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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.
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.
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
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.
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.
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
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.
"Your notation sucks!" -- Serge Lang (1927-2005)