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Good Physics Books For a Math PhD Student?
Posted by
kdawson
on Mon Nov 17, 2008 12:37 AM
from the queen-of-the-sciences dept.
from the queen-of-the-sciences dept.
An anonymous reader writes "As a third-year PhD math student, I am currently taking Partial Differential Equations. I'm working hard to understand all the math being thrown at us in that class, and that is okay. The problem is, I have never taken any physics anywhere. Most of the problems in PDEs model some sort of physical situation. It would be nice to be able to have in the back of my mind where this is all coming from. We constantly hear about the heat equation, wave equation, gravitational potential, etc. I'm told I should not worry about what the equations describe and just learn how to work with them, but I would rather not follow that advice. Can anyone recommend physics books for someone in my position? I don't want to just pick up a book for undergrads. Perhaps there are things out there geared towards mathematicians?"
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Submission: Good Physics Books For Math PhD Student by Anonymous Coward
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I'm gunna say this once.. (Score:5, Funny)
Get back to writing your thesis.
Slacker.
Re:I'm gunna say this once.. (Score:4, Funny)
I'd rather not follow that advice.
Parent
Re:I'm gunna say this once.. (Score:5, Funny)
That proves it! Only a PhD student would say that.
Parent
Re:I'm gunna say this once.. (Score:4, Interesting)
> So what kind of university are you at?
An American one. They specialize *much* later than the English [who are down to 3 subjects, eg Maths, More Maths, and Physics] for A levels [at 16]. Once at university Americans still have to take a relatively broad range of classes for the first 2 years. So, even in good US universities the first year or two only gets them up to A level standard. Top performers make up for it later on because after being held back from focusing only on what interests them for so long its such a relief to be able to concentrate that they really get stuck into it.
Parent
Re:Standards have slipped then... (Score:5, Informative)
This might come as shocking news to you, but the typical undergraduate PDE class only scratches the surface of a rather deep and broad subject. From the examples you list, it seems that you only worked with equations for which global existence and regularity are trivial, and you have lots of conserved quantities. Many aspects of PDEs are fields of active current research, including heuristics for fluid mechanics modeling, theoretical questions concerning geometric structures on manifolds (see Yang-Mills or Seiberg-Witten equations), and integrable hierarchies. I'm not a specialist in PDEs, but I'm sure there are others who can list much more, and describe interesting open problems in detail.
Also, I should point out that the lack of a required PDE class does not necessarily mean standards have slipped. If you look at the requirements for a major in the top math departments in the US, you'll find that they have few required courses, and many options. I think these departments have decided that students should have freedom to focus on their interests after they have learned some fundamentals, and that there are other areas of mathematics, such as abstract algebra, topology, and combinatorics, that may hold their interest. I have met many mathematicians who have little experience with even the heat and wave equations, and they have done fine, because their work was not related to these questions. It is possible that the OP has taken a similar educational track.
Parent
Books (Score:5, Informative)
They Feynman Lectures on Physics would probably be a good place to start. It'll be basic to advanced.
http://www.amazon.com/Feynman-Lectures-Physics-including-Feynmans/dp/0805390456/ref=pd_bbs_sr_2?ie=UTF8&s=books&qid=1226900482&sr=8-2 [amazon.com]
If you want something more specific, to a topic, there will be a slew of books. I found some pretty good ones following links on Amazon from one to another and reading reviews.
Re:Books (Score:4, Insightful)
Parent
Re:Books (Score:5, Informative)
I just thought of another one. It's Mathematical Methods for Physicists by Arfken. I wouldn't necessarily recommend buying it, but find one you can flip through (most university libraries have it, as do most math/physics department libraries. and I can almost guarantee that someone you know has this book).
http://www.amazon.com/Mathematical-Methods-Physicists-George-Arfken/dp/0120598760/ref=sr_1_5?ie=UTF8&s=books&qid=1226903092&sr=1-5 [amazon.com]
It's a math text, but since it's geared as a math text for physicists, the explanations may have the right amount of physics in them.
