Good Physics Books For a Math PhD Student? 418
An anonymous reader writes "As a third-year PhD math student, I am currently taking Partial Differential Equations. I'm working hard to understand all the math being thrown at us in that class, and that is okay. The problem is, I have never taken any physics anywhere. Most of the problems in PDEs model some sort of physical situation. It would be nice to be able to have in the back of my mind where this is all coming from. We constantly hear about the heat equation, wave equation, gravitational potential, etc. I'm told I should not worry about what the equations describe and just learn how to work with them, but I would rather not follow that advice. Can anyone recommend physics books for someone in my position? I don't want to just pick up a book for undergrads. Perhaps there are things out there geared towards mathematicians?"
PDEs now? (Score:5, Insightful)
You are in your third year of a PhD program and are only now studying PDEs? Aren't they more of an undergrad topic, or have schools gotten weaker? :)
p.s. First post!
Making the math tangible does help (Score:5, Insightful)
Yes, stick to the mathematics. (Score:3, Insightful)
Some recommendations from another Math Ph.D (Score:5, Insightful)
Re:PDEs now? (Score:5, Insightful)
There can be a world of difference between graduate and undergraduate PDE courses; it's not like everything that's known about PDEs can be taught in a couple of undergraduate semesters. I expect most undergrad PDE courses are geared towards showing you the methods that work for a few classes of linear PDEs; a graduate course might be concerned with the analytical underpinning of those methods, or maybe about numerical and analytic techniques that are useful in solving classes of nonlinear PDEs, etc.
That being said, though, from the way the original question is worded, it sounds like it's the first time this person has seriously encountered PDEs. Not having this happen until the third year of a PnD program does seem a little odd.
p.s. No, you're not.
Re:Books (Score:4, Insightful)
Vector Analysis (Score:2, Insightful)
Re:Seriously (Score:3, Insightful)
Re:What? (Score:3, Insightful)
Wow, the level of ignorance here is astounding, that you would get moderated so highly. Real PDE (as mathematicians study it) is HARD, and requires a heavy background in analysis. This is not the same as undergrad "PDE" courses.
This is like the high schooler saying "Why are you taking algebra as an undergrad" to a math major studying abstract algebra. Its the same word and the topics are related, but its not even close to the same thing.
Re:Some essentials (Score:2, Insightful)
Re:PDEs now? (Score:5, Insightful)
PDEs are not normally part of a math degree. They do form the central basis to applied math degrees. People in the engineering and physical sciences have a great understanding of applied math, but they have little to no understanding of pure math. If you get a BS in a physical science or a BE from any decent university, you will basically have a minor in applied math (adv. calc, ODEs, PDEs, probability, statistics, nonlinear dynamics, complex analysis, and calculus of variations). But you have not even scratched the surface of pure math. Mathematicians worry primarily about pure math. To teach PDEs would be insulting to them due to its lack of generality. As many physicists and engineers have learned over time, if you have a difficulty in understanding mathematics that applies to your field, the worst person you can go to for help would be a mathematician that hasn't studied applied math. The best person you could go to would be a mathematician who specialized in applied math.
incorrect (Score:5, Insightful)
Most areas of science strongly rely on philosophy, and most scientists understand it poorly, usually to the detriment of the technical quality of their work. You can see this all the time, from physicists publishing embarrassingly poor papers on how quantum mechanics "disproves free will" (apparently without even an undergraduate understanding of free will), to AI researchers with little background in philosophy of mind, to statisticians rediscovering the problem of induction every few years. Not to mention the very naive understanding of the "scientific method" that an intro course in philosophy of science might be useful in addressing.
In any case, pure (as opposed to applied) math has not very much to do with the hard sciences. And there is furthermore just not enough time to fit in everything people need. A good understanding of computer science is, for example, required for most technical fields these days as well, and also fairly under-taught; probably I'd put it ahead of physics in importance to most non-majors.
He said "Mathematician" (Score:5, Insightful)
The mathematician goes off for three weeks filling in all the gaps and "leaps of faith".
He comes back to the book, and reads page three.
Mathematician flings book against the wall, and goes off and finds something more rigorous to read.
As I remember them, the Feynman lecture series were finely crafted instruments of torture for those who delight in rigor. Personally I think he entitled the wrong book "You must be Joking!"
Re:Books (Score:2, Insightful)
In the same vein, the following two might be worth checking out:
Mathematics for Physicists by Philippe Dennery and Andre Krzywicki, Dover, 0-486-69193-4, (1995)
Mathematical Methods of Physics, Jon Mathews and R.L. Walker, Addison Wesley (1971) 0805370021
Along with Arfken's book, these two were used in an upper level Mathematical Methods in Physics class when I was an undergrad.
Re:Books (Score:4, Insightful)
All the best, are.
Re:PDEs now? (Score:3, Insightful)
My question is:
How does a PhD student get that far without any physics courses?
Re:Some essentials (Score:2, Insightful)
Re:Partial differential equations (Score:3, Insightful)
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.
Re:Books (Score:4, Insightful)
I'll admit that my mathematical intuition isn't the greatest, though I can't help but think that this was intentional on Feynman's part, as to weed out those with weak mathematical skills from his freshman lectures.
Bear in mind, the classes these lectures were delivered to were at Caltech in the 60s, I believe. Those with weak mathematical skills didn't get in.
Also realize that many of the undergrad lecturers at Caltech take it as a badge of honor to see how much they can shovel at the undergrads, equating density and difficulty with learning. You might find that nearly all of the students in the class were spending quite some time poring over those lectures to figure them out - not because the profs wanted to weed them out, but because that's simply how things were/are done at Caltech. On the other hand, that's something you didn't want/need to do, valuing your own time and sanity, and not staring an "F" in the face if you didn't.
I'm in the same boat, I wouldn't have stood a prayer in that environment either.
I keep a copy of all 3 volumes on my bookshelf, as they are occasionally handy. However, I wouldn't dream of using them as my only reference.
Yeah, Feynmann wouldn't make a good reference but he's definitely entertaining and insightful. Probably about like Ambrose Bierce in that regard.
Re:PDEs now? (Score:1, Insightful)
abstract algebra for java programmers... (Score:3, Insightful)
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.
Re:Standards have slipped then... (Score:2, Insightful)
The PDE that you took as an undergrad is indeed intermediate level calculus. That's not the PDE the OP is talking about. Graduate level is an entirely different creature. Full of scary things like Sobolev spaces, weak derivatives, semigroups, and the myriad types of existence and uniqueness proofs, I'd bet you would not recognize the majority of the questions on the OP's exams.
Re:Standards have slipped then... (Score:3, Insightful)
You're exactly right - undergrad DiffEQ is more of a "Survey of Differential Equations". It's an overview of "safe" equations - most all of the work has answers that are trivial to find. My M410 professor always joked that his job was to protect us from differential equations. That being said, 300 or 400 level DiffEQ serves as a good foundation for more advanced classes in the subject.
My area of expertise is in three-dimensional electric field modeling. It's very frustrating and enlightening at the same time. And difficult to craft mathematical models that can converge on a solution. My feeling, from a non math major (my degrees are in electrical engineering), is that a career involving differential equations will be one that requires tenacity and perseverance.