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I enjoy reading up on new topics in physics and astronomy, and learning the concepts and ideas in the major theories, like relativity and quantum mechanics. Unfortunately, one thing that I lack completely (and I envy others for) is the mathematical understanding. I would like to start studying the mathematics as well, from Newtonian gravity to more advanced problems, but I was wondering what the best progression of topics would be. How should I learn the mathematics of physics?

I know that there are other questions on how to learn physics, and I apologize if this seems like a duplicate. However, I don't think that the mathematics have specifically been addressed in previous questions.

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voithos: It would help if you would give us the level of your current mathematical level. –  Michael Luciuk Jun 1 '11 at 18:48
    
@Michael: It's quite pathetic, in fact. High-school level trigonometry and algebra, along with some college-level pre-calculus. Math as a subject has never been difficult for me (yet), but I haven't been able to study much after high school. –  voithos Jun 1 '11 at 19:00
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voithos: There's a non-rigorous but "friendly" site you should try, khanacademy.org. It'll give you some help though calculus. Then go for some more formal texts or courses. –  Michael Luciuk Jun 1 '11 at 19:22
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I added this KhanAcademy.org to this PSE-List of freely available physics books has a lot of free online resources. Tanks Luciuk. –  Helder Velez Jun 2 '11 at 0:59
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5 Answers

up vote 2 down vote accepted

One particularly fun book that I really enjoy is "Road to Reality" by Roger Penrose. However it is most certainly not as self contained as it describes itself, and should be supplemented by other sources (both math and physics) if you want to have a good understanding of the material. But still, I personally love his writing style and he still keeps the material interesting even if you get lost in the math. Also, I find the book very inspiring and motivating as each topic I get stuck on, I delve into with other books and once I have a decent understanding, I am one chapter closer to the front lines of physics.

The basic idea of the book is the first half is dedicated to mathematics, starting from the most basic definitions of real and complex numbers, through calculus, complex analysis, group theory, and differential geometry. From here he goes on to the physics, starting with spacetime, Lagrangian mechanics, quantum mechanics, the standard model, and going to QFT, Cosmology, Supersymmetry, string theory, and, of course, Penrose's favorite, twistor theory.

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Ah, thank you, that sounds highly interesting. I will most definitely take a look at that. From your description, it seems like the physics portion of the book start somewhat mid-level and doesn't cover Newtonian mechanics, or nuclear physics, etc. If that is the case, then I would probably use a more general-purpose book at first before jumping into this. In general, though, it sounds like a highly enjoyable read. –  voithos Jun 1 '11 at 19:09
    
The book "claims" that you don't need any background knowledge to understand it. In practicality, it would be useful to already have some background in physics, but I wouldn't hold yourself back from looking at it if your interested. Penrose, in my opinion, was pretty successful in allowing readers without much or any background still get a lot out of the book. But I would agree it shouldn't be a substitute for the normal progression of physics books (Classical Mechanics, E&M, QM, Statmech, etc). –  Benjamin Horowitz Jun 2 '11 at 2:21
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Basic areas of mathematical physics are:

  • analysis
  • linear algebra

Depending on specific physics area like QM or ART u have to enhance ur knowledge with functional analysis or differential geometry. But without basic areas knowledge u will not see any light at the end of the tunnel ;)

Sites like hyperphysics try to keep the math as simple as possible. Also watch some online lectures on MIT for example for basic areas courses. Some universities teach thier physics student by physics Profs., so the examples and lecture is not too abstract as hearing a math. Prof lecture.

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One great place to start (assuming a knowledge of calculus) is a college upperclassman level E&M book like Griffiths. While Electrodymamics is an interesting field in its own right, and gives a whole lot of motivation for later topics, its place in a physics curriculum is also the first place that students are introduced to the machinery of solving partial differential equations in several dimensions, including vector calculus, separation of variables, spherical harmonics, etc. A grounding in these things is also essential before moving on to something like differential geometry.

As stated above, linear algebra is something that is also univerally necessary in higher-level physics. You can find a ton of resources on this by searching around (I can't recommend a good textbook, I grabbed a library textbook and just did all the problems that had answers in the back of the book, and that was years ago). Linear algebra is pretty much a prerequisite for a lot of the higher level math, too--you're not going to make much sense of tensors or mapping operators in topology if you don't have some sort of grounding in linear algebra.

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I would start with any good high school or introductory college text on physics. The textbook will not only show you conceptually what is going on, but will, out of neccessity of fulfilling the purpose for which it was written, give you the math along with it.

One caveat to that is that physics will not teach you calculus, and a college-level text will assume some calculus exposure. If you have not had differential and integral calculus, then stick to a high school level textbook, or... take a calculus course (or get the book and self-study).

Any good textbook will be structured to take you through the topics in a logical order. You will encounter free-body mechanics before you get to hydrostatics, hydrostatics before aerodynamics, electricity and magnetism before optics, and etc.

General guide for picking a GOOD text book:

1.) Look for texts which have more than one author (a collaborative effort) and are NOT the first edition. Something that is in its, say, fifth edition, is tested and proven and still in demand: an excellent sign of underlying quality.

2.) Look up community college physics courses on the web and write down what text they use, then surf for book rating comments on Amazon and elsewhere.

3.) Make sure the text book has LOTS of pictures.

4.) Make sure the text book has answers to odd-numbered problems (or answers to even-numbered... whatever) in the book. There is NO better way of verifying you grasped the material than working out a problem. You want to know that you got the right answer; and you have no teacher to go to.

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You might also try working through the math-physics relationship from the other end: start with a math book. Larson and Stewart's Calculus textbooks are some pretty general books that cover basic calculus--starting from algebraic/geometric principles--to elementary multi-variable calculus and differential equations, with some other cool stuff thrown in between. Tons of problems/answers/diagrams/examples, and generally some very good explanations, as long as you're willing to think about the ideas. You may even find these books too easy--I don't know.

Although these books will not necessarily give you much about the physics concepts that you are interested in, I mention this entryway into studying physics because:

  1. The understanding of many mathematical concepts, especially those involving higher-level calculus and linear algebra, is so very helpful to a thorough understanding of physics. However, I am not talking about pure number-crunching or calculation power--but rather, an understanding of the relationships in orders of magnitude, or the nature of functions' behavior over time (differential equations), or the effects of dimensions and units. The concept of simple harmonic motion--in an oscillating spring, for example--grows so much clearer when you have a good mathematical conceptual understanding of sinusoidal functions, even without any numbers or physical parameters to plug in. And that is one of many examples.

  2. The books that I mentioned, along with most general college math textbooks, have some great example sections with applications in physics. With these problems as a springboard, perhaps you will be able to use your conceptual knowledge of a physical phenomenon to understand a similar idea in math--and vice versa.

One more suggestion--One, Two, Three...Infinity by George Gamow has some great explanations of all sorts of physical phenomena, with some good math to back them up (though the math in this book is often more conceptual than technical). It is one of my favorite books of all-time.

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