(I've always liked it as my math reference).
Though, I don't think this will be at your level (probably below), but it may help with the ground work. As I said, don't buy it, but find a copy to flip through.
Parent
Re:Books (Score:4, Informative)
My wife, who's a Math PhD who does tons of PDEs in her field (Fluid Dynamics) seconds the Charlie Harper book.
She didn't even look up from her work to answer when I hollered the question to her. She says "Matthews and Walker" too, but doesn't remember the title. I don't see it up on the shelf, so it might be in her office, which means I can't relay the exact title, but if you look for "Matthews and Walker" you will probably find it.
Parent
Re: (Score:3, Informative)
Re:Books (Score:4, Interesting)
As a 4th-year Physics undergrad, I have to voice my opinion that I absolutely can't stand Feynman's texts.
They're nice to glance at, but approach the subject in a considerably different manner than any of the other renowned physics texts.
Similarly, his proofs were terse to the point of being difficult to follow. I'll admit that my mathematical intuition isn't the greatest, though I can't help but think that this was intentional on Feynman's part, as to weed out those with weak mathematical skills from his freshman lectures. This makes them rather frustrating to use as a general reference. Similarly, the texts are largely theoretical, and offer little advice with regard to problem-solving.
Personally, I've had good experiences with the Landau/Lifshitz series of texts, and it's hard to go wrong with Griffith's books on EM and QM. Goldstein's text on Classical Mechanics is also a well-known classic.
That's not to say that that Feynman's texts are all bad. Some sections are outright brilliant, and he actually takes the time to explain himself rather extensively in many sections, which many physics (and math) writers frequently neglect to do. I keep a copy of all 3 volumes on my bookshelf, as they are occasionally handy. However, I wouldn't dream of using them as my only reference.
Parent
Re:Books (Score:4, Insightful)
I'll admit that my mathematical intuition isn't the greatest, though I can't help but think that this was intentional on Feynman's part, as to weed out those with weak mathematical skills from his freshman lectures.
Bear in mind, the classes these lectures were delivered to were at Caltech in the 60s, I believe. Those with weak mathematical skills didn't get in.
Also realize that many of the undergrad lecturers at Caltech take it as a badge of honor to see how much they can shovel at the undergrads, equating density and difficulty with learning. You might find that nearly all of the students in the class were spending quite some time poring over those lectures to figure them out - not because the profs wanted to weed them out, but because that's simply how things were/are done at Caltech. On the other hand, that's something you didn't want/need to do, valuing your own time and sanity, and not staring an "F" in the face if you didn't.
I'm in the same boat, I wouldn't have stood a prayer in that environment either.
I keep a copy of all 3 volumes on my bookshelf, as they are occasionally handy. However, I wouldn't dream of using them as my only reference.
Yeah, Feynmann wouldn't make a good reference but he's definitely entertaining and insightful. Probably about like Ambrose Bierce in that regard.
Parent
Re:Books (Score:4, Insightful)
All the best, are.
Parent
He said "Mathematician" (Score:5, Insightful)
The mathematician goes off for three weeks filling in all the gaps and "leaps of faith".
He comes back to the book, and reads page three.
Mathematician flings book against the wall, and goes off and finds something more rigorous to read.
As I remember them, the Feynman lecture series were finely crafted instruments of torture for those who delight in rigor. Personally I think he entitled the wrong book "You must be Joking!"
Parent
Re:He said "Mathematician" (Score:4, Interesting)
You've got to remember that there was an awful lot that was obvious to Feynman - hell he won the Putnam without breaking a sweat. He ran into a classmate who wondered why he wasn't taking the Putnam exam and Feynman told him he'd finished the exam. The interchange took place when there were a couple of hours left on the exam clock and none of the other contenders completed the test in the allotted time.
