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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?"
Get back to writing your thesis.
Slacker.
How we know is more important than what we know.
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.
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!
I've read through at least some of both Halliday and Giancoli, but sometimes it's nice to have someone explain things to you instead. I happened to have some very good physics professors who always explained where every equation came from (although sometimes I couldn't figure out what they were getting at until they said, "Trust me on this math here" and suddenly wrote equations on the board).
If you're a practical sort of person then it really helps to understand what the math means in some sort of physical context. The academic purists be damned!
Engineering is the art of compromise.
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.
I'm in the exact opposite situation: I'm in a PDE class now with little grasp of the math but understand what they describe pretty well. I would hope you learn the material though, as I'd rather be able to get a solution from a mathematician. I don't know why you're snubbing undergrad books though - there are many that start to delve in the more advanced mathematics, enough so that it sets up the context for the PDEs. I'm a junior in nuclear engineering though - what's a 3rd year math PHD doing in PDE? Were you a Spanish major before? :)
Nuclear engineers build weapons. Civil engineers build targets.
Seriously, the discussion of mathematical models in good PDE books is crisp and clear. The discussion in physics books is woolly and imprecise. That's because physicists rarely know enough mathematics to be able to express themselves precisely. So I would say: Just stick with the explanation of physical phenomena which you find in the mathematics books. It doesn't get much clearer than that, if you read the PDE books which I used to read.
Having taken PDE's last year as a Nuke-E undergrad for intro to quantum, I can tell you that all the physical phenomena PDE's model are generally 'wave' based in _concept_. I also took our Physics 340 on "Heat Waves and Light" which is most of the stuff relevant to PDE's.... The textbook for that course was "Selected Chapters from 'University Physics', Young and Freedman, 11th edition." Where selected chapters were all the ones dealing with heat, waves, light, and a teeny bit of relativity. It's a pretty standard university physics textbook.
Why are you taking partial differential equations as a graduate student?
When you are studying an undergraduate topic?
Most of the previous comments have been far too elementary. I too am a math Ph.D. student and I understand what you are looking for as for while I was working in mathematical physics on loop quantum gravity. Here are some big ones; -classical mechanics has one resounding answer http://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1226901309&sr=8-1 -for quantum theory and such use http://www.amazon.com/Quantum-Physics-Stephen-Gasiorowicz/dp/0471057002/ref=sr_1_1?ie=UTF8&s=books&qid=1226901473&sr=1-1 -for GR and such http://www.amazon.com/Gravitation-Physics-Charles-W-Misner/dp/0716703440/ref=sr_1_1?ie=UTF8&s=books&qid=1226901528&sr=1-1 I dont know a good thermal book, but I am sure you can come up with one. By the way, there was a very similar ask slashdot during the summer from an astronomer asking for the same thing. good luck and I dont know what you research field is, but in general a great read if you are in algebra is the book on quantum groups by Majid. This has a nice physical perspective on the objects. http://www.amazon.com/Foundations-Quantum-Group-Theory-Shahn/dp/0521648688/ref=sr_1_4?ie=UTF8&s=books&qid=1226901678&sr=1-4
She must be a TIGER in the bathroom... I mean bedroom... ~Ryan
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.
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.
Karma: Non-Heinous
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.
When I was still in school, we use the Quantum Mechanics from Richard W. Robinett
http://www.oup.com/us/catalog/general/subject/Physics/QuantumPhysics/?view=usa&ci=9780198530978
http://www.amazon.co.uk/Quantum-Mechanics-Classical-Visualized-Examples/dp/0195092023
After that would be books on solid-state
============
Mathematics will always come back to hunt you down, in so many ways
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.
* Thirring - Classical Mathematical Physics * Landau - The Classical Theory of Fields * Arnold - Mathematical Methods of Classical Mechanics * Sakurai - Modern Quantum Mechanics * Carroll - Spacetime and Geometry
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
as a recently graduated engineering student, and having taken my share of advanced calc and physics, i would actually recommend an undergrad physics book geared towards engineers. this is probably the best place to start in understanding how the equations you mentioned apply...
... in what University can you get a Doctorate in Mathematics, without having taken any physics classes?
Part of the responsibility of a University is to see that you get a broad education. If you have had no physics, you do NOT have a broad education. Period.
You're right. Because *everything* that a person needs to know about PDEs is taught in that undergrad class. This must be some sort of joke! The outrage!! We shall not stand for this!
Of course, the other option (even though it's completely ridiculous) is that--like most colleges--there is more than one level of PDE class, just as there is more than one calculus class. But I know, it's crazy (that's why we threw that out at the start!)
Since you want intuition, an introductory undergrad book might actually be a good idea. Higher level books will often assume you have seen the subject before.
Quantum Chemistry by McQuarrie is a good first book for quantum mechanics and the Schrodinger equation. Dirac's book is more advanced but also good (much harder to read). Much different focus though.
For electricity and magnetism a good first book is Griffiths Introduction to Electrodynamics. Here you'll see applications of the Poisson and Wave equations. Jackson is the classical "second" course textbook. (Upper level undergrad, beginning grad).
A good introduction to applications of the diffusion (i.e. heat) equation is Random Walks in Biology by Howard Berg. One benefit is that it is a very short book too!
For nonlinear equations there are too many references to know where to begin... There are millions of books on just the Navier-Stokes equations... Generally I'd just poke around Amazon and browse some of the books with good reviews.
Anyways if the original poster wants references for a specific PDE or area of physics please post a followup...
