How did you learn the required math for physics?

[Etranger]

Hey,

I already know the "monkey calculus" that I was taught in high school and would only need some 2 weeks' worth of work to start tackling a book like Boas' Mathematical Methods in the Physical Sciences. However, some people have argued that, even as a physics major, it would be beneficial if I were to study from both volumes of Apostol and Spivak. Heck, a friend of mine suggested Artin's Algebra and learning all the algebra I need from there.

Now, I welcomed the opportunity to learn the math in a formal manner but I'm starting to doubt my initial position. Progress is very slow and going through it alone, while certainly possible, can on occasion be somewhat tedious. I've completed most of Spivak's first chapter and have started working through some problems. (only the first 3 questions so far; they have many parts) While I find those things interesting, at some point I can't help but ask myself when this will end and I stop working altogether.

So, I'm asking those who are/have been grad students and/or physics majors: do you feel that such formalism is necessary? Is it useful when doing physics? I like studying pure math but at this point, I just want to get my hands dirty with some physics. I have this book and I just want to get started already!

Thanks

[WhoaNonstop]

I would just start with the Physics and when you encounter the math, look it up then.

-Riley

[Etranger]

Am I correct in assuming that by that, you mean "pick up the math as you need it and proofs while nice, aren't a necessity"? If yes, when taking grad courses or doing your research in condensed matter, did you ever feel that this somehow held you back? Is studying math from the perspective of a pure mathematician useful in your work?

I'm just slow when it comes to pure math...

[blighter]

Am I correct in assuming that by that, you mean "pick up the math as you need it and proofs while nice, aren't a necessity"? If yes, when taking grad courses or doing your research in condensed matter, did you ever feel that this somehow held you back? Is studying math from the perspective of a pure mathematician useful in your work?

I'm just slow when it comes to pure math...

Proofs aren't a necessity when it comes to most physics. They are only there to convince you that the result indeed holds. But when it comes to topics like the String Theory, you might find yourself in the need to develop your own math. But as long as you stay away from those, you won't require proofs.

[bfollinprm]

To be a physicist, you need a great deal of mathematical intuition. It's more useful to be able to intuit whether something is true (and the conditions under which it holds) based on some rough sketch of a method or some toy model than know the formal proof, which often solves the problem in a direction that isn't very helpful in making constructive use of a theorem. The reason being the primary skill of a scientist is to know what hunches are worth turning into research projects quickly; if you need a theorem later the proof can be found in a textbook, journal, or farmed out to the math department. To obtain this intuition it's sometimes useful to put some time into learning the mathematics, since if you don't know anything you obviously will have very bad educated guesses, but knowing the proofs is in no way essential. I'd shy away from heavy mathematical treatments in favor of model-based treatments, like those found in mathematical methods texts for the physical sciences. To be clear though, this is my own opinion, there are lots of physicists who think knowing the proofs are really important to them, though I'd posit that when they're using them they're really being mathematicians as well as physicists.

An example of what I mean here I always give is the idea of completeness of a basis in spectral theory. Almost all analytical physics rests on Fourier analysis--any function can be broken into a sum of sines and cosines of different wavelengths of varying amplitudes (fourier coefficients). The proof that you only need sines and cosines (and a rigorous accounting of the conditions) requires at least half a year of graduate-level analysis, a re-definition of the integral, introduction of abstract topological spaces, Sturm-Louisville theory, and Green's functions, but all a physicist need know is this:

We know it works for many functions, since we've tried it many times and get plots that look exactly the same. Let's thing about the general case. Large wavelengths capture large-scale fluctuations. Small wavelengths capture small-scale fluctuations. If we first fit the general shape of a function using large wavelengths, we should be able to correct for local inconguencies by using small wavelengths, regardless of the shape of the function (we know from example it works for functions that don't look even close to sine waves, so there's no reason to suspect this wouldn't be true). There might be an issue if the function changes its value too quickly--that is, has a large slope. Large slopes mean a large derivative, so we probably need a condition on the derivative. Since we can go as small as we want on the wavelengths, that condition is probably only that the derivative doesn't become infinite--in other words, that the derivative exists at every point. If the derivative doesn't exist somewhere, we're probably ok unless we get very close, since everywhere else around that point the function behaves totally fine. Since again we can go as small as we want on wavelengths, it seems reasonable to suppose that we can trust the fourier transform except exactly at the point where the derivative becomes singular--that is, if we cut off our function before the singular derivative, we should be totally fine.

That argument is piss-poor, and any mathematician who reads it will be outraged (rightfully, if I were calling it a proof), but it gives motivation to, and intuition for, the methods of Fourier analysis and the main conditions upon the fourier transform existing, which is all you really need to use it.

