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Good Physics Books For a Math PhD Student?
Posted by
kdawson
on Mon Nov 17, 2008 01:37 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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Firehose: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: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: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
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
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: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: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
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
Some recommendations from another Math Ph.D (Score:5, Insightful)
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: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: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
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