anybody want a practice test in calc 3? (vector calc)


Math 2500 sp08 Test 4, 4/25/2008 NAME
Review of operator symbols: Dx means differentiation wrt x, so “multiplication by Dx” means differentiation wrt x. Thus: If f is a function, Dxf = partial derivative of f wrt x;
and if we define “del” = ∇ = (Dx, Dy, Dz), then ∇f = (Dxf, Dyf, Dzf) = grad(f); and
if F = (M,N,P) is a vector field, then ∇×F= (DyP-DzN, DzM-DxP, DxN-DyM) = curl(F); and ∇•F = DxM + DyN + DzP = div(F).
Recall also dxdy = (dx/ds dy/dt – dx/dt dy/ds) dsdt.

(15) IA. Important theorems:
a) If C is a smooth curve going from point p to q, and f is a smooth function on C, what does the fundamental theorem of one variable calculus give as the value of the path integral (i.e. “work” for a force field, “flow” for a velocity field) of F = ∇f, along C?

b) If C is the boundary curve of a smooth surface S, and F = (M,N,P) is a smooth vector field on S, state Stokes’ thm. relating the path integral of F along C, to a surface integral.

c) If S is the smooth boundary surface of a bounded region R in 3 - space, and F = (M,N,P) is a smooth vector field on R, state the divergence theorem relating the flux integral of F across S, to a volume integral.

(15) IB. Important facts: True or false? (and briefly why or why not)
a) If f is a smooth function in a region R in space, then curl(gradf) is always = 0 in R.

b) If F = (M,N,P) is a smooth vector field in a region R in space, curl(F) = 0 in R, and C is a closed curve in R, the path integral of Mdx +Ndy+Pdz along C is always zero.

c) If G is a smooth vector field in a region R in space with curl(G) = 0, and U is a simply connected subregion of R, there is a smooth function f in R, with gradf = G.

d) If F is a smooth vector field in space, defined on two smooth surfaces S,T having the same (oriented) boundary curve, the (flux) integral of ∇×F over S, or over T is the same.

e) If G is a smooth vector field defined in all of 3 space, and div(G) = 0 (everywhere), then the (flux) integral of G over the surface of any sphere is zero.

II.a) Let R be the plane region inside the ellipse C: (x/2)^2 + (y/3)^2 = 1.
If F = (0,x), the flow of F around C is computed by the path integral ∫C x dy.
Compute this integral using the parametrization x = 2cos(t), y = 3sin(t), 0 ≤ t ≤ 2pi.

IIb) If we apply Green’s theorem to the path integral above, what double integral does it equal, over R? Compute that double integral, changing variables by the parametrization x = 2s cos(t), y = 3s sin(t), for 0 ≤ s ≤ 1, 0 ≤ t ≤ 2pi, and the “recalled” formula for dxdy.
(You should get the same result. What geometric quantity have you computed?)

III. Let H be the hemisphere of radius 2, x^2 + y^2 + z^ 2 = 4, z ≥ 0, and
Define the vector field F = (xz, x + yz, y^2).
a) Compute ∇×F =

b) Show the flux of ∇×F outward through H equals 4pi, in one of these ways:
i) Explain, with minimal computing, why it equals the area of the circle x^2 + y^2 = 4.
ii) Compute it as a path integral using Stokes.
iii) (last resort) parametrize H and actually compute the flux integral.

IV. Let S be the boundary surface of the solid tetrahedron T with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1), and let F = (xz cos^2(z), yz sin^2(z), yx).
a) Compute div(F) = ∇•F = ?

b) Compute the flux of F outward through S either directly as a surface integral (masochists only) or by using the divergence theorem.

 

[k3N70n]

Hi Mathwonk.

I'm looking to pick up a book in Algebra to work through over the summer. I took a course last fall where we used "Contemporary Abstract Algebra" by Gallian, so I was curious if you could recommend a good text to follow up. Thanks in advance.

-kentt

 

[Cincinnatus]

k3n70n, mathwonk already answered that question like 800 times on this board. it's even answered on this very page of this thread!

He's answered it enough times that I can do it for him. He doesn't have a high opinion of Gallian's book (which I share). He usually recommends Artin's book for undergraduates.

 

[k3N70n]

Sorry about that. I should have looked. Thanks Cincinnatus.

 

[tgt]

Mathwonk, what do you think of attending seminars when you know you will not understand a thing? It probably dosen't happen to you much but for beginning grad students, this can happen a lot. Would you advise to not go and do something more productive instead like one's own work? So only go to ones that you have some idea of?

 

[Cincinnatus]

Sometimes the free food makes it worth going.

 

[mathwonk]

actually it still happens every seminar i go to, but it is still worth it if you understand even one thing. and as just observed, there is always the cookies and coffee. and sometimes homemade brownies.

 

[k3N70n]

k3n70n, mathwonk already answered that question like 800 times on this board. it's even answered on this very page of this thread!

He's answered it enough times that I can do it for him. He doesn't have a high opinion of Gallian's book (which I share). He usually recommends Artin's book for undergraduates.

Is Gallians book really so bad that I should read Artin's. If I've already gone through most of Gallians how much of the material will be rehashed again in what I'm sure is a better book? Isn't there a better book that would lead itself to someone who's gone over the basics? Or is Artin's book really that much better?

 

[eastside00_99]

Is Gallians book really so bad that I should read Artin's. If I've already gone through most of Gallians how much of the material will be rehashed again in what I'm sure is a better book? Isn't there a better book that would lead itself to someone who's gone over the basics? Or is Artin's book really that much better?

I personally don't like Artin. I'm using it in a graduate sequence in abstract algebra. I think it would be better to use Artin (starting at chapter 1) if you have never had any algebra. I am including here linear algebra. If you do it that way, then I think Artin's Algebra is a great book. But, if you have already had a course in linear algebra and abstract algebra, I think it would be best to use something else. Dummit & Foote seems to be the standard.

 

[mathwonk]

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, perhaps such as Dummit and Foote (or certainly 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 contains 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.

 

[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.