Other Should I Become a Mathematician?

[Gib Z]

Why don't you accelerate if you feel that confident that your so good? I know that the university of Sydney has an accelerated program but that was just for students who was in one of the three science Olympiads (Physics, Biology and Chemistry). But i would imagine that if you feel confident then you can take some of the course tests (should be on the internet) and if you score quite well, i would suggest you see the dean or physics head or what ever.

I went for work experience at USYD's math department and specifically asked if they had an acceleration program and they told me no! :( Why would they offer such a program to those who are in a science Olympiad program?

 

[tgt]

Just wondering how you like to tell others (when asked what you do) that you are a maths professor. What are their reactions? I find that the general public are impressed enough by a maths student. I just like to know what the reaction is when they hear you are a maths professor. Do you find that you are very respected being a maths professor?

 

[mathwonk]

I don't have a huge social life, so do not often tell it.
oh yes, and because i try to minimize the shock, i usually tell people i am a "math teacher".
this does not impress too many of them, since i do not use the "professor" title, unless pressed.

in fact when i started posting here, i declined to say i was a math professor for a long time, until quizzed about it.
up until then lots of people argued with me over my statements about math, and it bugged me that afterwards my opinions on math received more weight than they had before.

i.e. i started out believing that anonymous correct answers to math questions would impress people just by being correct, but eventually found that more people think i must know something about math because i am a professor, than think i am a professor because i know something.

 

[tgt]

I don't have a huge social life, so do not often tell it.

oh yes, and because i try to minimize the shock, i usually tell people i am a "math teacher".

this does not impress too many of them, since i do not use the "professor" title, unless pressed.in fact when i started posting here, i declined to say i was a math professor for a long time, until quizzed about it.

up until then lots of people argued with me over my statements about math, and it bugged me that afterwards my opinions on math

received more weight than they had before.

i.e. i started out believing that anonymous correct answers to math questions would impress people just by being correct,

but eventually found that more people think i must know something about math because i am a professor,

than think i am a professor because i know something.

By stating that you are a maths teacher, it could mean a primary school maths teacher so that would lessen the effect drammatically. It's funny because the average Joe might think more highly of a uni maths student then you, a 'maths teacher'.

Do you get treated really well when they do finally find out that you are a maths professor?

But I tend to be like you and don't like to show off too much. Maybe all mathematicians are like that?

 

[mathwonk]

well i have become more modest as I got older. maybe i realize that i have good reason to be modest.

Physical anomaly, I apologize for teasing you. You are in a position of needing guidance. Your ability is a blessing. There are many good books you can enjoy and be challenged by.

best wishes.

 

[mathwonk]

yes math professors do enjoy a certain fear/respect from many strangers. it does not last much past getting to know you though. then you get the treatment your personality commands or fails to command. i.e. people assume you are smart until you open your mouth too often.

 

[RufusDawes]

Just curious if any of you are planning to become a high school mathematics teacher ?

 

[mathwonk]

i would like to become one after i retire from university teaching, but i don't know if i can get hired, and I may not have the stamina to deal with teenagers.

 

[eastside00_99]

Hey mathwonk, I need some advice about the qualifying exams. I am going to CUNY, and there are 6 areas (out of which one choose three) that offer qualifying exams. You have two years to complete the exams, they are given three times a year, and you can only fail one exam twice before having to pick a different area. Here are the six areas:

1) Real Analysis
2) Complex Analysis
3) Algebra
4) Differential Geometry
5) Topology (starting with general topology)
6) Logic

At my undergraduate school, I took the graduate qualifying sequences in Algebra, Algebraic Topology, and Real Analysis. I also took half the qualifying sequence in Differential Topology. I have not had ANY complex analysis or Logic. So, basically, the way I see it, I have three options:

1) Take the sequences that would most prepare me for specializing in Algebraic Geometry (which is quite a big subject I know). As I see it, if I follow this plan, this would mean take the following sequences: Algebra, Topology, Differential Geometry.