What do you mean you are a third year PHD candidate in mathematics and you are only now taking PDE!? I took that sophomore year in my undergraduate engineering program, before we got into any of the serious engineering classes. If I remember correctly, it was the same time as we studied relativity in physics. What have you been doing all this time...
is where to start when it comes to deriving PDEs. The heat equation and the wave equation fall easily out of vector analysis, as do a number of other familiar PDEs. I'd start with a vector analysis book.
are of course a natural first place. For textbooks, it really doesn't matter all that much. At the level of generality you're operating in, textbooks are textbooks.
Don't be too hard on them, the engineering majors never have to get into the really heavy math (they just think their math is heavy).
My god, I had to learn that crap as a freshman UNDERGRAD!!! Now grant it I was an electrical/computer engineering major at the time, but still, I can't believe that a third year math PHD candidate would not have had partial diffs... I mean, seriously, it is the only way to do some stuff, especially anything in the real world (hence all the physics basis on the questions).
We were all warned a long time ago that MS products sucked, remember the Magic 8 Ball said, "Outlook not so good"
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.
I think his problems may be the result of how the questions are being given to them. They probably won't be your standard undergrad, here is an equation, give me the answer, type, but more of the here is the situation, figure out the equation, then solve it type.
We were all warned a long time ago that MS products sucked, remember the Magic 8 Ball said, "Outlook not so good"
Many of the standard introductory undergraduate and graduate physics textbooks have been mentioned by other posters, but I'm surprised that no one has mentioned Michael Spivak's Elementary Mechanics from a Mathematician's Viewpoint , which is based on his Pathway Lectures at Keio University.
"It take 9 months to bear a child, no matter how many women you assign to the job."
The wave equation and diffusion equation are technically partial differential equations because of the 3 space dimensions and time, but these are simple PDEs because the three space dimensions are basically the same and the derivatives usually only appear as the Del operator, which treats each direction equally, and the boundary conditions are usually such that the constant of integration is just zero.
In thermodynamics, you actually have serious PDEs which involve variables that aren't all the same, and the constant of integration must be found by matching arbitrary functions to each other and boundary conditions.
This probably isn't a book for someone new to physics, but it does use some PDEs.
I am currently taking a pde course as an undergrad and am using Partial Differential Equations: An Introduction by Walter A. Strauss. While this book does have some faults it does an excellent job of relating pdes to their physical interpretation.
I can recommend "The road to reality - a complete guide to the laws of the universe" by Roger Penrose. The guy undoubtedly knows what he's talking about (being a famous physician himself) and the book is very math-centric. First the mathematical concepts are explained, then based on that the physics of our universe.
The difference between engineering and math is that engineering focusses on real-world problems and the bit of math required to solve them. Because there are too many other things to learn - and engineering centers on practical applications. A lot of math appears to be intellectual masturbation unless you have proper training - and lacks any trivial practical application. Until suddenly, someone might find use for it to describe something in physics. Or not. A lot of the riddles you solve as a geek are applied math. Think topology. :)
Why would an engineer have to bother with abstract algebra? Or why should he be able to derive about everything in math from aimple set of axioms?
Engineers don't know math. Much.
(Disclaimer: Here speaks a CS guy who used to date a lovely Math PHD. And I thought MY mind was warped...)
There are fabulous books by many different Russian authors called (mainly) "Equations of Mathematical Physics". They may help you...
You're right. Because *everything* that a person needs to know about PDEs is taught in that undergrad class. This must be some sort of joke! The outrage!! We shall not stand for this!
Of course, the other option (even though it's completely ridiculous) is that--like most colleges--there is more than one level of PDE class, just as there is more than one calculus class. But I know, it's crazy (that's why we threw that out at the start!)
The material he is describing is what is covered in the undergrad PDE course. Its frequently given as both an undergrad course number and a graduate course number: same book, just more work for the grad level class.
If someone is passing you on the right, you are an asshole for driving in the wrong lane.
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.
I hope some math professors are reading this. They always seemed to think that they only needed to teach the "how", as "why" would already be obvious or would become clear. It didn't, not for me. More like that was the excuse, because actually "how" alone was much easier to teach. I studied PDEs in calculus classes, but never used them for anything. When they did come up with example uses, they were pretty contrived, and often could be solved with plain old algebra. Or they were so small that hand application of numerical methods could pin down the answer. Took only a few iterations of the Bisection method to get that zero, or you'd hack up a quick and dirty program to push some data into a linear algebra library function and get back results, something like that. And what's a student to think on hearing that although faster, Newton's Method, which is based on calculus, isn't as reliable as Bisection, which is simple algebra. Not good examples when trying to show students how useful and valuable calculus is.
Books? There's more than books alone out there. Lots of material on the web. Lots of combined material. Here are some books associated with Sage. Are you making use of mathematical software: Sage, Matlab, Mathematica, Maple, or some such? Or are you at least able to code up something in a general purpose language if needed? Much math is to the point where you can't advance without computers. Maybe I'm a bit behind. These days, I suppose all math students use such software.
I've noticed also that people with backgrounds in pure math don't have a good basic understanding of Computer Science. You know all about Fourier Transforms, you've heard of the Fast Fourier Transform, you've heard of big O, but you don't see what the big deal is about the FFT-- to you FFT is just one of many ways to do a Fourier Transform, one specific to computers which a person would not use if working out such a transform on paper. Do you have an appreciation of the algorithmic complexities of the math problems you are encountering? The way multiplication is done in grade school is just fine for relatively few small numbers, but when you want to do millions of multiplications of large numbers (1000 digits, say), you'd better use a computer, and you'd better program the computer to use FFT. A textbook on Numerical Methods could be worth checking out.
