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Good Physics Books For a Math PhD Student?
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?"
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
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.
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.
Try Quantum Chemistry by McQuarrie for quantum theory--one of my favorites. It will get you up to speed on waves. I would have never thought there could be such a thing as a gentle introduction to the Schroedinger Equation, but McQuarrie is the closest there is. You can't go wrong with Atkins's Physical Chemistry for thermodynamics. For electrodynamics, there is Jackson. The classic on Information Theory is Cover and Thomas. For gravity, read Gravity (I've never read it though)--beware that its so thick, it has its own gravitational field. But I guess you don't mean relativistic physics. Decent Newtonian mechanics books are a dime a dozen because you don't need more than calculus to learn it.
Just callin' it like I see it.
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.
When things get complex, multiply by the complex conjugate.
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.)
Off the top of my head I would say... Introduction to Partial Differential Equations Applications - E. C. Zachmanoglou & Thoe; mostly math already, but has applications. For introduction to the wave equation try The Physics of Vibrations and Waves - Pain. The Shrodinger equation is explained well in Quantum Mechanics - Griffiths.
An excellent Physics book that is very math heavy but assumes no prereqs is Penrose's Road to Reality. 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.
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.
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
Also had a bit from http://science.slashdot.org/comments.pl?sid=1031405&op=Reply&threshold=-1&commentsort=0&mode=nested&pid=25782785
It wasn't too bad.
Hard for me to say if either of those are really "good" texts as I hated Thermal.
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 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
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.
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.
A 4- or 6-year degree in math or science should include both math and science. If not, you are NOT receiving the education you need to really understand your field. Regardless of how you feel, mathematics actually relates to (and is constrained by) our physical universe. If you do not understand that, then you are not well versed in either.
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.
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.
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).
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.
Mathematical Methods in the Physical Sciences, by Mary Boas. This is the book I used when I read my Physics degree - give it a try.
It says right there in TFS that he hasn't studied the heat or wave equations.
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.
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
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).
"Malo periculosam, libertatem quam quietam servitutem." -- Jefferson
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.