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I'm a freshman undergraduate. I've got my sight on mathematical physics. I love math but I don't have the talent nor the inclination for purely abstract mathematics. I also love physics.

The only math for physics I know is cook-book calculus 1. (I have deep interest in mathematics, I have finished an intro to proof book, some introductory set theory, cardinality of infinite sets. Right now I'm working on mathematical logic.)

I have this compulsion to prove every result and understand why some math thing is defined so. As such, I am very slow in learning mathematical methods. Cook-book calculus sucks. But where I come from we're now allowed to take math classes. I'm worried that if I don't study the pure maths version of calculus I might miss some important result or the understanding which might be really useful.

Self-studying mathematics really takes time. I want to be as efficient as possible.

My question is, how should I learn the math behind the math methods? What mathematics should I learn for mathematical physics? Is learning the mathematics really that important?

If possible, please recommend some good books.

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closed as not constructive by David Z Apr 15 '13 at 7:10

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Your question is a bit open-ended, maybe you can define a bit more precisely what you are looking for. For the math necessary in standard physics course have a look at "Mathematical Methods in the Physical Sciences", by M.L. Boas which gives an overview but does not contain a rigorous axiomatic path through the topics. –  Alexander Feb 22 '12 at 15:15
    
@Alexander: I want to understand as much mathematics as possible. But it would really take too much time, so I want to know which would really be useful. –  Ron Feb 25 '12 at 12:16
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3 Answers 3

Skip to the last paragraph for the bottom line, otherwise read on:

First you need to take more of the core math sequence to learn vector calculus, and at every chance you get you should try to deepen your intuition for it (what does it mean physically when you evaluate the divergence of a vector field, etc). Then you'll probably want to take classes (often required) in linear algebra, methods of solutions to both ordinary and partial differential equations, and the applications of fourier analysis to just about anything you can find (including its application to the solutions of differential equations). No matter what type of physics you do, knowing those topics cold will be crucial. It's also handy to gain some familiarity with special sets of orthonormal functions that come up often in solutions of differential equations (Legendre, Bessel, Hermite, etc).

In general I recommend you master the math mentioned so far in the context of physics applications, before diving in to the more abstract realm if it interests you - at least, this is how I learn best, by first seeing the concrete applications before abstracting. Most books with titles along the lines of "mathematical methods in physics" are good for this purpose, just follow some recommendations and then choose your favorite based on your own experience. I'll second the recommendation of Boas in the Alexander's comment.

At this point you'll probably have enough of a foundation in mathematical methods to work in many areas of physics, but it would still be a wise investment to learn some abstract algebra and group theory in particular, ideally with an emphasis on Lie groups. This is crucial for gaining a deeper understanding of quantum mechanics, and will be useful regardless of whether you go on to do experiment or theory in particle physics, condensed matter / solid state, or atomic physics / optics. Again, I'd recommend you seek out classes along the lines of "group theory for physicists" first, although if you really like it you could probably benefit from an additional class from the pure math perspective, too. Elementary ideas in what's known as representation theory come up frequently here and it's good to be aware of the connections.

Next, if you're thinking of working on either particle physics or gravity, you might consider a class in differential geometry (or better yet just take General Relativity, since the first few weeks of a GR class are typically a crash course in differential geometry). You'll probably have learned pieces of this subject already by this point (i.e. what is a manifold, from your vector calculus or if you've learned about Lie groups), but a course in diff. geometry can help tie things together and teach you new conecpts.

By that point you'll have a better idea of what it is you want to research, and can plan future math classes (or the lack thereof) accordingly. Popular choices are more specialized topics in representation theory, topology and algebraic topology. And I'm sure other people might chime in with their favorite math electives.

Bottom line: Beyond the core calculus classes and linear algebra, I'd recommend taking math methods classes from the physics department first before allocating too much time to self-study on your own, and generally your knowledge of how the math "really works" will just grow with experience. If your department doesn't offer a math methods class, then make sure you have access to a math methods textbook to consult as new things come up. To learn more advanced topics (group theory, representation theory, other parts of abstract algebra, differential geometry, topology and algebraic topology), again look for classes before going it alone, but now some of them might be back in the math department. One last thing to keep in the back of your mind: you'll probably want to develop computer programming skills, too.

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For the Lie group and differential geometry part, I strongly recommend Fecko: "Differential Geometry and Lie Groups for Physicists". –  Tobias Diez Feb 23 '12 at 14:53
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If you really wish to understand physics at the fundamental level, start with Group Theory and Topology. In fact, I will say that there is no other way to do this. This is my opinion (bring on the drive-by downvotes!) and I am sticking to it. :)

As for books to start out with: (cheap and good)

  • A book of Abstract Algebra- Charles.C.Pinter, Dover Publishing- $10.
  • Introduction to Topology- Mendelson, Dover Publishing- $5.50.
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By "topology" you should specify that you mean homology and bundles, which are useful, and not open-sets and perfect sets, which are not. –  Ron Maimon Feb 23 '12 at 15:52
    
Yes, you are right. Good point. –  Antillar Maximus Feb 23 '12 at 17:23
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You may also want to have a look at "Mathematical Methods for Physics and Engineering" by Riley, Hobson and Bence (ISBN 978-0521679732). It can be used for reference; and includes a separately-bought students solution manual.

I say look at it before you buy/use it. And don't use any single book only; taking classes is invaluable, because sometimes you can read a chapter/paragraph over and over again and get confused and, therefore, having someone else overcome a hurdle can go a long way.

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