Intellectual Property is a monopolistic, selfish, and defective concept. It is "tyranny over the mind of man"
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.
Since you are set on the math, but weak on the physics, a great book for a conceptual understanding of physics is called Conceptual Physics: http://www.conceptualphysics.com/
Some years ago as an math undergraduate I could not get to classes in a physics course (for reasons that are not important now). Instead, i learned quantum mechanics from the books, and enjoyed several other parts. (I then did some old exams to prepare for the final.)
Yeah, Amazon are evil, but if you like, you can abuse the system. Find the book you need, and then go somewhere else and buy it. Of course, you are exposed to the ads, but it's a small price to pay. (I know, they make money off of ad views. It's hard to be subversive, these days.)
The general idea is straightforward. Partial derivatives are just the concept of a derivative generalized to higher dimensions. Just as a derivative is a tangent to a curve, a partial derivative is is a tangent plane to a surface.
I recall studying PDEs in a 3rd year undergrad course. How you can get to Ph.D level in maths and not have at least a working (basic) understanding of them is beyond me.
I drink to make other people interesting!
This kind of reminds me of the comments I got from Business Calculus students when I was carrying around my graduate Algebra book, which was appropriately titled "Algebra". "Oh, Algebra! I had that in High School. It's not so hard..." If only they knew what was inside that bright lemon-yellow cover...
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.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
especially anything in the real world
So why would this be in a math program, again? ;-)
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
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!"
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).
What kind of a joke undergraduate degree do you have? What university accepted you into a graduate program? You are in your third year of your Ph.D. in math, and you never took physics? And as a Ph.D. student you have to ask Slashdot how to learn physics instead of actually researching? I pray you don't become a teacher or work on anything critical in the future.
It is written in a mathemical language (Def, Theorem, Proof...) and is highly structured to help line out the mathematical basics behind classical mechanics and electrodynamics (some differential geometry is needed for the latter).
The second volume on quantum mechanics requires a pretty solid knowledge of functional analysis.
Regardless of the fact you're a math student or not, I recommend Motion Mountain, the free physics book. It covers pretty much anything (up to the most recent stuff) and it is beautifully written.
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.
You're rather condescending. Why do you need to to do physics to do a maths degree? There are lots of Uni's that'll accept you into such a degree without Physics. I'll list a few at the top of my head - Princeton, Harvard, Standford, UCLA, Oxford, Cambridge, Warwick, Australian National Uni, etc etc. Physics is important on a set of measure zero. Ie., why should someone interested in, say, Model theory do physics. It's ridiculous to ask that. I've got a few degrees, but in my Science my minor was in Astro. At heart, I'm a pure mathematican, that's what I'm starting my PhD on next year. But there is really very little relevance to even the physics kind of stuff I've done in pure (ie., even in PDE theory, the physics is of very little importance). Rigourous mathematics is very different from the way things are done in the physics world. This guy is actually approaching things from a good perspective - having the mathematical machinery before attacking the physics is a much better thing. The physics is relatively easy to learn, the maths is the hard part. There's nothing wrong with asking Slashdot. I mean, he could ask his supervisor, but maybe his supervisor doesn't honestly know. What's wrong with reaching out to the broader community.
I don't know if it has been mentioned here, but V.I. Arnold (Lectures on Partial Differential Equations) might be a starting point. Arnold emphasizes physics in his writing. His introduction to classical mechanics is an absolute must for everyone interested in this kind of topics! He really blows away the fog.
Roger Penrose: The Road to Reality
Shows you how lots of that maths, even the abstract stuff, applies to trying to describe the universe. And ponders right from the start on why you are doing it.
Zill & Cullen's "Differential Equations with Boundary Value Problems" has good PDE applications (9th chapter onwards), including the heat/wave/etc. equations mentioned. It should be a good starting point to bridge where you currently are with a physical basis (they uncover what the equations physically represent with some mini-derivations). I used 4th ed. I would be surprised if Arfken didn't hit this.
I think all the PDE (partial diff. eq.) like heat equation, wave equation, gravitation should have been covered in high school. Just get a basic high school text, and spend a few minutes. These equations should at any rate be so self evident that you should not have any problem to understand this intuitively.
You will probably need a textbook on quantum mechanics for the Schrodinger equation, mostly because of operator formalism or bracket notation. I would recommend just adding a class in quantum mechanics rather than finding a book. That class will be all PDE.
don't cut it off www.mgmbill.org
You gotta be kidding right? ODEs is typical for math/physics undergrads, but PDEs is almost always optional for math undergrads and sometimes optional for Physics undergrads. Mathematicians often specialize in areas that have no need for PDEs.
Advanced Mathematical Methods for Scientists and Engineers. Carl M. Bender
This discussion reminds me a little of my first graduate-level class in magnetohydrodynamics. I was the only person in the room who'd never taken a class in fluid dynamics. Oops. Weirdly enough, I ended up doing a PhD in ... magnetohydrodynamics.
Not to rant, but why do people post 'ask Slashdot' questions that are so vague a 20 second search seems to answer them? Editors!!???
Not to be rude, but it seems your utter and absolute reliance on what lives in Google and Wiki questions your value on the human factor. This is basically the reason anyone posts questions here, to get a focused response that has a bit more meaning and is a hell of a lot easier to parse real information from than 916K Google hits.
I have their fluid mechanics and deferential geometry books and found them quite helpful (and cheap ~$15, hard price to beat :)).
Halliday & Resnick - It's the standard 1yr majors (w/calculus) undergrad intro to Physics. Any old used edition should be fine.
Heat Transfer by Carslow and Jaeger, 1956.
