What Math Topics Should I Study Next for Advanced Physics Applications?

[WiFO215]

I have learned

Calculus (single and multi-variable)

Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. I have not learned series solutions nor special functions which I see is the next step in this chapter)

Linear Algebra (self-taught from Hoffman and Kunze. I have not completed this however. Am still continuing)

Have done the Calculus of Variations chapter from Boas Math Methods

My math is physics oriented. I hope to start reading Goldstein's book this fall, followed by E&M, QM, Stat Mech etc. in the following years. What math should I learn now? Tensors? Group theory? Pick up where I left off from ODEs and continue to PDEs? Complex analysis? Could you also suggest textbooks for the respective chapters?

 

[maverick_starstrider]

Well ultimately it depends on what field you want to study but Complex Analysis and PDE's is important, I'd also maybe do some more linear algebra. If you have Boas look at everything in there and make sure you have it. Make sure you've gone over special functions and such (Bessel and Von Neumann at the very least).

For GR: You want differential geometry and tensor analysis
QM is Linear algebra and complex analysis.
If you want to go on and do QFT and stuff I'd spend more time on calculus of variations (which is really a subfield of functional analysis).

 

[WiFO215]

Any other book to learn Linear Algebra you refer to apart from Hoffman and Kunze and the usual other texts suggested in other texts around the forum (Shilov, Axler, Friedberg et al).

EDIT: for complex analysis, would this book by Rudin called Principles of Analysis be suitable? It seems to be hailed all over the forum.

 

[maverick_starstrider]

Any other book to learn Linear Algebra you refer to apart from Hoffman and Kunze and the usual other texts suggested in other texts around the forum (Shilov, Axler, Friedberg et al).

EDIT: for complex analysis, would this book by Rudin called Principles of Analysis be suitable? It seems to be hailed all over the forum.

If you can get through Rudin, my summation is that, you will be in great shape. I've never tried Rudin and most of what it goes through will not be of relevance for a physicist (which is not to say that it isn't great/useful to learn it). But in general I can't say I could recommend a "standard" math text for a given field. I took courses in these things but I can't say I found the lin alg or complex analysis books so noteworthy and didactic that I'd recommend them (or remember them for that matter). I tended to view my math learning in the following perspective: I got a couple of books for the math subject test GRE preparation and undergrad math contests and I tried to do practice problems up the wazoo from those. Of course, I personally, lost interest before completing my goal but I've always been a notorious slacker. But if you can ace the math subject GRE (which really only covers approximately the first 2 years of a math education) you should be laughing for all of a bit of what you want to do (functional analysis (i.e. calculus of variations will not be part of it)).

 

[maverick_starstrider]

Although in terms of CONTENT I could probably help. For complex analysis you need to know contour integrals very well as well as a strong understanding of cauchy-riemann, residue theorem and branch cuts (and laurent series come up every once in a while to). In terms of lin alg, just get yourself up to hilbert spaces (which is an amalgum of inner product spaces, vector spaces, operators and cauchy sequences).

 

[snipez90]

The Rudin book you are referring to doesn't actually have complex analysis in it. It is more or less a rigorous (but terse) introduction to analysis text.

 

[WiFO215]

Ok. Will it be useful?

(((I like learning math too. I considered doing a BSc in that for quite some time. Possibly I could read that in my spare time. Will it help me solve previous Putnam paper problems? I ask this as I try doing these problems in my spare time now)))

 

[thrill3rnit3]

Rudin's PMA is real analysis, not complex analysis.

If you want to learn the math needed for physics, I suggest you look into Arfken's book.

 

[WiFO215]

I think I diverted the topic by asking if rudinis useful for putnam. Getting back to topic could you guys suggest books for the topics you mentioned besides arfken? Something specific for those topics?

 

[thrill3rnit3]

I think I diverted the topic by asking if rudinis useful for putnam. Getting back to topic could you guys suggest books for the topics you mentioned besides arfken? Something specific for those topics?

Complex Analysis by Lars Ahlfors:

https://www.amazon.com/dp/0070006571/?tag=pfamazon01-20

Solid intro to complex analysis (and real analysis) is this book by Shilov:

https://www.amazon.com/dp/0486689220/?tag=pfamazon01-20

PDE by Stanley Farlow:

https://www.amazon.com/dp/048667620X/?tag=pfamazon01-20

 

[Daverz]

https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20 is some of the most beautiful mathematics you'll see in the undergraduate curriculum.

 

[WiFO215]

OK thanks guys. What about group theory and tensors? I saw these topics were used in Goldstein's. Don't I need to know this before I learn anything else, considering I am going to be using that text this fall?

 

[Daverz]

OK thanks guys. What about group theory and tensors? I saw these topics were used in Goldstein's. Don't I need to know this before I learn anything else, considering I am going to be using that text this fall?

I think your stated mathematical knowledge is quite enough for Goldstein. But how much exposure have you had to classical mechanics previously? My usual recommendations at the undergraduate level are French's Newtonian Mechanics and Fowles's Analytical Mechanics. At the same level as Goldstein, but with much less verbiage, is Landau and Lifschitz, Mechanics.

 

[WiFO215]

I've done Newtionan Mechanics from French and K&K.