tgt
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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?
Should I Become a Mathematician?
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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.
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.
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asdfggfdsa, classic texts in algebra are listed earlier in this thread, e.g. artin's algebra, and jacobson's basic algebra.
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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.
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.
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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.
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.
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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
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mathwonk said:
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?
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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.
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.
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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.
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
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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.
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.
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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?).
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?).
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I cannot advise on applied math, but perhaps others will?
PowerIso
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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.
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.
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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".
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
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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.
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
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mathwonk said:
Thanks for the book recommendations - I have picked up Algebra, by Lang, from the uni library.
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well lang is good but not sufficient, as it is all theory and no examples.
i recommend you add hungerford to it.
i recommend you add hungerford to it.
tgt
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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 doesn'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 doesn'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
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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.
there's 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.
there's 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.
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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.
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.
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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.
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
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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
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
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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.
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
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Sorry about that. I should have looked. Thanks Cincinnatus.
tgt
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Mathwonk, what do you think of attending seminars when you know you will not understand a thing? It probably doesn'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
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Sometimes the free food makes it worth going.
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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
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Cincinnatus said:
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
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k3N70n said:
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.
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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.
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.
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