-- Cave quid dicis, quando, et cui
> As a third-year PhD math student, I am currently taking Partial Differential Equations.
What are you saying here? That you are in your third year of a PhD program? At Michigan Tech in the 1960s diff. eq. was a lower division undergrad course.
> Can anyone recommend physics books for someone in my position? I don't want to just pick
> up a book for undergrads.
Well, do it anyway. Read "The Feynman Lectures on Physics".
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
To follow up with someone else who responded to your post....
There is also the added fact that there usually hundreds of books on any one topic. How would you know which one to read? When someone is given so many choices, it is natural to ask someone else what they think of book X or Y and which one is better. I am sure you have done something similar as well on occasion. Considering the submitter is in grad school, he likely doesn't have time to read through multiple books on the same topic, just to find out one of the ones he read was garbage and that there are better books out there.
Not to get too side tracked, but it is also this phenomena that makes "word of mouth" such a powerful device. People will advocate anything for you if they like it enough.
I recall studying PDEs in a 3rd year undergrad course. How you can get to Ph.D level in maths and not have at least a working (basic) understanding of them is beyond me.
I'm finishing my PhD in math, and I know almost nothing about PDEs. It's not relevant to my field of research.
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
Hungerford, right? I'm an engineering grad student who once picked it up to learn something about group theory, but didn't even get past the first chapter because it was so dense and (from my perspective) esoteric. I know that rigor is important and all, but could you enlighten me as to what's so great about algebra? I mean, I can obviously see why analysis is important, but algebra still escapes me.
I agree with most of the above suggestions. I'll add: Classical Mechanics - Taylor An Introduction to Thermal Physics - Schroeder A Modern Approach to Quantum Mechanics - Townsend I also recommend avoiding "Math Methods" books, since they're only going to give you more math, with less of the physical applications. Once you start to get a bit of physical intuition though, definitely check out Arfken and Weber. It was a bit too math-heavy for me when I used it as a physics student, but I think it'll be a nice bridge for you.
An algebra? You mean there's more than one?
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The Feynman lectures. The MIT video physics courses (free on the internet)are very good.
If you take that sentence to mean he someday wants to get a PhD and is a Junior it seems pretty much spot on. Some 20+ years ago where I went, 6th semester (3 Calc, 3 DE) was primarily linear PDE's.
Since he doesn't differentiate I was left assuming he's an undergrad with PhD intentions... kind of a big difference but maybe I just misunderstood.
That said, he at least has the right approach. The whole view of 'don't worry about what these equations mean, just memorize them' is completely f*cked up.
By all means get books, but if you're really serious take physics courses like Thermodynamics etc (not the chem variant). If you're really a PhD you should breeze through them but they'll help in the long run.
Belthize
Michael Spivak is writing Physics for Mathematicians, of which the beginning can be downloaded. Videos of the lectures are supposed to be available, but I couldn't get them to load. Has anyone heard recent news of this book?
Lots of applications for PDE's in modelling non-linear optics in various materials (both cw and pulsed light evolution), but particularly relevant to modern telcom infrastructure is (silica) optical fiber also interesting is the newer photonic crystal fiber. For a good ground work on all the phenomena in the former fiber check out any edition of "Nonlinear fiber optics" by Govind P. Agrawal (Academic Press).
Correction: it's going to be Mechanics for Mathematicians, so it won't help with the heat equation, etc.
Description about those groups and fields are like Java interfaces. These are just a collection of facts that allow you to prove theorems without knowing the particular implementation of an algebraic structure (e.g. natural numbers, matrices, geometry); or in the case of Java, being able to write a class method to use another class without looking at the actual source code of the other class.
Abstract algebra is exactly that, abstraction.
I once had a signature.
If you're concerned about these things, I think you're in entirely the wrong field, buddy. Why don't you take a few physics classes and see if that stuff is more your speed? :D
Physics is all about linearizing those DEs. Even at the PhD level, you'll almost never find anyone dealing with anything non-linear. Go talk to your engineering friends. They will be only doing numerical analysis, but they will at least be using those equations.
There are applications, too. The operators in quantum mechanics form a C*-algebra acting on a Hilbert space. Learning the properties of a C*-algebra is easier than trying to deduce what the properties of the momentum and position operators might be and then attempting to generalize from there to other operators.
If you ever hear someone talk about symmetries in physics (immensely important and useful, BTW), they are talking about groups. A symmetry in physics shows up when you can take any solution, transform it in a well defined way, and get another solution. Ok, so now you have another solution. You can transform that in another way, and get another solution. So we see that these transformations compose to form another transformation. Take a glance at the other group axioms, and you find that your symmetry operations form a group. So, the results of group theory are useful to deduce properties of systems that have certain symmetries.
Outside of theoretical physics: True and False, together with AND and OR as plus and times (I can't remember which is which) form a field. You can make a vector space over any field you like, and once you can make a vector space, you can make matrices. Once you can make matrices, you can use them to solve coupled linear equations: for instance, take a set of Boolean equations. You can either work out what the solution is tediously by hand, or you can just pack them into a matrix and invert it.
SIGSEGV caught, terminating
wait... not that kind of sig.
Halliday and Resnick, _Fundamentals_of_Physics_ (I, II and III) Then watch Star Trek TNG, in order. At some point, the episodes follow the text chapter by chapter, showing an example for each. This was incredibly helpful for the more difficult to grasp topics (such as degeneration).