He felt that Mathematicians spent an awful lot of energy developing stuff that was obvious, and hence a waste of his time. He used to harangue math graduate students that if they could clearly state what they were working on, he could reproduce and finish what they were doing within the evening. The thing was, he could do it. He was far more interested in why things worked the way they did rather than proving that the math he was using was correctly applied - the results mattered to him far more than the technique.
He used to say that the renormalization techniques he used developing QED which won him the Nobel Prize probably weren't kosher math but they produced the right answer to the tenth decimal place.
In the end, that's what doomed the Feynman Physics Undergraduate books - they were simply too advanced for the vast majority of their intended audience. While he was giving the lectures, the undergraduate attendance declined while the graduate attendance increased thereby keeping the room full which misled him as to how clearly he was teaching his intended audience. It wasn't until the mid terms came in that he realized something was amiss. If the average Caltech student couldn't suss what he was saying, it's a fair bet few other physics undergrads would be able to. The graduate students, and other faculty, on the other hand, loved the class because it gave them insights into topics they thought they completely understood.
Parent
PDEs now? (Score:5, Insightful)
You are in your third year of a PhD program and are only now studying PDEs? Aren't they more of an undergrad topic, or have schools gotten weaker? :)
p.s. First post!
Re:PDEs now? (Score:5, Insightful)
There can be a world of difference between graduate and undergraduate PDE courses; it's not like everything that's known about PDEs can be taught in a couple of undergraduate semesters. I expect most undergrad PDE courses are geared towards showing you the methods that work for a few classes of linear PDEs; a graduate course might be concerned with the analytical underpinning of those methods, or maybe about numerical and analytic techniques that are useful in solving classes of nonlinear PDEs, etc.
That being said, though, from the way the original question is worded, it sounds like it's the first time this person has seriously encountered PDEs. Not having this happen until the third year of a PnD program does seem a little odd.
p.s. No, you're not.
Parent
Re: (Score:3, Interesting)
No. ODE's are typical of Undergrad. But, PDE's are typical of Masters. That isn't to say that PDE's are taught in Undergrad, period. Rather that PDE's in Undergrad is atypical. At least in North America. Other parts of the world either have vastly superior high-school/Undergrad or skip a lot of the, necessary for actually understanding, stuff. Germany and China are respective examples.
Re:PDEs now? (Score:5, Insightful)
PDEs are not normally part of a math degree. They do form the central basis to applied math degrees. People in the engineering and physical sciences have a great understanding of applied math, but they have little to no understanding of pure math. If you get a BS in a physical science or a BE from any decent university, you will basically have a minor in applied math (adv. calc, ODEs, PDEs, probability, statistics, nonlinear dynamics, complex analysis, and calculus of variations). But you have not even scratched the surface of pure math. Mathematicians worry primarily about pure math. To teach PDEs would be insulting to them due to its lack of generality. As many physicists and engineers have learned over time, if you have a difficulty in understanding mathematics that applies to your field, the worst person you can go to for help would be a mathematician that hasn't studied applied math. The best person you could go to would be a mathematician who specialized in applied math.
Parent
Re: (Score:3, Informative)
Nope, I'm currently doing a second degree in my spare time and did this stuff last year (my second year) and my degree isn't even a full maths degree, only 50% of it is maths.
What on earth is a 3rd year Phd maths student doing only doing PDEs now??? This really is undergrad stuff. I understand these topics can go more advanced but the stuff described sounds like the basic undergrad stuff.
I wonder if perhaps the person asking the question actually means they're a 3rd year undergrad student who wants to do a
Re:PDEs now? (Score:5, Informative)
You both probably studied how to solve certain simple PDEs in simple geometries (like the heat, wave, and Poisson equations). At a graduate level one normally learns how to prove existence and uniqueness of solutions to PDEs, how smooth those solutions are (i.e. how many derivatives do the solutions possess), and how to define weak forms of PDEs for which non-classical solutions exist (solutions that are not necessarily even continuous). Then there is the whole area of non-linear equations which is a very active research topic... (See the Navier-Stokes Equations.)