2) Take the sequences that I know the least about as I probably have enough (not sure how to qualify that word enough though) knowledge of the above three areas to specialize in Algebraic Geometry. This would mean take: Logic, Complex Analysis, and Differential Geometry.

3) Just take the exams that I know the most about without necessarily taking the corresponding classes. For instance, I could study all summer for real analysis and algebra, and before the semester starts, take the exams. I would be using one of my chances, but the good thing is that you are not kicked out for failing an exam twice--you just have to choose a different sequence.

I think I may attempt the real analysis exam at the end of the summer regardless of which plan I take. Also, I need to talk to a few people about this. I know a lot of students who find out what exams are the easiest and then take those. But, I don't want to do that. Of course, I would not punish myself by taking the hardest exam just because it is hard and no body passes. I am open to advice which ones would you recommend?

 

[asdfggfdsa]

Hi, mathwonk:

I've been thinking about becoming an algebraist (after completing a course in Galois theory) - are there any texts which you would consider classics in algebra?

asdfg

 

[tgt]

When you lecture at uni, do you need to look at your notes once in a while to keep track or can you walk in without any notes teaching a full lecture without referring to any notes?

 

[mathwonk]

eastside, it is good advice to just get the quals out of the way as quickly as possible. so i would take them in the areas i knew best, and can prepare for soonest.

it is also good advice to learn something about complex analysis, since beginning with riemann it has been a key tool in doing and understanding facts from, algebraic geometry.

 

[mathwonk]

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

 

[mathwonk]

i seldom look at notes. i find it better to go through a calculation without notes, since that forces me to actually see what i am doing, and then maybe the class will see too.
once i had a post class evaluation that criticized me as follows:

"this man comes to class with just a box of chalk and a sponge to erase the board,
no lesson plan at all!"

of course the lesson plan is in my head, and i have filled up many pages with calculations the night before, which there is no need to consult again in class.

usually the only time i have notes, is when i do not understand what i am presenting, but sometimes i write out and copy a complicated calculation, or at least I may copy the problem so it will be one with numbers that will come out nice.

 

[mathwonk]

i am usually not trying to present a canned set of information for people to memorize, but to show a way of thinking about the topic.

i try to show what to do first, then second, and so on,...

i am always trying to prepare people for that moment when they are alone with a problem.

i.e. where do we begin? how do we remember key formulas? how can we recover them if we forget? how can we shortcut the work in special cases?

usually this can only be done by remembering what the calculations mean.
e.g. some books teach multiple integration, and then how to compute them by repeated integration, then they state greens theorem but say they will not prove it.

In fact they have already proved it, since just looking carefully at what repeated integration says, shows that it may be stated as greens theorem.

i.e. greens theorem computes a path integral as a double integral, but repeated integration computes a double integral as a moving family of single integrals, which is just a path integral around the boundary of the double integral region, i,.e. greens theorem.

even earlier, seeing that repeated integration works is just seeing that the derivative of the moving volume function, is the height function. but to see this one must know the meaning of the derivative as a limit of ratios [in this case volume/area = height] , not just know a bunch of derivative formulas.

 

[mathwonk]

the first 4-5 pages of this thread have a lot of book recommendations, but the specific cheap copies i located then are surely gone by now.

 

[tgt]

i seldom look at notes. i find it better to go through a calculation without notes, since that forces me to actually see what i am doing, and then maybe the class will see too.


once i had a post class evaluation that criticized me as follows:

"this man comes to class with just a box of chalk and a sponge to erase the board,

no lesson plan at all!"

of course the lesson plan is in my head, and i have filled up many pages with calculations the night before, which there is no need to consult again in class.

usually the only time i have notes, is when i do not understand what i am presenting, but sometimes i write out and copy a complicated calculation, or at least I may copy the problem so it will be one with numbers that will come out nice.