I am an aerospace/electrical engineer. When I took PDEs, I found I needed to review ODEs because you have to be able to solve ODEs in your sleep, to begin to solve PDEs. But since you are a math major... I can understand your frustration at the way they just throw bits and pieces of physics at you while teaching you PDEs. But you will find that you really need to ignore that nagging feeling that there is something you are missing and try to understand the phenomena only in terms of the PD equations only. These equations were developed hundreds of years ago in some cases. They are based on somewhat hazy ideas like The Caloric Fluid that turned out to be right in application even though they now seem wrong in reality. Enjoy the somewhat ephemeral, soft nature of the physics concepts presented in PDE class. Ultimately, the only reality you are interested in, is the one described by the bare PDEs you work with. After you can do that, you then have the tool to learn physics. Of course, the only thing that really cures that nagging feeling is to fail a few tests.
V. I. Arnold, Mathematical Methods of Classical Mechanics. He has what is obviously the One True perspective on the math underlying classical physics (symplectic geometry). Complement your reading with a modern math book on symplectic geometry and you'll really understand what's going on. For QM, I've been told that Mackey's "Mathematical Foundations of Quantum Mechanics" does things in a mathematician-understandable way. You can find illegal scans (and most other math books) of this on the usual Russian websites.
Being a Physics student myself, I found this gem: Mathematical Methods in the Physical Sciences by Mary Boas It covers a huge array of information in the methods needed for Science. It gives great explanations for them, what they are used for, how to use them, and gives lots of practice examples at the end of each chapter. Super book. Nearly indispensable.
Scientist world wide still haven't figured out how Hunter S Thompson managed to stand upright.
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Course of Theoretical Physics. Covers everything in several volumes from mechanics to statistical theory to quantum and relativity, all in a frighteningly mathematical framework that should appeal to the maths grad student. Enjoy!
As an applied physicist wanting a better grasp of the mathematics I use, I'm working through Mathematics of Classical and Quantum Physics by Byron and Fuller. It's a Dover book, so it's inexpensive, and so far it has been enlightening. For instance, it uses the example of vortexes in a swirling bucket of water to help visualize the curl of a vector field.
"Scientists discover the world that exists; engineers create the world that never was." --Theodore von Karman
I have a seismology PhD. We really didnt LEARN the physics equations we had in undergrad class until we had to implement them in computer programs and simulate or reduce real data. Then you learn the equations, numerical implementations, and data are all approximations with limits you WILL RESPECT.
An equally challenging alternative is to teach physics to someone else. Then you discover the holes in your learning too.
In the early 1970s Arizona State University, like many others, reorganized its Engineering curricula. Engineers have similar needs for advanced mathematics as physics majors and after PDE we were required to take a course called "Engineering Mathematics". It did quite a thorough job of relating the theoretical forms and formulas to the physical world. You did have to survive into your junior year to experience that reconciliation.
and the Physics will reveal itself to you. But really, an excellent place to start would be a math text for physics students, e.g., http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710
I can whole heartedly recommend picking up a copy of Advanced Engineering Mathematics by Michael Greenburg. It's a serious book, and it's especially helpful if you're trying to piece together the mental constructs of what the math means to what the physics means. I bought it to understand the math that that describes the physics, but it works both ways. It covers everything from ODEs, PDEs, linear algebra, vector field theory, Fourier methods, complex variables, etc. It basically covers everything from mid-level undergraduate math to some graduate stuff (it's used as a grad textbook at my school). I seriously can't recommend this book enough. ISBN 0-13-321431-1
The thing about it is that it's presented in a way that's digestible. I'm not saying that you can read the book like it's the next in line of the Ender series, but you can get through the damn thing.
I'm a recent graduate with a PhD in mathematical logic; I can totally relate to this problem of having a non-standard background. Before Grad School I went to a liberal arts college, where my math major consisted of something like 9 courses. When my advanced studies began I felt totally lost. But you have to ask yourself: Do I really have the time and energy to commit to a high level exposition of physics at this point? The answer to this question will depend on whether you intend to specialize in PDE's. If the answer is no, then I believe you should buy some good popular explication of the topics you're covering. Unfortunately, I do not know what such a book would be, but having read extremely good expositions of several high level mathematical concepts (Prime Obsession; Unknown Quantity; "e", the Story of a Number, and Incompleteness) I have some confidence that a book of similar quality may exist in this area. Of course, if you are intending to specialize in PDE's, it will be worth the time and effort you will need to invest in reading a serious text. Even so, keep in mind that you can never learn everything, even about a small subdiscipline. My advice is to find a particular area of PDE's, become an expert on that, and get out of graduate school as soon as possible.
If I recall you need PDE for any wave equation ( of more than 1-D? ). Maxwell's equations are pdes. This is second year physics for all Caltech undergrads. Phys majors will see Navierâ"Stokes in the third year of undergrad.
1. The understanding of the laws of physics hasn't changed much since Newton was describing those laws hundreds of years ago. So, if the only point of getting a book is to put info into your head as apposed to passing a specific class, get an old one. A 5, 10, 20 year old book will have all the knowledge you seek and cost A LOT less.
2. Go to a site that specializes in used text books and has review information.
You could start at:
http://www.betterworld.com/list.aspx?SearchTerm=Advanced+Engineering+Mathematics+by+Michael+Greenburg
As you can see, a little value shopping, and you can get all the knowledge you seek and save 90% on the price of the book.
Also, :-)
Save the earth! http://www.betterworld.com/custom.aspx?f=impact
I got my BS (physics) from Texas A&M (cue the Aggie jokes in 3... 2... 1...) and every physics major had to get their PDE merit badge before going on to linear systems/matrix math. So the grand math total was 3 semesters calc, 1 ODE, 1 PDE, and 1 linear systems. To my knowledge, that's pretty routine for physics programs - in fact, I don't see how you could get through stuff like QM without it. I'm not sure what math majors took for math, but surely it had to be at least that in-depth.
