Textbook/study plan questions

[jb188397]

I am self studying physics as a hobby with the very broad goals of eventually learning theoretical analytical mechanics, EM, general relativity, etc. I have no specific deadlines. My two questions are:

1. With regards to learning the required math, if there is no particular rush, is it better to work with through dedicated math texts for linear algebra, ODE, differential geometry concurrently with physics texts or just get through the standard methods texts like Boas and Arfken to get the basic techniques down and refer to the individual math texts as needed for enrichment?
2. Regarding learning physics, is it better to work through one text at a time on a particular topic or juggle several to see different perspectives?

Assume that I have studied mechanics, EM, multivariable calculus/linear algebra in college a long time ago, and more recently have studied single variable analysis I.e. Rudin and am comfortable with proofs.

Thanks!

 

[fresh_42]

I am self studying physics as a hobby with the very broad goals of eventually learning theoretical analytical mechanics, EM, general relativity, etc. I have no specific deadlines. My two questions are:

1. With regards to learning the required math, if there is no particular rush, is it better to work with through dedicated math texts for linear algebra, ODE, differential geometry concurrently with physics texts or just get through the standard methods texts like Boas and Arfken to get the basic techniques down and refer to the individual math texts as needed for enrichment?
2. Regarding learning physics, is it better to work through one text at a time on a particular topic or juggle several to see different perspectives?

Assume that I have studied mechanics, EM, multivariable calculus/linear algebra in college a long time ago, and more recently have studied single variable analysis I.e. Rudin and am comfortable with proofs.

Thanks!

I truly believe mathematics used by physicists and mathematics used by mathematicians are two different languages. And I mean language in a literal sense: different vocabulary, foremost different perspectives, and often different formulas. E.g. physics requires measurements, and measurements require coordinates aka frames. Hence you will meet coordinates in physics all the time: linear, affine, even projective, curved, local, in unusual metrics, and so on. The average mathematician hates coordinates. So if you don't want to spend time learning both, I recommend specifically searching for texts, books, pdf that address physicists particularly. E.g. look for "Differential Geometry for Physicists" instead of just "Differential Geometry." Here is a list of tips and tricks https://www.physicsforums.com/threads/how-to-use-the-w-in-www.1062388/ on how to use the internet. Almost all universities publish lecture notes on their servers nowadays so that you can find everything online. And if you want to buy books, try to figure out whether they fit your purposes beforehand. Many editors have reading probes for their textbooks or on Google Books. Or you can download those lecture notes and find a copy shop to make a paperback out of it.

 

[jtbell]

Assume that I have studied mechanics, EM, multivariable calculus/linear algebra in college a long time ago,

At what level, for the physics specifically? Only first-year level, e.g. Halliday/Resnick for mechanics and E&M? Or intermediate level, e.g. maybe Symon, Taylor, or Fowles/Cassiday for mechanics, and Griffiths for E&M?

Different textbooks on the same subject usually present things somewhat differently, in different sequences, so I think it would give me a headache to switch randomly from one to another. I'd pick one and try to use it as a primary source, with one or two others as supplements for getting alternate presentations on specific topics that confuse you.

Of course, you might find that your initial choice for primary textbook isn't working out, in which case feel free to switch to a different book for that purpose.

 

[Frabjous]

What’s your end goal? Since you have some background, you can probably get to junior-senior level physics with a minimum of dedicated math study. Physics texts teach a lot of applied math.

You should work through 1 book. It can help to have other books as supplements.

 

[jb188397]

I truly believe mathematics used by physicists and mathematics used by mathematicians are two different languages. And I mean language in a literal sense: different vocabulary, foremost different perspectives, and often different formulas. E.g. physics requires measurements, and measurements require coordinates aka frames. Hence you will meet coordinates in physics all the time: linear, affine, even projective, curved, local, in unusual metrics, and so on. The average mathematician hates coordinates. So if you don't want to spend time learning both, I recommend specifically searching for texts, books, pdf that address physicists particularly. E.g. look for "Differential Geometry for Physicists" instead of just "Differential Geometry." Here is a list of tips and tricks https://www.physicsforums.com/threads/how-to-use-the-w-in-www.1062388/ on how to use the internet. Almost all universities publish lecture notes on their servers nowadays so that you can find everything online. And if you want to buy books, try to figure out whether they fit your purposes beforehand. Many editors have reading probes for their textbooks or on Google Books. Or you can download those lecture notes and find a copy shop to make a paperback out of it.