I find that a lot of younger presenters need notes. The older professors don't need them. Even if you know how to prove any theorem and do any problem, how do you keep track of the order in which you want to present the material? Or is there a natural order in your head which come to you easily?

I guess the ultimate test for your knowledge of some material is if you can present it without referring to notes?

 

[mathwonk]

kurt,

well that would be a different world from the one i inhabit. i struggle with many of my students to get them to even think about math as a process of reasoning rather than computation.

since anyone can teach strong students, the older you get and the more experienced you become as a teacher, it can happen that the more you are asked to teach weaker students, and leave the teaching of more creative ones to younger colleagues.

IN my whole life I have only had one teacher, a great inspiring graduate algebra teacher, maurice auslander, do something like what you said, but even then he only handed out very terse notes in which he had sketched the proof of a very deep result he was proud of, (all regular local rings are ufd's, 1965), and made it the goal of our semester to read and understand the proof.

as to presenting a problem and arriving at a proof of it during the semester, i proposed that once in a faculty seminar, and even there some audience members were astonished at the optimism of the idea.

bott on the other hand, at harvard, used to present hard problems in grad classes, and according to lore, once challenged a class including john milnor with an unsolved problem that milnor actually solved as if hw.

i myself also was in a class at harvard where hironaka challenged us with a hard but preliminary version of an open problem, that was soon solved by his future phd student mark spivakovsky.

but i am usually so isolated from such students that recently when i wrote an honors calc exam, from long habit i made it too easy, and left off some thoughtful questions i later wish i had asked.

here is one i decided would be too theoretical for my undergrads, to my regret, as i would have liked to see what they did with it:

Assume f is differentiable on some interval [a,b], that f '(a) > 0, and f ' (b) < 0, but not that f ' is continuous.
i) Use the definition of derivative to prove there is some e >0 such that f(x) > f(a) for all x in the interval (a, a+e), and f(x) > f(b) for all x in the interval (b-e, b).
ii) Assuming standard theorems from diff calc, prove f '(c) = 0 for some c with a < c < b.

you see i am only asking them to understand the meaning of differentiability, and use that understanding to derive the intermediate value property for possibly discontinuous functions which are known to be derivatives of other functions. but i lost my nerve about asking even this of a group of honors level undergraduates. in hindsight however i should have done so, as they had already seen many of the more standard problems i did ask, and some of them were very creative and insightful, and i would like to have seen how they handled this slightly offbeat problem.

 

[mathwonk]

tgt, it is usually only possible to do one thing in one class, so the order of topics is not too important.

usually the order is as follows:
introduce and motivate the topic with an interesting problem.
take guesses as to how to solve it.
either run with any good ideas wherever they lead,
or at some point lead the discussion to the tool you want to present, and present it,
making it as precise as necessary.
give examples of the workings of the tool, with specific numerical computations.
give homework to reinforce it.

 

[Cloud]

Applied Mathematics?

Hi,

I finished with Computer Engineering and Electrical Engineering for my undergraduate degree. Thinking about pursuing MS and may be PhD if I can totally absorb into it.
But I find it difficult to choose among engineering/applied Mathematics/Physics. I roughly aim for applied mathematics for now and applying schools. Can you please advise me on this matter? Thank you in advance.

 

[mathwonk]

at my university we struggle to teach students to stop expecting us to use class to carry out model calculations for them to imitate later, and to begin to appreciate that we are there to help them understand the meaning of the calculations, and the theory behind them. the specific calculations are for them to practice at home.

at some schools, the teachers just read and explain the book in class, at others they expect the students to do this at home, and in class they show what the material is good for, and how it can be extended. the teacher at a school like harvard introduces material in class that he/she knows from their own expertise, that is not found in the books.

there is a constant struggle to increase the depth of the students' experience, without submerging their heads under more than they can absorb.

of course occasionally i have students so strong i myself cannot keep up with them, but only occasionally, (every decade or so?).

 

[mathwonk]

I cannot advise on applied math, but perhaps others will?

 

[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.