The Logic of Science is a great read. You would need to have a basic knowledge of PDEs but it is more about statistics. It has a wonderful advanced applications section that gives a great review of applied science in many fields.
If you are looking for some general graduate level books, I would suggest either Boaz, Arfkin, or Butkov. All 3 have a "Mathematical Physics" textbook.
Gravitation by Misner has good applied Differential Geometry problems.
I have secretly hidden some mispelled words in this post. Can you find them?
You've got to be fucking kidding me. Name one high school in the US that's covering PDE's. Calculus I could believe - differential equations of any flavor? Not a chance.
Yes, I agree that PDEs probably should have been introduced before the 3rd year of a PHD program (maybe they were, TFA isn't totally clear on that point). But saying that they are being covered in high school is a ludicrous exaggeration.
I went to a regular high school in Norway. Diff. eq. was a required class. It included complex functions, higher order as well as systems of equations, but fairly elementary. This covered all the basic physical models described. Most (at the time) 16 year old student actually got a good grasp of the material as well if i recall correctly.
don't cut it off www.mgmbill.org
Have you considered Visual Complex Analysis by Tristan Needham? It might be too low-level, I don't know. I also second the suggestion of Penrose's "Road to Reality"
I hate to be a "pedant." (but then again, perhaps mathematicians are all unnecessarily pedantic to a physicist or engineer.)
A partial derivative of a scalar-valued-function f in two variables is NEITHER the tangent plane (if it is defined) to the graph of the function at a given point, nor a line on said plane. It is the SLOPE of a line which is parallel to the tangent plane.
More generally, a partial derivative is the rate of change of a function in a given direction. It is a SCALAR, not a geometric object.
In fact, in most undergrad textbooks, what I have been calling a partial derivative is called a "directional derivative." But to me it is justified to call such a derivative a partial derivative, since we can apply a change of basis, so that said direction is aligned with a coordinate axis in Cartesian space.
Pedantry is underrated.
How you can get to Ph.D level in maths and not have at least a working (basic) understanding of them is beyond me.
Simple. Take the "non-applied" mathematics track. The mathematics that is used in physics and chemistry is called "applied". The rest of math, including algebra, logic, algorithms, and many other topics is called "non-applied".
Anything by Dr Michio Kaku. I am not even a physics student and I always find his stuff a good read.
two books that stand out from my undergrad physic studies that may not give you a practical appreciation for ODE (but really, ODEs are so mind-numbingly rote, why would a math PHD be studying them?) but these are towering works of genius founded on beautiful mathematical arguments:
The Principles of Quantum Mechanics
Dirac
Classical Mechanics
Goldstein, Poole, Safko
An equally challenging alternative is to teach physics to someone else. Then you discover the holes in your learning too.
I think that applies to just about any subject. As a student, you learn enough to write the papers and answer the tests. But as the teacher, you have to know it well enough to take an hour-long oral exam three times a week.
I know you said you don't want an undergraduate text, but as a physicist (BS+PhD), trust me, you do want an undergraduate text. The textbooks used in introductory classes (100-series, or equivalent) won't go beyond basic Calculus, because the students haven't had time to learn anything more, and even the intermediate classes (200-series) won't go past ordinary differential equations. Also, looking at the topics you mentioned (heat equation, wave equation, etc), I think what you want is not just a physics textbook, but specifically a mechanics textbook, since that's where these topics are normally covered.
So why do I say you want an intro text? Because you already know the mathematics; what you're trying to learn are the basic concepts of classical mechanics. These are what you'll find in an introductory textbook. More advanced book, on the other hand, will assume you already the difference between a force, an energy, and a momentum, and therefore they don't bother explaining it; they'll instead move on to more sophisticated treatments of the subject building on the student's previous exposure (variational methods, field theory, and the like).
My advice, then, is to go to the university library and check out a random introductory textbook. "Halliday and Resnick" [those are authors, not the title] was the standard intro text when I was a student, and it's a good choice. "Marion" and "Webster" are even older (and a bit more advanced) but they would also be good choices. Anything with the title "Intro to Mechanics" is probably a second-year book, but you might luck out and find something you like there too. "Landau and Lifschitz" is good too, but the series is probably too advanced to just pick up and read. Also, I would avoid the Feynman lectures at all costs -- they're great after you know the subject but piss-poor if you're trying to actually learn physics from them.
If you're bound and determined to use a graduate level text, then Goldstein's Classical Mechanics and Jackson's Classical Electrodynamics would be the standard graduate-level textbooks and will find plenty of partial differential equations in either one.
-JS
Vanity of vanities, all is vanity...
Check out university course web pages from good schools. I specialized in astrophysics but don't work in it but I like to stay on top. I check out class syllabuses for to date and new reading. There are so many books out that I like to see what has recieved the stamp of approval for professors at respected universities. Find a subject that really interests you and the math will always be there. Personally, black hole are always fun, you get a little of everything from them.
Some styles of presentation are useful for learning the material first time through, while others are useful for reviewing the information after you've mastered for a while. I consider Penrose an excellent choice for the latter.
For pedagogy you want material ordered from basic to complex, ditto for underlying mathematical tools. For review work I love to read historical narratives of the inventing physicists. That adds personal color to the ideas. And it also shows false avenues explored before coming up with the best answer. A basic physics course lacks time to do this.
Feynman once said, Physics is to Math, what sex is to masturbation
After taking the 10th undergraduate engineering math course (trimester system) I met all pre-requesites for my two advanced math electives.