Parent
Re: (Score:3, Informative)
It says right there in TFS that he hasn't studied the heat or wave equations.
Re:PDEs now? (Score:4, Informative)
You both probably studied how to solve certain simple PDEs in simple geometries (like the heat, wave, and Poisson equations). At a graduate level one normally learns how to prove existence and uniqueness of solutions to PDEs, how smooth those solutions are (i.e. how many derivatives do the solutions possess), and how to define weak forms of PDEs for which non-classical solutions exist (solutions that are not necessarily even continuous). Then there is the whole area of non-linear equations which is a very active research topic... (See the Navier-Stokes Equations.)
Clearly graduate level approaches to PDEs differ from undergrad approaches.
However, the topics you suggested as grad level were mostly introduced to us at undergrad level (year 3 of 4 year course) in Chemical Engineering, and that was 30 years ago. Yes, we studied existence, uniqueness, and smoothness of PDE solutions. We also studied the diffusion/heat equation with moving boundaries (diffusion with reaction), and coupled instances of the diffusion equation (interphase transfer).
The Navier-Stokes equations were introduced, but not studied until graduate level. I think that generic numerical solvers are used nowadays for simple NS problems (they were PhD stuff in those days), but analytic underpinnings are reserved for grad school.
Parent
Making the math tangible does help (Score:5, Insightful)
Some essentials (Score:5, Informative)
Goldstein, Classical Mechanics. Standard grad level mechanics, solid book, mathematically rigorous yet still intuitive.
For EM and Quantum, even a math grad should read the advanced undergraduate books by Griffiths:
Introduction to Electrodynamics
Introduction to Quantum Mechanics
For thermodynamics, I don't know the best text.
For General Relativity, the standard undergrad book is Hartle's Gravity. But since you're a math PhD, you can go straight to the finest first grad level Relativity book by Sean Carroll:
Spacetime and Geometry
If you're looking for intuition, the indispensable and invaluable books are Feynman's Lectures on Physics.
I can recommend mathematical physics texts, but I get the impression you want the missing background for understanding. Hope this is helpful.
Re:Some essentials (Score:5, Informative)
I'd like to second all of these recommendations, but for Quantum Mechanics if your linear algebra is sharp, I might suggest Principles of Quantum Mechanics by Shankar.
Griffifhs' Quantum Mechanics is an excellent introduction, but it assumes relatively little math knowledge, and tends to gloss over some of the assumptions being made. This is good for a student who's going to spend most of his effort trying to learn the practical aspects of doing Quantum Mechanical calculations, but not ideal for someone who grasps the math quickly and easily, and wants to really understand how things work.
Shankar is a little more difficult mathematically (and is thus often a poor introduction for an undergrad) but it very clearly lays out the assumptions being made, and how the math relates to the physics.
I haven't actually read the Sean Carroll book, but I took a course from him, and I can't imagine the book is anything but excellent.
Parent
Re: (Score:3, Informative)
Griffiths QM book is absolutely terrible. All it does is skim the surface. Greiner is vastly superior. Griffiths E&M book is good though.
A survey of the best (Score:3, Informative)
My favorites (Score:3, Informative)
I think the best book for what you are asking (and I am 95% sure this is the right book, but I've lent it out so I had to look it up from dover) is "Vector Analysis" by Homer E. Nowell. It develops the theory of vector calculus using an intuitive approach and builds up the theory of electromagnetism simultaneously.
You might also look into the Feynman lectures. I do not normally recommend them as 'learning' material because, while excellent, I'm not aware that they come with any problem sets. But for you they may be a good supplement.
And, just to throw it out there, but it seems to me that most technical schools have enough overlap between physics degree requirements and math degree requirements that if you have a reasonable interest in the other it might not be out of the question to work that into your curriculum.
