What mathematics courses should one take for physics?

[Vitani11]

I'm a double major in physics and math. I have taken Calc 1-3, ODE, and linear algebra. This summer I'm taking PDE and in Fall I'll be taking complex variables. What other courses would you recommend?

 

[fresh_42]

I'm a double major in physics and math. I have taken Calc 1-3, ODE, and linear algebra. This summer I'm taking PDE and in Fall I'll be taking complex variables.

Sounds reasonable from the physics point of view.

What other courses would you recommend?

To achieve what?

Considering the overlap of the combination of mathematics and physics I suggest differential geometry (if not already included in the other diff courses), maybe topology and algebra as fundamental concepts like linear algebra is. If available these topics could be braced in a course about Lie groups which includes parts of all of them. However, this "recommendation" still depends on my question: To achieve what?

 

[Vitani11]

Sorry I should have been more specific. I would like to develop as many tools as possible to tackle physics problems and "get deep" into topics such as quantum or relativity. So for example as you said differential geometry (I am assuming for relativity) and topology. I will be taking quantum in Fall so if that was a naive statement forgive me because I do not yet know the mathematics involved other than what I have learned in math physics. What do you mean by "..maybe topology and algebra as fundamental concepts like linear algebra is."? Are you saying that topology is fundamental? Here are the courses I must choose from: Advanced Calculus 1/2, Complex Variables, Numerical Analysis 2, Vector Calculus, Game theory, Optimization, Calc of variations, PDE 2, and Linear Algebra 2. I would love to take calculus of variations because I know it's important for things like deriving the Lagrange and it is a gateway to a lot of simpler physics (correct me if I am wrong). I've learned enough of calculus of variations in order to understand the lagrange and how to optimize things such as finding the maximum volume which can be inscribed in an ellipsoid etc. but not much more than that. To take calculus of variations I must first take Advanced Calculus 1 as it is a prerequisite. I can basically only pick 3 of these. Complex variables I will take regardless because I know it is important which leaves two. I would also like to take vector calculus but if I did that I could not do Calc of variations since I wouldn't have the advanced calc class down and I'm limited on how many courses I can take. Wow, I wish I could just take all of these math courses, lol. On the other hand I can switch from applied to regular mathematics and then the door would open to courses such as abstract algebra and topology but I don't know if that is worth the switch.

 

[fresh_42]

Sorry I should have been more specific. I would like to develop as many tools as possible to tackle physics problems and "get deep" into topics such as quantum or relativity. So for example as you said differential geometry (I am assuming for relativity) and topology.

For relativity, particle physics, Noether's theorem and things like that. It's the natural analogue to Euclidean geometry if you don't require an embedding into a surrounding space.

I will be taking quantum in Fall so if that was a naive statement forgive me because I do not yet know the mathematics involved other than what I have learned in math physics. What do you mean by "..maybe topology and algebra as fundamental concepts like linear algebra is."?

I mean that topological concepts are a) a generalization of all the metric stuff, which is only a part of all topology and b) include geometrical aspects which might be needed to understand e.g. cohomologies, differential forms and terms like AdS or Yang-Mills theory. The same is true for algebra: group, ring and field theory (mathematical fields, not physical) are fundamental structures like vector spaces are.

Are you saying that topology is fundamental?

Yes.

Here are the courses I must choose from: Advanced Calculus 1/2, Complex Variables, Numerical Analysis 2, Vector Calculus, Game theory, Optimization, Calc of variations, PDE 2, and Linear Algebra 2. I would love to take calculus of variations because I know it's important for things like deriving the Lagrange and it is a gateway to a lot of simpler physics (correct me if I am wrong).

I think this is true and a good choice among the listed courses. To give a more detailed advice, it would be needed to know what these courses actually cover with respect to the realms I noted. Also Game theory might cover some useful aspects about probability theory and statistics, which are very much needed in experimental physics, like e.g. particle physics. But in doubt I personally would prefer a stochastic course over Game theory. LA2 is also likely a must-have.

I've learned enough of calculus of variations in order to understand the lagrange and how to optimize things such as finding the maximum volume which can be inscribed in an ellipsoid etc. but not much more than that. To take calculus of variations I must first take Advanced Calculus 1 as it is a prerequisite. I can basically only pick 3 of these. Complex variables I will take regardless because I know it is important which leaves two. I would also like to take vector calculus but if I did that I could not do Calc of variations since I wouldn't have the advanced calc class down and I'm limited on how many courses I can take. Wow, I wish I could just take all of these math courses, lol. On the other hand I can switch from applied to regular mathematics and then the door would open to courses such as abstract algebra and topology but I don't know if that is worth the switch.

There are many open questions that heavily depend on what you already have learned and where you want to go to. In the end it is more important to learn how to learn than what you learn, so you will always be able to get a good textbook on a subject and learn it from there.