All the choices still had names including you-are-an-idiot words like, "Introductory ... Basic ... Elementary ... Beginning ... Essentials ... Starter ... Early"
Oh please, who modded this up?
Sure, the maths in maths books is great because mathematicians know better - so says a mathematician.
Alternately, mathematicians have their heads so far up their arses they completely miss the real physical significance of whatever they are supposed to be doing - so says a physicist.
If you want to know physics, read a physics book: Essential Pre-university Physics by Patrick Michael Whelan is a nice quick one since I assume you don't want to put aside 6 months of your time to wade through Feynman.
It has been a while since I did my PhD in (Theoretical) Physics. In that time, I did read some 'math books for physicists'. ... corollary ... theorem ... corollary ... proof ... '. Not really simple to read, and indeed (for me) not simple to learn from.
I do recall reading some 'mathematics for mathematiciancs' books as well. They were of the form 'theorem
Indeed, I happen to think that the UNDERSTANDING of mathematics comes from doing physics (indeed, many (most?) advances in mathematics comes from physicists trying to understand/explore some theoretical model. String theory, anyone?).
Off course, you are welcome to disagree with me.
However, I do not understand the quest for a 'physics book for mathematicians': I happen to think there is no such thing!
A good physics book on a mathematical subject is probably what you need. My suggestion: speak to a PhD student in physics at your University (note: I am assuming that you are at some University that has PhD's on both mathematics as well as physics), and ask him (her?) for a good book on PDE's. Good chance, this is what you are looking for.
The MIT post a lot of excellent material on PDE. http://search.mit.edu/search?__EVENTTARGET=&__EVENTARGUMENT=&site=ocw&client=mit&getfields=*&output=xml_no_dtd&proxystylesheet=http%3A%2F%2Focw.mit.edu%2FOcwWeb%2Fsearch%2Fgoogle-ocw.xsl&proxyreload=1&as_dt=i&oe=utf-8&departmentName=web&courseName=&q=partial+differential+equation I presume you are not a student of MIT, but the material is open to the public and excellent for the studies you indicate you need. Better than just another text book and more focussed.
Pearls of mathematical wisdom.
Get this book
No suggestion other than to forget about it. I would stick with an undergraduate book in Physical Chemistry (see Quantum Mechanics) or one in Physics (GA Tech had a good book coming out 10 years ago, so I can only assume it is still around...and I mean good).
Having both a math degree and a Master's in Quantum Chemistry, you will get too bogged down in the language of Chemistry (particles in one dimensional boxes and all of that funky stuff). Chemists make horrible mathematicians (which is why I'm in Computer Science now).
So stick with a good undergrad book and team up with a good physical chemistry student (or physics student) who has a good grasp of Schroedinger's equation. You can swap ideas and help each in turn.
BTW - It was Diffe-Q's that finally helped me make since of Wave Motion (ie Schroedinger's equation). And stay away from those damned laplace transforms and be a real mathematician!
Good Luck.
James
The truth is usually just an excuse for lack of imagination.
As a third-year B.Sc. EE student, I had to take Partial Differential Equations. Of course, double-e's also had a special class that crammed four different Math department classes into a single semester. I remember Fourier transforms being one of them, but my memories of the other three have faded.
Nothing for 6-digit uids?
Introduction to Quantum Mechanics by Pauling and Wilson. Wonderful reference to basic QM, a true classic. This text goes into more detail of the math of QM than later texts. Amazing to contemplate that this text was published (1935) only 10 years after Schroedinger published his equation and shows how much had already been accomplished.
As an aside, it has never been clear to me how Schroedinger came up with his equation in which he combined classical wave mechanics with the earlier work of Bohr and others on electrons "orbiting" the nucleus. Also not clear to me is how it was derived, other than by observation, that the product of the wave equation and its complex conjugate is proportional to the probability of finding a particle at a particular location.
What an exciting time it must have been to be a physicist at Gottingen in the twenties. I doubt if there will ever be a ~50 year period (1903-1954) when so much basic scientific information is discovered -- Relativity, quantum mechanics, DNA, and the transistor (solid state physics).
maybe Transport Phenomena by Bird, Stewart, and Lightfoot. It uses PDEs a lot. Covers diffusion, heat transfer, etc.
...the future crusty old bastards are already drinking the Kool-Aid.
In case nobody's mentioned it yet, you should take a look at The Geometry of Physics by Theodore Frankel. The goal of the book is to provide a "working knowledge" of exterior differential forms, differential geometry, algebraic topology, Lie groups, vector bundles, etc. The applications include thermodynamics, electromagnetism (in curved space, of course), soap films, Kirchhoff's laws, relativity, Tensors, Dirac spinors, gauge fields, winding numbers, etc.
For example, the chapter on the Dirac equation starts with SO(3) and SU(2) groups, Clifford algebras, and the Dirac operator, then moves on to spinors, bundles, and eventually the Dirac operator in curved space-time.
The book is fun to read, and places emphasis on geometric intuition before developing more abstract notions of differential geometry. Very intuitive and insightful, full of figures and cartoons, yet also sticks to the "theorem-proof" format found in standard mathematics textbooks.
Sussman & Wisdom's "Structure and Interpretation of Classical Mechanics" is a must-read. It presents the very general Lagrangian formulation of classical mechanics using a clear, unambiguous Scheme-like language. It's a gem. (Sussman's "Structure and Interpretation of Computer Programs" was the intro CS book at MIT for years.)
Dirac's "Principles of Quantum Mechanics" is a highly readable introduction to the field. Dirac didn't (and couldn't) assume much prior knowledge. This book is the origin of the supremely clever bra-key notation.
