I am becoming a secondary math teacher, and have completed Calculus III, and proofs. Within about a year to a year and a half, I will have combinatorics, abstract algebra, an advanced course in statistics and probability, Diff EQ, Linear algebra, and a couple of others completed. I have been reading physics books for non-physicists like Brian Greene (as an example) for a number of years now. Since however, I now have the mathematical underpinnings to understand physics on a more formal level, that is what I would like to do. I also want to study physics because I enjoyed Calculus III a lot, and don't want to lose track of it. I know where I would like to get in terms of having a command of physics. I don't however, really know where to start. I would like to eventually (emphasis on eventually) understand why Quantum mechanics and relativity don't jive with each other, and then be able to learn some of the physics behind string theory. I don't know if that is realistic for a laymanor not. I would however, like to try...with the full understanding that it ovbiously won't happen overnight. So while I know I could buy a college physics textbook to start with and churn out all of the chapters over a long period of time, I am not sure how profitable that would be to get me where I want to be. So if anyone has any ideas on how I should proceede, then they would be much appreciated. Thanks.
I've been doing the same thing, with a similar starting background. In part, finding your own path is a good chunk of the fun, IMO. So here's my path, for what its worth. I started with popular accounts (like Greene) as well textbooks (Feynman Lectures on Physics). But I didn't start doing problems. I just read various popular accounts and texts for a bit to see where my interest lay. But eventually (again IMO) I needed to actually work as many problems as possible from the real text books to gain a deeper understanding of what is going on. My A-List of books that inspired my path (and which I tried to work as many problems as possible): Shankar - Principles of Quantum Mechanics Arnold - Mathematical methods of Classical Mechanics Rudin - Real and Complex Analysis Rudin - Functional Analysis Currently working thru - Thirring - Classical Mathematical Physics Supporting books I worked thru to understand the above: Shankar - Basic Training in Mathematics Lee - Intro to Smooth Manifolds Bamberg/Sternberg - Course of Mathematics for students of physics Rudin - Principles of Mathematical Analysis Artin - Algebra Munkres - Topology
Cool! Is the Rudin book on real and complex analysis a textbook for the math class "real analysis"? Or is that something different. I am going to take real analysis eventually, but I will wait until I am working. My first goal is to get highly qualified under N Child Left Behind to teach mathematics, then I will get a job, and then take real analysis. I dig calculus! :-)
Those are good thoughts. I am laboring under a couple of assumptions, and please feel free to correct me if I am wrong. One is that calculus is essentially the framework for classical physics. So regardless, I am going to be working my way through a college physics textbook. I figure it sure cant' hurt. My interest in physics definitely leans however, to 'the weird stuff', which leads to my second assumption. I am presently a musician (a professionally trained classical guitarist). I know that student composers who skip over traditional composing to try and jump right into the weird stuff are generally terrible composers. To really know how to break traditional rules of composition, you have to know them intimately well. I am assuming there is a similar analogue in the world of physics. That is to say I am assuming that truly appreciating the pretty far out realms of theoretical physics is predicated upon having a solid command of classical physics, Einsteins work, quantum mechanics and then some of the more exotic theories last. I certainly will never have the command of this stuff that a post-doctoral fellow in theoretical physics at Oxford has. I can sure try to get as close as I can though over the course of my life.
It sounds like we have very similar goals. For instance, one of my goals is to get as rigorous understanding as possible for quantum field theory. I tried jumping into the deep end and reading the semi-technical accounts, like "How Is Quantum Field Theory Possible?" by Auyang. (It wasn't ~200 bucks when I bought my copy years ago :) ) But (IMO), the only thing that really worked and gave lasting understanding is to take no short cuts, and build up the foundation a piece at a time, and do the problems. Your music analogy is very accurate (I'm a professionally trained oboist. I also played classical guitar back in grad school). Rudin's "Real and Complex Analysis" is a common grad school text for analysis. The advanced undergrad one is "Principles of Mathematical Analysis". Another resource I use is ocw.mit.edu. Lecture notes/syllabi/suggested books/etc. Take a look at the topics in a standard mathematic methods for physics book. For instance: http://www.amazon.com/Mathematical-...=sr_1_1?ie=UTF8&s=books&qid=1260324507&sr=1-1 That'll give you an idea of the subjects to be fluent in. I'm personally biased towards the math treatement (theorem/proof) style than the physics style (ala Arfkan). Time will tell where your preferences lie.
