Good Physics Books For a Math PhD Student? 418
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?"
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.
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.
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.
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.)
Physics/Astronomy Graduate student perspective (Score:3, Informative)
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.
Re:Some recommendations from another Math Ph.D (Score:2, Informative)
Here's a good thermal book I used in my Undergrad.
http://www.amazon.com/Thermal-Physics-2nd-Charles-Kittel/dp/0716710889/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1226902024&sr=8-1 [amazon.com]
Also had a bit from http://science.slashdot.org/comments.pl?sid=1031405&op=Reply&threshold=-1&commentsort=0&mode=nested&pid=25782785 [slashdot.org]
It wasn't too bad.
Hard for me to say if either of those are really "good" texts as I hated Thermal.
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: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.
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.
Here are a few books (Score:1, Informative)
For classical mechanics you definitely want Goldstein. (http://www.amazon.com/Classical-Mechanics-3rd-Herbert-Goldstein/dp/0201657023)
Another good supplement is The Variational Principles of Mechanics by Cornelius Lanczos of functional analysis fame (http://www.amazon.com/Variational-Principles-Mechanics-Physics-Chemistry/dp/0486650677).
For Electrodynamics, please partake of Griffiths. (http://www.amazon.com/Introduction-Electrodynamics-3rd-David-Griffiths/dp/013805326X)
However, you will also want something on thermal physics and I have no awesome suggestions for that. But in classical mechanics you should get a lot of nice PDEs (such as the wave equation) which will be covered by the sources I mention. In electrodynamics you will get Laplace's equation (which will also show up in gravitation in classical mechanics). There are no really good books on QM that have been published, so I would just not worry about getting the physics behind the SchrÃdinger equation.
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.
Re:Books (Score:3, Informative)
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 you will not understand the subject.
What I was getting at is that it actually does work both ways. An understanding of our real world (physics), often constrains what real mathematicians do once they leave the university. You will not make it very far as an actuary, for example, if you do not understand at least the basic physics of what happens when someone experiences an automobile crash or a myocardial infarction.
Psychology adds to a broad education, but that is not even remotely related to what I was saying. Nor philosophy, nor accounting. I was not suggesting a educational free-for-all, just that physics and mathematics often go hand-in-hand.
I would not require it, but I do believe that it would benefit most people if they did have at least a little of each. I have. More than a little, actually.
But all that aside: math and physics are closely related "hard sciences". Philosophy, psychology, and accounting (we might as well include sociology and art history here), are all valuable education (at least I think they are), but they are NOT hard sciences, nor are they related to the subject at hand. In future, please stick to the matter under discussion.
Re:Some essentials (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.
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.
Aerodynamics? (Score:2, Informative)
It's for senior undergraduates, but "Fundamentals of Aerodynamics" by John Anderson progresses from inviscid flow all the way through to tacking the
Navier-Stokes equations using numerical methods. I'm but a humble engineer and looking at those equations hurt my head, so it might be OK for you.
Oh, and you'll get $1M if you so happen to solve the Navier Stokes equations (or simply prove a solution exists).
Re:PDEs now? (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 Phd in maths one day maybe? I know some unis do leave PDEs until 3rd year for some reason.
Book Recommendation (Score:2, Informative)
Re:PDEs now? (Score:3, Informative)
It says right there in TFS that he hasn't studied the heat or wave equations.
Re:They're All Targeted for Mathematicians (Score:2, Informative)
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.
Re:Books (Score:3, Informative)
My academic background is in physics, so I'm likely more on the other side of the fence than you are, and still have little idea of what your expectations are book-wise.
Anyway, here's a few just to get you started that I would recommend looking into:
Hope this helps!
Re:Books (Score:3, Informative)
With that said I generally found that doing problems was far better at building intuition than any text. The texts out there all have their own take and generally speaking you have to either find the one that is right for you or just get a lot and make the best of it. I often found that older papers were also invaluable in their insight and simplicity.
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.
Standards have slipped then... (Score:3, Informative)
I graduated in 1985 with a BS in Math & Chemistry. Partial Differential Equations was a required course back then, and the school I attended was nothing special in terms of what they required.
PDE is intermediate level calculus.
But to address the OP's question, try finding an advanced physical chemistry text. There are plenty of uses for PDEs in pchem. Your average introductory level texts won't bother to go that far into the math, probably just through you a few simple related rate equations, but when you get into multiple competing reactions and non equilibrium dynamics then you're pretty much in PDE land for sure.
Nice thing about pchem, it is pretty easy to visualize what they're talking about. A lot easier IMHO than when you're discussing electrodynamics, which is a lot less tangible (at least to me).
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.
Some other books (Score:2, Informative)
I'd recommend that you start with Sagan, Boundary and Eigenvalue Problems in Mathematical Physics. II.1 The Vibrating String (with derivation from principles). II.2 The Vibrating Membrane (with derivation). II.3 The Equation of Heat Conduction and the Potential Equation (with derivations).
I'd also include Crank, The Mathematics of Diffusion. You have to get all the way to eqn. 1.9 on p. 5 before starting to treat anisotropic media. This derives from and extends Carslaw and Jaeger, Conduction of Heat in Solids.
You will want to eventually read (but not during your class), Frankel, The Geometry of Physics. Bridging the gap between the the Exterior Calculus and what you will see in a PDE class is too much work. However, much like the algebra-based-physics student taking differential calculus realizing how many equations he could have *not* memorized if only he had known how to take a derivative, realizing how much second order differential physics follows directly from the properties of certain forms/bundles/et c. is very enlightening (although somewhat opaque at first).
Running my finger down my math/phys shelf (and skipping those that won't provide much physical basis for the setups):
Jackson, Classical Electrodynamics
White, Fluid Mechanics
Ozisik, Boundary Value Problems of Heat Conduction
Segel, Mathematics Applied to Continuum Mechanics
Shankar, Principles of Quantum Mechanics
Boon and Yip, Molecular Hydrodynamics
Hayes and Probstein, Hypersonic Inviscid Flow
and a seemingly endless supply of books by Greiner.
Misner, Wheeler, and Thorne, Gravitation is probably more index gymnastics than you want to try to absorb for PDE. But it's a fun read, is all about PDEs, and they more than completely ground their derivations in the physics.
You might also want to thumb through Brouwer, Studies In Logic And The Foundations Of Mathematics: The Axiomatic Method With Special Reference To Geometry And Physics, Part II.