Yes, stick to the mathematics. (Score:3, Insightful)
What? (Score:4, Funny)
Why are you taking partial differential equations as a graduate student?
Re:What? (Score:4, Funny)
Parent
Re:What? (Score:5, Informative)
Contrary to what most people seem to think, the material taught in most Calculus and Differential Equations courses has very little resemblance to what most Mathematicians study. These fields actually all fall under the heading of Analysis, which is just one of several major branches of mathematics. A student not interested in analysis could easily spend most of his math career working in another area.
For the most part, differential equations courses are aimed at non math majors, such as physicists, chemists, engineers, and the more analytically minded biologists and economists, so even a Math major specifically interested in analysis isn't necessarily going to take classes on partial differential equations.
I myself double majored in Physics and Math, and every single course i took about differential equations was for the Physics major rather than the math Major, so I think that Math grad student could quite easily end up with a PhD without ever dealing with differential equations unless they interested him.
Parent
Re: (Score:3, Insightful)
Wow, the level of ignorance here is astounding, that you would get moderated so highly. Real PDE (as mathematicians study it) is HARD, and requires a heavy background in analysis. This is not the same as undergrad "PDE" courses.
This is like the high schooler saying "Why are you taking algebra as an undergrad" to a math major studying abstract algebra. Its the same word and the topics are related, but its not even close to the same thing.
Some recommendations from another Math Ph.D (Score:5, Insightful)
Re:Some recommendations from another Math Ph.D (Score:4, Interesting)
Parent
Re:Some recommendations from another Math Ph.D (Score:4, Interesting)
Jumping Jesus on a pogo stick, you're pointing him to The Black Death straight out of the gate? Why not give him underwear made of wolverine chow? Wheeler would have died ten years ago if not for the life-giving tears of those who opened that book unprepared. That is to say, everyone.
Seriously, dial it back a bit. First, hit the Feynman lectures (stop when you get to 'partons'.) Then, for someone coming from a mathematical bent, I'd suggest starting with Sokolnikoff's book "Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua", which covers a lot of ground besides GR. Due to the absence of a just and loving god it is out of print, but surely one of the profs in a math department with a PhD program has a copy (or at minimum the library.) And there's always copies on Alibris.
And, seconding suggestions from other posters, Kittel and Kroemer's "Thermal Physics" is a good starting point on thermo, As for quantum, in the absence of all knowledge in the field I'd start with Tipler's "Modern Physics", with the goal of ramping up to Cohen-Tannoudji, Diu, and Laloe's "Quantum Mechanics".
Parent
Physics/Astronomy Graduate student perspective (Score:3, Informative)
Enter the Physics vs. Math Holy War. (Score:5, Funny)
I love watching this one happen.
It's funny because no matter what, the only thing a physicist and a mathematician has ever been able to agree on is magic mushrooms.
Road to reality (Score:5, Informative)
An excellent Physics book that is very math heavy but assumes no prereqs is Penrose's Road to Reality [amazon.com]. This pretty much covers all of the main theory/formulas in cosmology, and he has 350 pages of math (much of it graduate level) to get there.
The Feynman Lectures on Physics (Score:5, Informative)
I can not recommend these books enough. Feynman does a brilliant job of bringing the concepts of physics to life.
All together, they are quite extensive, but the individual topics are brief enough to digest in one sitting. Wether you only have a passing interest in physics, or a graduate degree in the field, you will find that there is much to appreciate in these lectures.
Even for those simply taking physics as requirement, I think that these would give you a real appreciation of the field, and probably make the classes a lot easier at that.
They're All Targeted for Mathematicians (Score:5, Informative)
I've a couple of degrees in Physics, and I assure you, half the print in the _vast_ majority of Physics books is equations. Most physics texts seem to assume a math minor. Most Physics majors first see partial differential equations, special functions, and group theory as undergraduates. A couple of friends took partial diffeq for fun. Yeah, that's one way to know you're a nerd.
