- #1

jordi

- 197

- 14

One possibility to learn the necessary maths to study physics, from a "mathematical physics" point of view is to first study a degree in maths (and maybe also a Master, too).

However, not everybody has the time to do that.

What I am doing now is the following:

First, go to the basics, and study mathematical logic and set theory (including the construction of numbers). I like both books by Goldrei, for that, especially for self-study.

Then, when one understands the completeness theorem, and how the numbers are constructed, one can go to mathematical analysis. I enjoy Apostol's Mathematical Analysis, but for particular-historical reasons. Probably, other books could do the job equally well. In particular, Lebesgue integration is not required. There should be multiple integration (Riemann), though.

The previous steps are intended to give maturity to the student. It is especially important to solve many problems by oneself, and be able to "mentally reproduce" all the theorems and proofs.

So far, everything pretty standard. The key issue of my proposal is to go straight ahead into Szekeres and Hassani books on Mathematical Methods for Physics. Probably Szekeres would suffice, but Hassani is good in some things, too.

I would say that these two books are "rigorous enough", especially if one has accomplished enough mathematical maturity with the Analysis book. Some people say these books are not rigorous enough, but I would disagree. For sure, they have less material in each chapter they cover, than the equivalent "pure math" book, but usually the material there is the one that is going to be used in physics. And, as said, it is rigorous. They key issue of this approach is many proofs of the theorems are not provided (which is an anathema in most math books). But this should be fine for many students, after having accomplished mathematical maturity with analysis. This maturity should lead to search for rigorous theorems, without needing the proof of them, necessarily. And this is exactly where these two books excel. If one needed more material or explicit proofs, sure, go to a math book, but often in physics it is not necessary (at least, up to undergraduate level).

Studying these two books allows to cover a massive amount of math material in a relatively short period of time. Even better, Szekeres deals with classical mechanics, electrodynamics, special relativity, thermodynamics and quantum mechanics, using math vocabulary. As a consequence, a math oriented student can understand better the typical physics books, after studying Szekeres.

Modifications of this recommendation could be: add a calculus book before Apostol (say, the calculus books by Lax), or even substitute completely the analysis book by the calculus book. Also, one could get rid of mathematical logic and set theory, if one is comfortable enough with the explanations on logic and set theory that most calculus / analysis books have, at least summarized in a chapter 0 or in an appendix. Finally, one could use Boas before Szekeres, too, but I do not see this as essential.