# What math topics to learn next

I've been self-learning math and physics topics for the fun of it, and I'm not quite sure where to go from here.

So far, I have a firm foundation in differential, integral, multivariable, and vector calculus. I'm working on differential equations. I don't know much linear algebra at all.

My purpose in learning math is so that I can understand physics topics. So far I've covered pretty much all of classical mechanics, including thermodynamics, and I have a firm understanding of special relativity. I'm currently working on electromagnetism. My goal after E&M is to move onto general relativity and quantum mechanics.

This in mind, by question is: mathematically, what should I work on next? I know I'm going to need to learn things like tensors, differential geometry, topology, and stuff like that for general relativity, I just don't know what specific order would be best to learn the topics in.

So what order would it be best to learn those in? Thanks for the help!

chiro
I've been self-learning math and physics topics for the fun of it, and I'm not quite sure where to go from here.

So far, I have a firm foundation in differential, integral, multivariable, and vector calculus. I'm working on differential equations. I don't know much linear algebra at all.

My purpose in learning math is so that I can understand physics topics. So far I've covered pretty much all of classical mechanics, including thermodynamics, and I have a firm understanding of special relativity. I'm currently working on electromagnetism. My goal after E&M is to move onto general relativity and quantum mechanics.

This in mind, by question is: mathematically, what should I work on next? I know I'm going to need to learn things like tensors, differential geometry, topology, and stuff like that for general relativity, I just don't know what specific order would be best to learn the topics in.

So what order would it be best to learn those in? Thanks for the help!

Hey agent_509.

I would recommend you learn PDE's and also complex analysis and group theory to start off with. Also it might help if you understand some probability and statistics in some depth since many parts of physics are represented in models that are probabilistic and if you have to do an experiment and show evidence for a hypothesis, then you will need to understand statistics, how it works, and how to use it properly (properly is the key word here).

For GR you will have to do differential geometry and depending on how deep you want to go with that you can attack it mostly from a small theory to large computational point of view or a large theoretic point of view.

You could learn more math than this but I have a feeling it might not be needed if your perspective is largely physical rather than mathematical.

The other suggestion I have is to learn any kind numerical analysis possible and also enough programming that you can model systems or do what you need to do, to analyze data and produce output.

I disagree with the above poster. Although more advanced classes in physics and whatnot are good, you really need to learn linear algebra, and learn it well. It will come up later in quantum mechanics, and as well in GR. You can't learn QM at all without linear algebra, and some functional analysis couldn't hurt either. I would suggest learning linear algebra at the same time as more ordinary differential equations (series solutions, stability, etc.) and then move on to complex variables, introductory analysis and PDEs. You probably should be doing physics at the same time, like EnM, quantum, and then statistical mechanics, analytical mechanics and the like. But in the short term, just do EnM and linear algebra. lots of linear algebra. :D

chiro
I disagree with the above poster. Although more advanced classes in physics and whatnot are good, you really need to learn linear algebra, and learn it well. It will come up later in quantum mechanics, and as well in GR. You can't learn QM at all without linear algebra, and some functional analysis couldn't hurt either. I would suggest learning linear algebra at the same time as more ordinary differential equations (series solutions, stability, etc.) and then move on to complex variables, introductory analysis and PDEs. You probably should be doing physics at the same time, like EnM, quantum, and then statistical mechanics, analytical mechanics and the like. But in the short term, just do EnM and linear algebra. lots of linear algebra. :D

Yeah you're absolutely right. For some reason I assumed that the OP knew linear algebra but after re-reading the post he hasn't. No idea how I interpreted that.

The following is the sequence I would recommend given your background.
First and foremost: 1) Learn Linear Algebra! If you want a proof based approach I would recommend reading the chapter you need from Hoffman/Kunze, otherwise you can learn the computations and basic ideas from Gilbert Strang's MIT OCW Course: http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/
This is available in the MIT OCW "scholar" version with posted assignments, exams, etc. to supplement the lectures. I highly recommend you work through the exercises if you seriously want to understand this stuff.

2) After learning Linear Algebra, I would recommend you learn the following somewhat at the same time. Differential Equations (again, great MIT OCW scholar course offered here: http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/) They just posted this one up recently and I've worked through the entire thing to review my differential equations and learned a few things I didn't know. (Never took a full course on differential equations, I just saw them in my physics classes and briefly touched on them in Calculus). Meanwhile, I recommend you familiarize yourself with the basic notions of topology and real analysis. For this, I recommend you read Chapter 2: Topology of Rudin's Real Analysis, or if you find that treatment too difficult, then try Munkres. In either case, in order to understand the formal definition of a smooth manifold you will need to have a decent understanding of some topological notions.

3) Once you know Linear Algebra, some Differential Equations, and some basic Topology I would suggest you head towards some Abstract Algebra and Group Theory. There are multiple textbooks on these, but you will in particular need to understand the general idea behind group theory, what a group action is, the orbit of a group action, etc. In particular you will need to know this because you will use the general linear group GL(n, R), special linear group SL(n, R), orthogonal group O(n, R), and special orthogonal group SO(n, R) all over the place.

I disagree with chiro's suggestions and I think you should learn complex analysis and PDE's when you run into them, and not sooner. The outline I just gave will give you the basics for finding out what side of physics you are more interested in. Personally, I recommend going all the way with the topological/geometric side of things and understanding manifolds and their applications to General Relativity! This will lead you into some interesting areas of Quantum Gravity research.

Oh, did I mention, learn some differential geometry whenever you end up needing it you figure out how manifolds are being applied in GR.

chiro
I disagree with chiro's suggestions and I think you should learn complex analysis and PDE's when you run into them, and not sooner. The outline I just gave will give you the basics for finding out what side of physics you are more interested in. Personally, I recommend going all the way with the topological/geometric side of things and understanding manifolds and their applications to General Relativity! This will lead you into some interesting areas of Quantum Gravity research.

One thing that I have observed personally is that for some people, math takes a while to sink in for it to make sense in not only a theoretical context, but also for a practical context as well.

For this reason I recommend that you get exposed to the ideas early so that it's easier to think about them in more detail later on.

I know that the perspective between physicists and mathematicians (applied or otherwise) are completely different, but again I have found that things not only make more sense but are able to be more useful when there is some familiarity there.

What makes it difficult for people studying things like physics and engineering is that there is a lot of 'catch-up' because all these things are often done concurrently rather than sequentially and although this is largely necessarily, I still think it's a good idea to get acquainted with the ideas so that they can for lack of a better word, 'settle' in your head and be in the background when you do the 'physics' or 'engineering' part which is the real focus.

I agree with the previous posters that you should do linear algebra right away. Hoffman & Kunze is a pretty good book, but if you're not familiar with proofs then it might be too difficult. There are good alternatives though.

If you're planning to study math rigorously, then you might benifit from learning proofs and logic as well. If you just want to use the tools, then you might not care about the proofs.