Roger Penrose's "Road to Reality" is an engaging coverage of much of modern physics as well as the prerequisite mathematics.
there's a great book called Schroedinger's kittens, some good back story on fundamental physics, etc. Author John Gribbin Talks about Quantum mechanics, historically, and even ...in regards to recent developments, and how we got there.
http://www.amazon.com/Schrodingers-Kittens-Search-Reality-Gribbin/dp/0316328383
This is sort of a sideways answer to your question.
I went the other way - physics first then grad math on ode's and pde's. My experience was that things I had had to really slog through when presented by the physics dept suddenly became crystal clear when I looked at it from the math dept point of view. Now it may be that this was the result of taking the physics first - i.e. it wouldn't have been so easy in the math if I hadn't taken the physics first. But I don't think so - the math courses just did a better job of explaining them than the physics course... in fact my reaction at the time was something like "holy crap, two years of physics courses just dropped out as examples in one math course... why didn't they make us take this first!!!!"
The tyrant will always find a pretext for his tyranny - Aesop
College isn't cheap - and in most cases certainly isn't free (even if it is, someone's paying) - so why the heck are you turning to slashdot when you've got someone who teaches the course you could ask. If they're too busy to field the question then you should be yelling at someone about not getting your money's worth.
Just my 2Â,
Nick
RandomAndInteresting.comdefending the world from stupidity since 1979
I challenge you to find an instance on a physics paper claiming to "disprove free will" published in a reputable journal. I doubt you can find one, let alone myriad instances which I might expect if I took your "all the time" statement seriously.
I'm going a little off-topic here, but with a solid gathering of minds such as this, I hope someone can answer (and maybe elaborate) a few questions for me. I'm a high school student, soon to be attending University (hopefully U of M), and I have to ask (very naively) - is higher level mathematics especially tough?
Let me explain a little. Throughout all of high school (remember those days?), I've taken probably 6 math courses (up to pre-calculus (should have been calculus, but that's another story)). I have always found math particularly easy (despite failing a trig class, but that's yet another story). Honestly, I personally feel I'm very good at math, and especially understanding mathematical concepts (just like anyone, I'm still prone to silly mistakes). For all of my academic life, math has always just worked. I listen to the lecture, look at a few examples, and in general, the math just opens up to me; it's just a matter of setting up the problem and reaching the end.
So, I guess what I'm really asking is: are the concepts introduced in higher level math courses concepts that just "click" for some (i.e., math oriented) people, and then getting the concept down is just simple repetition? Or is it that once a student hits undergrad, and higher levels of maths (calc 1-4, etc), the concepts truly become tough?
Simply put, just hearing about these tough math courses scares me. When math is challenging, I find it extremely fun (I love the "lightbulb" moments), but hearing about these very complex problems/concepts makes me wonder if I'll ever be able to understand this stuff.
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.
I was thinking of this one when I made that comment. The entire discussion, including a back and forth with a Nobel Prize winner, is embarrassing to read, and clearly evinces a total lack of understanding of undergraduate philosophy in the area. In particular, they seem to assume a simple form of incompatibilism is universally accepted (i.e. that free will and determinism are inherently incompatible), when it's actually probably a minority view, with compatibilism (e.g. recently defended by Daniel Dennett) being more widespread.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
Resnick on Physics
I've seen several very good books on physical applications of advanced mathematics. I'm guessing queen-of-the-sciences has enough heavy reading to work through. Might I suggest something along the lines of a nice bed time story. Ok, Physics of Supper Heros was written as a text book for kids but I throughly enjoyed reading it. How many physics books can one say that about? It's not a bad introduction to physical concepts. There is a web site that will provide reasonable in site to the depth of the book. Best of all, it's quick reading and available by Inter Library Loan in most areas.
You could try the two-volume Bamberg and Sternberg, A Course in Mathematics for Students of Physics. I think it could conversely work as a course in physics for students of mathematics. And Arnold, Mathematical Methods of Classical Mechanics. Or Nayfeh and Balachandran, Applied Nonlinear Dynamics Also Lawrie, A Unified Grand Tour of Theoretical Physics. Terse and gives you the equations up front.
If you're into differential geometry you could try The Geometry of Physics by Frankel, or Schutz's Geometrical Methods of Mathematical Physics. But that's pretty advanced physics, general relativity, gauge theory, quantum field theory and such.
I dislike algebra. Calculus, I love, but algebra...[shudder]. Calculus always made a lot more sense to me than algebra. I get algebra, even linear algebra, but anyone (taking calculus) who scoffs at algebra clearly does not understand math very well.
;-) and have used linear algebra in my multivariate statistics courses.
You may wonder, "What's a clinical psychology student doing talking about algebra and calculus?" I used to be electrical engineering (and took linear algebra) until I saw the light
I don't delude myself that I know a LOT about ANY area of mathematics, lol.
What I do know has been vastly useful though. Just seemed like the OP was professing general ignorance about the topic in general, but you're probably right.
I also doubt there are any significant areas of mathematics which aren't subjects of current research either. Kind of the nature of math, there are always more questions you can ask about any area, new ways to look at things, etc. Truly the one topic which is genuinely inexhaustible.
"Malo periculosam, libertatem quam quietam servitutem." -- Jefferson
in which case, in my experience, you do not understand that there is any sort of disconnect between correlation and causation at all. This has often been true even of articles in "peer-reviewed, science-oriented" periodicals. And worse.
In my opinion, anyone working for a newspaper or other major news source should be required to take a basic course in symbolic logic, with an emphasis on just exactly that: correlation vs causation.