When do I start seeing cool math in physics? It's been nothing

huh?

>>8453828
I don't know, exciting math beyond the basic calculus sequence and elementary linear algebra

>>8453823
You can start researching advanced mathematics whenever you decide to do so. Often the textbooks will include challenging problems for you to solve.

>>8453838
I guess that's true

>>8453823
Implying that calc III isn't cool af

>>8453823
I think differential equations are pretty cool, they show up everywhere in physics. Same with linear algebra >>8453835 and complex analysis will become very useful for you eventually. Baby steps though.

>>8454057
>>8454069
They are I'm just getting bored with them

I haven't had to use the integrating factor method, lagrange multipliers or anything yet btw it's most "take this derivative. do a trig sub integral" boooo

Where would I use complex analysis tho?

>>8454081
Lagrange multipliers are actually extremely useful in mechanics. You don't really use the integration factor in physics because any differential equation that can be solved with them is usually one where you'd just memorize the solution to. Have you gotten to oscillations yet in diff eq? It gets even more interesting when you move onto things like the heat equation and laplace equations.

>Where would I use complex analysis tho?
In some problems, it's a lot easier to consider the variable as the real or imaginary part of a complex number, then take advantage of exponentials to solve your problem much more quickly. Study oscillations in more detail if you want the full story.

>>8453823
Never in undergrad physics. The physics major is biased against math and theoretical physics. It's only in modern theoretical physics that any real math comes up. The little math you do see is linear algebra in QM and very simple differential geometry in GR.

Here's what you need to learn to do theoretical physics.

I'd recommend you read Landau&Lifshitz Mechanics and Schutz's GR books immediately.

Now learn QM from some book. I like Weinberg's book the most but it's nonstandard. Shankar or Townsend would be more common choices I recommend. I hate Sakurai.

Learn thermodynamics from somewhere. It's the only classical theory from centuries ago which has been so successful at describing phenomena that no one calls it a theory. Maybe once we find a single situation where it doesn't work we'll revise it, but don't bet on that happening.

You'll have to learn a bit of EM but I don't think you need a book on it. Jackson covers the little you need and a bunch of stuff you don't.

Now you should be able to study QFT. Also take a look at Wald's GR book and Gauge Fields, Knots, and Gravity by Baez.

Now on math. First a point. Rigorous math IS useful and necessary to learn if you want to do much of anything.

Linear Algebra. Comes up literally everywhere. If you want a good example, look up normal modes of coupled oscillations. Basically the problem just turns into finding eigenvectors. The best books are Hoffman and Kunze, then Curtis's Abstract LA. Jordan's Linear Operators for QM is also good.

Real Analysis. It's rigorous calculus that turns into much more. Rudin's book. Some don't like it but I swear by it.

Complex Analysis. An example: Integral from -inf to inf of 1/(1+x^2). Consider this integral as one being taken along a path in the complex plane, 1/(1+z^2) along x-axis. Cauchy's theorem says we can distort this path without changing the value of the integral as long as we don't pass singularites(infinities). Here the singularites are at z=i and z=-i.

>>8453823
>>8454554

Cont.

Contort the integral into a half circle except for stretching it around the singularity z=i. Now you can basically consider this as two contours, the half circle and full circle around z=i. Take the half circle to infinity and the value of 1/(1+z^2) disappears on it. Now we just need to evaluate the path around the singularity. The basic idea is that if we expand a complex function in terms of 1+1/z+1/z^2+..., the 1/z term is the only one which does not disappear when integrating in a circle. The value of dz/z along a circle is the integral from 0 to 2pi of iRe^(it)/Re^(it) dt = integral from 0 to 2pi of i dt = 2pi*i. So we conjecture of the value of a function along a circle is the coefficient of 1/z in it's series expansion times 2pi*i. This is known as the Residue theorem. In this situation 1/(1+z^2) = 1/(z+i)(z-i) so it's coefficient for 1/(z-a) is 1/(z+i). Around the point z=i this is 1/(2i) so by the residue theorem the value of the integral along a circle about z=i is 2pi*i(1/2i)=pi. So the integral from -inf to inf of 1/(1-x^2) is pi, which you can check on Wolfram or trig methods. Try doing this for 1/(a^2+x^2) as well.

A less elementary example is the integral from -inf to inf of e^(iwt)/(w^2-w0^2) dw. Which is a sinusoidal. It's something like 2pi*i[e^(iwt)/2w0^2 - e^(-iwt)/2w0^2].

Both the underlying theory and the application of this are a big deal. Ahlfors is the classic but it sucks for anything beyond intuition. Look at Conway, Lang, Stein&Shakarchi, or Narasimhan instead.

Abstract Algebra. Artin, Lang,Herstein, Dummit&Foote. Focus on group theory at first. Eventually you'll need everything, including categories.

Some very advanced topics are algebraic geometry, differential geometry, differential topology, algebraic topology, and functional analysis. Of these I think the most useful are DG and AT. The least useful from what I've heard from the theorists I've asked is FA but I know very little about it myself.

>>8454554
>>8454555
thanks man!

>>8454555
>Some very advanced topics are algebraic geometry, differential geometry, differential topology, algebraic topology, and functional analysis. Of these I think the most useful are DG and AT. The least useful from what I've heard from the theorists I've asked is FA but I know very little about it myself.
Not OP but I love the fields you named (Im in pure maths), Im considering going into applied maths but given I love these isnt theoretical physics a better option?

>>8453823
Quantum physics II or III

>>8453823
Mathematical physics is fun math, also classical mechanics II is awesome.