Thank you for your response and recommendations. Yeah I understand they are basically different languages. My goals really from the mathematical perspective are to be not only adept using the mathematics involved in higher level physics but also to understand why the techniques themselves work. My initial path through math started because I was reading some book on the classical mechanics utilizing change of variables for multiple integration. While I had a good intuitive understanding of how and why it worked from learning multivariable a while ago, I wanted to see the nitty gritty mathematical justification. So my goal was to read the related chapters in Spivak COM, and to prepare for that I worked through a good chunk of Rudin's analysis (which don't get me wrong was very interesting and I really liked), but by the time I got to the multiple integration chapter in Spivak I was just sort of frustrated with how long it took for me to get to that point just to understand the rigorous proof behind this basic technique (although again it was cool to see in its own right) and wasn't really learning any physics. And at the end of the day my intuitive understanding of change of variables isn't much different after all that. So I am guessing that approach is not very effective in general. Thanks.

 

[jb188397]

At what level, for the physics specifically? Only first-year level, e.g. Halliday/Resnick for mechanics and E&M? Or intermediate level, e.g. maybe Symon, Taylor, or Fowles/Cassiday for mechanics, and Griffiths for E&M?

Different textbooks on the same subject usually present things somewhat differently, in different sequences, so I think it would give me a headache to switch randomly from one to another. I'd pick one and try to use it as a primary source, with one or two others as supplements for getting alternate presentations on specific topics that confuse you.

Of course, you might find that your initial choice for primary textbook isn't working out, in which case feel free to switch to a different book for that purpose.

Thanks for you response. I want to say it was Kleppner and Purcell but it was so long ago I don't remember. I am basically starting over so I have good foundation.

 

[jb188397]

Thank you for your response. I have a couple of broad goals. For example, I want to get to the point where I can study general relativity say at the level of Wald, really understand the mathematics behind it, and then maybe take it further into cosmology or Choquet-Bruhat’s more mathematical GR text. If I get this far maybe do the same thing with quantum field theory.

I understand on the physics side, I would at the very least need to build up strong foundations in terms of classical mechanics (Morin/KK -> Goldstein), EM (Purcell/Griffiths -> Zangwill/Jackson), and QM (Griffiths -> Sakurai) +/- texts in thermodynamics and waves so my work is cut out for me, but again have no specific time constraints other than I want to see steady progress on the physics side. Very much a long term hobby project.

My initial plan, as I enjoy pure mathematics as well, was to learn all the requisite mathematics from pure math texts and then approach learning physics from the mathematical physics perspective. I don’t think this goal is even remotely realistic, would take a very long time to get to anything resembling physics, and I don’t love pure mathematics so much that I would want to do the likely hundreds or thousands of proof based exercises from textbooks needed to get a firm grasp of even undergraduate pure math degree. Thanks.

 

[fresh_42]

And at the end of the day my intuitive understanding of change of variables isn't much different after all that. So I am guessing that approach is not very effective in general.

I had a few somehow similar experiences. My brain is algebraically wired. If physicists speak of lowering or raising indices, then it's basically a formal technique for physicists they use or don't use without thinking much about it. For me, it is switching to another completely different vector space which isn't even naturally isomorphic. I think it's better to learn such automatisms and look at the principles once instead of trying to get to the core of them every single time.

Or, it took me quite some time to understand what physicists mean by the word generator in the context of (Lie) groups. It means something completely different in mathematics. And I could go on. If physics is your goal, it's better to accept some words and techniques than to resolve them whenever you meet them.

 

[Vanadium 50]

The most straightforward thing to do is probably to pick a university, and go though its course syllabi more or less in the order students do. The time needed would of course scale.

You will want the scaling to be realistic. People come here intending to learn an entire BS in months or even weeks. Ir does not work out this way.

 

[WWGD]

Most of the Math you'll need, to refresh or otherwise, in a format that can help in both Math or Physics, can be obtained from books on Advanced Calculus. These are books that don't just do the standard Calculus fare, but also include the perspective of Derivatives as linear maps ( hence matrices), Vector Calculus, etc. There are some available online for free.