Math methods for modern physics

>>8576344
The math used in modern theoretical physics is beyond almost everyone. Vector calculus isn't actually used anymore, being replaced by modern multivaraible calc like that found in Spivak's calc on manifolds. Educators don't know that since most of them didn't get to the cutting edge modern math and physics, so they just teach the vector calc since it's simple and that's what they were taught. Another issue that comes up is the focus towards "applications" since many more students take math courses that won't actually learn the later math. Like engineering, CS, chemistry, etc. majors. Teaching them about manifolds would be pointless since they really won't need it. The math and physics majors who will need it are pulled back as well, since the educators in charge don't see the problem. This is the case in most universities except for the top ones like MIT which have separate honors math courses or accelerated programs which skip the lower level courses to get to the rigorous ones. These two things combined, the educators never learning the real stuff, and the need for applied courses cause the calc1-3,linalg,diffyeq phenomena to continue.

Some examples of strange material being put into this sequence for no good reason. Calc 2 focuses on sequences and series, but to no real depth like in real analysis. So you just memorize some ways to check convergence of some canned examples which have no real purpose and you'll easily forget them since you never understood them properly. Linear algebra is mostly fine except it doesn't get to many interesting and useful topics like tensors or actually applying them like in solving linear differential equations. Differential equations is the worst of the bunch. Literally just memorize a sheet of paper with all the methods on it and that's it. Worst of all is that they tell you about Laplace transforms without telling you how they are actually used.

>>8576344

Differential Forms, Tensors, Quaternions, Exterior Algebra, Clifford Algebra, etc etc

>>8576372
>The math used in modern theoretical physics is beyond almost everyone
citation needed. do you have an example? whenever I encountered mathematics that triggered the fuck out of me (due to appearing freakish) it turned out to be not that hard after all once you dove into the material properly

>>8576344
>>8576372
More on that last part on Laplace transforms. If you take the Laplace transform of a function,L(f), then you can get the original function f by summing the residue at all the poles of e^x*L(f). This is how it's actually used, not just looking up inverse Laplace transforms in a table. But they can't actually teach this, because no one will have gotten to complex analysis yet in the sequence.

>>8576376
I meant for people in general, not math and physics majors. But even so there is a huge amount that shows up. Just about everything in math is useful at some point. There's the stuff that comes up in physics at very clear points: differential geometry in GR, groups in QM, complex analysis in QFT. Then for modern research many different things get used. Algebraic geometry in conformal field theory, algebraic topology occasionally, functional analysis and measure theory everywhere, representations expand on groups from QM, even categories and heavy abstract algebra are useful(see Geroch's mathematical physics). Some books to give a taste are Bleecker's gauge theory and variational principles, Quantum Fields and Strings by a ton of people, and any serious books on string theory and M-theory(try looking at the chicago physics bibliography).

>>8576344

In a perfect world

>Grade School Geometry is replaced by Intro to Proofs and Discrete Math
>Precalculus should return to teaching limits, sequences, and series
>Calculus 1 should cover the infinitesimal approach
>Calculus 2 should cover Fourier series and transforms and approximation methods
>Linear Algebra should have geometric algebra
>Calculus 3 should cover differential forms and geometric calculus
>ODEs should always cover Laplace transforms and series methods
>Complex Analysis should be a required course
>And all these courses should be done before you finish high school

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>>8576372
>>8576393

Thanks for the recs.

In light of your posts and other revelations about modern curricula, I kinda figured out that while I am decently competent at math, I grossly misunderstand the larger perspectives and concepts of how it all fits.

Can anyone recommend a book that basically says "Hey, turns out you need to relearn all the shit they taught you in school, and here's how to fucking put it in perspective"

A "time to fuck your shit up SENPAI" book.


Ive been going through rudins complex analysis, and its been good so far, but it feels dated even for what it is, and it's relatively limited in scope, especially in regard to application in physics.

>>8576431
Stein and Shakarchi might be more your speed

A student who actually belongs in graduate school will "catch up" to what they missed in undergrad once they've realized that undergrad math wasn't enough. If you recognize that differential forms are superior to muh vector calculus and you are generally expected to know these things, you better damn well go learn it, otherwise grad school may not be for you. At least that's how it is for theory students.

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>>8576490
Looks decent will look into

>>8576502
Gotcha, are there many other perspective shifts that go on?

>>8576374
This anon gives a list, and I can't help but wonder what else is waiting around the corner that I'd have to go back and essentially rework how I frame all of it.

As of right now, the goal is working MHD or some other crossover between nuclear and electricity.

Other anons that ran into a similar problem, what is your area of study and what did you to do to ensure you were working with the proper tools and outlooks?

This shit is both exciting and upsetting, like I'm glad I see it now, but I'd have been fucking screwed without some of the advice I've seen in other threads.
I couldn't wish it on my worst enemy to have gone through grad school without others showing them what's worth their time and what's not.

a little different than OP, but what are some reference books that I can refer to for tools to solve my EE problems. Something that would be a good investment for a job

>>8576399
>before you finish high school
Sure

>>8576704

Any Numerical Analysis and/or Applied/Numerical Linear Algebra book