I suggest a library or a used bookstore, as these things are expensive. Here are some of the typical texts you see around on various physics topics (by author's name, because the titles are useless):
Electromagnetism:
Griffiths is a really great undergrad book, which is easy to read.
Jackson is the classic first semester grad-school book.
Math Methods of Physics:
Arfken is a classic.
Cantrell is an up and coming variant.
Thermodynamics:
Kittel is an oldie, but a goodie. Someone else prolly has a better suggestion.
General Undergrad Phenomonology:
The World Wide Web - Invented at CERN, y'know.
Halliday & Resnic is probably the easiest book to find.
Serway is newer.
Relativity:
Rindler is the standard.
Mechanics:
Goldstein is pretty easy to find.
Quantum:
Landau (yep, the same) and Lifshitz is a solid text that
hits on Shcrodinger's equation well.
Griffiths is easier to read, as is Eisberg & Resnick.
Modern Physics:
Less of an obvious choice, but it'll be a good source for more sexy topics.
A lot of partial diffeq is used in mechanics. IIRC, partial diffeq was invented to describe mechanical systems, so many of the examples are very intuitive (for you of course, not for 99.9% of the population.)
Interestingly enough, this Wikipedia link http://en.wikipedia.org/wiki/Partial_differential_equation [wikipedia.org] can take you many places, as it seems to come from the mind of a physicist more than a mathematician.
Alternately, you will probably have success finding a physics student at your relative level that has the intuitive feel, but is weak on math. You could quite a bit from each other in short order.
may the electromagnetic force be with you,
-Rick
Re:Man... (Score:5, Funny)
No, I'm in the exact opposite situation. I don't know anything about PhD level math _or_ physics.
Parent
Re: (Score:3, Insightful)
Re:What I want to know is... (Score:5, Funny)
If you have never taken any psychology classes, you do NOT have a broad education. Period.
If you have never taken any philosophy classes, you do NOT have a broad education. Period.
If you have never taken any accounting classes, you do NOT have a broad education. Period.
This is fun!
Parent
Don't be an ass. Oops, sorry, too late... (Score:3, Informative)
A degree in mathematics, from a responsible university, should include at least some physics. And of course a degree in physics requires a certain minimum of math, or
incorrect (Score:5, Insightful)
Most areas of science strongly rely on philosophy, and most scientists understand it poorly, usually to the detriment of the technical quality of their work. You can see this all the time, from physicists publishing embarrassingly poor papers on how quantum mechanics "disproves free will" (apparently without even an undergraduate understanding of free will), to AI researchers with little background in philosophy of mind, to statisticians rediscovering the problem of induction every few years. Not to mention the very naive understanding of the "scientific method" that an intro course in philosophy of science might be useful in addressing.
In any case, pure (as opposed to applied) math has not very much to do with the hard sciences. And there is furthermore just not enough time to fit in everything people need. A good understanding of computer science is, for example, required for most technical fields these days as well, and also fairly under-taught; probably I'd put it ahead of physics in importance to most non-majors.
Parent
Re:Partial differential equations (Score:5, Informative)
Good thing you weren't modded up. Basically nothing you said was enlightening or even correct, except for the contents of the first sentence.
You didn't even bother to correct the OP, you just sat back and decided to be a useless pedant. Yes, OP is technically incorrect, but your post is uninformative and completely worthless.
All possible partial derivatives of a point on a 3-dimensional graph fall on a tangential plane. Usually we speak of a tangent line, setting x or y constant, but if one redefines the coordinates, then any line on that plane that passes through that point is a partial derivative. So that "partial derivative plane" contains all possible partial derivatives of that point. This designation is intuitive and not particularly misleading, so there was little point in being an ass about it.
Parent
Re:3rd year PhD student taking PDE? (Score:5, Funny)
Parent
Re:3rd year PhD student taking PDE? (Score:4, Funny)
An algebra? You mean there's more than one?
Parent