[King Vitamin]

It really does come down to your style - some theorists are very phenomenological, some like to have an excellent grasp of mathematical results/intuition without learning heavy machinery (I'm guessing this is what bfollinprm is referring to), and some physicists are essentially pure mathematicians (if you want to be Witten). I personally am between the last two camps - I like to spend a lot of time studying very pure math not just for the intuition but also to understand topics from a point of view that requires more mathematical machinery (formalism) but once the formalism is there physics takes a (in my opinion) simple/elegant form. For instance, you can study the quantum Hall effect without using homotopy/Chern-Simons theory, but I like to understand it in terms of geometry because I like learning the formalism and interpretation. It's all based on your style - there have been nobel-prize winning theorists just about everywhere in the above spectrum.

With that said, it is true that you rarely use quite the same level of rigor if you're doing a proof in quantum field theoretic condensed matter compared with what a pure mathematician does. But I agree with bfollinprm's assertion that you gain intuition from learning proofs for general problem solving skills, and pick up important tools.

Personally, I wish I had taken even more pure math as an undergrad than I did. And I currently spend a lot of time studying pure math books.

[guy]

I think that one the essential components of competency in physics is having a physical intuition for what the math means. A stellar mathematician will be a much less successful physicist than a person understands what the math is communicating in a given context.

The points that have been made about learning the mathematics when it comes up and developing an intuition from studying formal mathematics form only an apparent dichotomy. The core concept is what I would describe as "relevant rigor". You can get away in physics without knowing all the details--this is what is often construed as physicists ignoring certain components of the mathematics. That fact is, the good physicists don't ignore the mathematical assumptions they make. Rather, they have learned when certain requirements, e.g. continuity, that have to be reasoned for in formal mathematics are simply a non-issue in (certain) physical systems.

I am certainly of the group who has learned mathematics from the physics. For example, I never formally learned series expansions and other topics from high school Calc BC except through physics courses. On the other hand, I learned linear algebra in a very formal setting and found that most of what I learned was of little value for a course in quantum mechanics (and other subjects heavily dependent on linear algebra). The terse appendix at the back of Griffiths is much more useful in this regard.

The reason why physics isn't "just math" is in the same spirit as the answer to your question. It's true that a lot of the mathematics used in physics is mechanical and an answer just pops out. However, the reason for this is that the problem has already been framed in the right way. I would try to learn the mathematics by understanding what assumptions are required for a certain theorem or method to be applicable--and which ones are trivially true for the physical system under consideration. Once the math is contextualized, it can be treated as an independent representation of the physics. That is as long as you adhere to mathematical laws as well.

I guess my answer is really a qualified "pick it up as you go". Just be careful to pick up the relevant bits.

[Etranger]

I appreciate all of the input. It's interesting to see your approaches. I've decided that as of now, I will stick to getting the math I need and then going forward. I still like pure math, so if I ever come across anything that I find cool, I'll try learn it.

Plans for the coming weeks: using OCW to brush up on Calc I-II. Then studying Mechanics from Kittel's book.

A related question: would one need to buy a book such as "Div, Grad, Curl and all that" or do 18.02 (OCW) to learn the required vector calculus for physics? Or would the first chapter of Griffith's E&M and that of Kittel (mechanics) suffice for these purposes? If one can get away with it now, would one be in serious trouble when doing grad courses in the subjects? I tried opening up the first few pages (preview somewhere on the net) of the Landau-Lifshitz mechanics book and it was a lot of scary differential equations. I guess what I'm asking is: would I also need to properly understand the theory of differential equations or if a cook-book approach will be enough.

I don't know why but the word "homotopy" looks so...elegant.

[bfollinprm]

You need differential equations as taught out of a mathematical physics book like Boas or Arfken. There's no need to learn all the existence proofs and things you'd focus on in a PDE class in a math department--if you're ever in the situation where you need to know if there's a solution, you'd also want the solution, and mathematicians (and the books they write) aren't always very interested in those.

If you've never learned vector calculus, you should take a vector calc class in a math department, since it's quite useful to actually understand continuous vector spaces rigorously. Barring that, pick up div, grad, curl and all that, it's only like 6 bucks and expands the first chapter of Griffiths out to about 200 pages. If you already have had vector calculus, then the first chapter of Griffiths should be fine.

[taraneh]

as a physics major I tell you, if you want to be successful in both math and physics you need to know mathematics and as you know more, everything will be easier for you. The start of it always is hard but as you progress you will enjoy learning mathematics.
in fact physics without mathematics in nonsense and it has no meaning but you don't need to learn everything. the book "Mathematical methods for Physicists" by George Arfken is a very good start if you have a moderate mathematics background.
you can go through the book and waive chapters which you don't need.