Those are some awesome books. By the way if you don't mind me asking, I've wanted to try Bamberg/Sternberg but because I don't have the solutions to the answers I didn't. Can you tell me the methods of studying you used for the book? If it's not so much to ask, maybe for all the books. I'm greatly interested in your study methods.
Nothing magical about my study methods. Just like everything: time and effort. Don't give up, and keep plugging away at it. For instance, for Rudin's RCA: read the chapter quickly, just the definitions, and theorems, ignore the proofs. That'll give me an idea of the main results of the chapter, possible applications, etc. Then start over, and work thru each theorem. Rudin's RCA is especially great (IMO) because each theorem is like a bunch of mini-problems where you have the answer. I.e. between each step in the proof, there is generally a gap that you need to fill in yourself. Any you always have the answer : the next line in the proof. A given proof generally has a fairly straightforward idea behind it: i.e an "executive summary" encoded in it, and then all the corresponding technical details. Make sure you see the forest, AND the trees. Draw pictures. Construct the simplest non-trivial example that satisfies the theorem. Remove various hypothesis in the theorem to see why the theorem becomes false. Work as many problems as you can. Work to become self-sufficient at determining when your proof is sound vs. where there are weak parts. Filling in all the gaps of the proofs as mentioned above is good practise in this area. Ask on the forum if you're not sure and have exhausted all other inspiration. Be sure to write down what you do. All the notes, examples you come up with, etc. Write it down in such a way that when you come back to it a year later, and re-read the proof/problem, you can quickly find your notes for that proof/problem. Another thing I do is use multiple books of different levels/different perspectives: For instance Rudin's RCA will motivate things (by providing applications) that were missing when the ideas were first presenting in Rudin's PMA. Or Munkres Topology will treat some topological point in more detail than Rudin. For Bamberg/Sternberg: I primarily used that for its nicely worked out examples of exterior calculus. My goal was working thru the problems in Arnold's book, and I leveraged Bamberg/Lee as needed in order to make sense of Arnold. For the more calculation/less proof oriented problems (Arnold for instance) if I didn't have the answer, I make up my own simpler problems for which i do have the answer. Ex: how vectors/forms transform under change of variable. Arnold didn't have a specific problem on it, but I knew that the answer had to be independent of the choice of coordinates, which is enough info to check my answer.
Calculus 3 is important when studying subjects like thermodynamics, classical mechanics and electrodynamics, but you also need a good foundation in partial differential equations (which i think is very important in classical physics). Since the stuff you learn in calc 3 (curl divergence..etc) are like the ABC's in E&M (Maxwell's eqns). To actually solve some of the simplest problems you need both PDE's and Calculus
Since you have a lot of the math and enjoy them. My best suggestion is study Partial Differential equation. They cover important stuff that is needed for electromagnetics, electrodynamics, quantum mechanics. I am a self studier, I have quite a few books on this topics and ODE. Books like: Differential Equations by Dennis GF.Zill and Michael R. Cullen, Partial differentation Equations by Asmar, Partial Differential Equations by Strauss. Elementary Differential Equations and Boundary Value Problems by William E. Boyce & Richard C. Diprima. I only starting to study for a little while, on the subject I studied, I like Differential Equations by Dennis GF.Zill and Michael R. Cullen of all books. It is primary a ODE book BUT it has quite a bit on PDE, Bessel, Legendre Strum Liouville sections that are pretty good. Particular if you are self study, this has more detail formulas for you to get into the topics. My main book is Partial differentation Equations by Asmar. It is a complete book but lack the basic derivation of formulas like the Zill book. I think this two will make a very good pair. I have solution manual for both. I am currently studying this topic. For self studying this make a very good pair. There are not as many PDE books as the other Calculus, choices are very limited. I would have bought more if I can find them. I'll try to update after I study more.