Maths for an undergrad to learn on the side

[barek]

I am currently a first year undergraduate student studying physics in the UK. I will have some time over the summer and was wondering what maths it would be useful to self study as I will have access to my university's lecture notes and problem sets for all of their maths courses.
By the end of first year I will have have had maths lecture courses on:

linear algebra
single, multivariable and vector calculus
complex no.s and ordinary differential equations
complex analysis

and next year I will have a mathematical methods course covering Fourier analysis, more ODEs and PDEs

We have also had physics lectures on coupled oscillations and waves, classical mechanics, electromagnetism, circuits, optics and special relativity.

Thanks for the help!

 

[fresh_42]

I have listed a couple of links about self study in general and a page where free pdf can be found, which cover the first time of a study:
https://www.physicsforums.com/threads/self-teaching-gcse-and-a-level-maths.933639/#post-5896947
Some of them might be of value to you. It's problematic to learn ahead, because a) there is a risk of learning it wrong and b) a likelihood of getting bored when you have to at university. On the other hand, you could either learn something which isn't a mandatory subject in the first two years, like group theory or topology, or you can learn calculus, which you will need on a daily basis as a physicist, so you can't do anything wrong here.

As a personal recommendation, I'd say learn everything you can find about coordinate systems and (inertial) frames: from ordinary vector spaces, to manifolds or whatever you can find on Wikipedia. 90% of all physics here is written in coordinates, resp. their abbreviations and notation codes (Einstein, Dirac). The fitter you are in this field, the better for you, and I think it is such a naturalness, that its importance is a bit neglected in the courses. But this assessment could easily be because I don't really like them.

 

[barek]

I have listed a couple of links about self study in general and a page where free pdf can be found, which cover the first time of a study:
https://www.physicsforums.com/threads/self-teaching-gcse-and-a-level-maths.933639/#post-5896947
Some of them might be of value to you. It's problematic to learn ahead, because a) there is a risk of learning it wrong and b) a likelihood of getting bored when you have to at university. On the other hand, you could either learn something which isn't a mandatory subject in the first two years, like group theory or topology, or you can learn calculus, which you will need on a daily basis as a physicist, so you can't do anything wrong here.

As a personal recommendation, I'd say learn everything you can find about coordinate systems and (inertial) frames: from ordinary vector spaces, to manifolds or whatever you can find on Wikipedia. 90% of all physics here is written in coordinates, resp. their abbreviations and notation codes. The fitter you are in this field, the better for you, and I think it is such a naturalness, that its importance is a bit neglected in the courses. But this assessment could easily be because I don't really like them.

Thanks so much for taking to the time to respond, it was super helpful! Is it a good idea to learn calculus rigorously as part of an analysis course? We some really basic groups in school (definition of a group, Lagrange's theorem, Klein group etc). Are there specific types of groups that are important for physics (I've heard of Lie groups but are there others as well?). It looks like the topology course assumes knowledge of a course on metric spaces which assumes knowledge of analysis so I might look at the analysis course.

 

[fresh_42]

Thanks so much for taking to the time to respond, it was super helpful! Is it a good idea to learn calculus rigorously as part of an analysis course?

I'm not sure. In physics you calculate a lot, so it is probably far more helpful to learn integration, and to solve differential equations, than it is to rigorously learn how to prove the error margins of Taylor series. Of course good physicists are also good mathematicians, but their language is often a bit different, so one could get confused.

We some really basic groups in school (definition of a group, Lagrange's theorem, Klein group etc). Are there specific types of groups that are important for physics (I've heard of Lie groups but are there others as well?).

Lie groups are the backbone of the standard model of particle physics, but they are not an easy stuff. They carry a topology and a differential structure, and they are manifolds, i.e. they have their own local coordinates, so good knowledge in calculus and better also in differential geometry and the basics of topology should come first. Of course you can learn about the groups $SU(n)$ as matrix groups, which only requires some linear algebra, and forget about the analytical aspect, i.e. learn it later. This way you'll have an example where you can test your knowledge in linear algebra and learn how to handle matrices. All this depends on so much more, which I can't know: where you currently stand at, whether you have already in mind where in physics you want to go to, how much time you want to spend on which topic etc. There can be written entire books which only deal with this special unitary groups. Well, I don't know a single one, and what I mean is usually spread over a couple of books with $SU(n)$ as an example for larger theories. In crystallography you have other groups, groups of geometric symmetries, which are closer related to the ones you mentioned.

It looks like the topology course assumes knowledge of a course on metric spaces which assumes knowledge of analysis so I might look at the analysis course.

Not really. Metric spaces are the origin of topology, but they are a very special example, and learning general topology, they even can be disturbing, because for metric spaces, topologies behave very nicely and everything is fine. Abstract topology however can be rather pathological when it comes to examples. It's good to have metric spaces in mind, but they are no requirement. Topology is about open sets (like open intervals) and continuous functions, which are defined in a way, which needs no metric ($f(x)$ is continuous if the pre-image of open sets is open). It's just that if there is a metric, then we have the usual analytical definition. So learning the basics about topology (est. pages 1-50 in a topology book) can be very useful.

Here's another good article: https://arxiv.org/pdf/1205.5935.pdf which you can study, which a) will be very helpful for your geometric understanding, which is needed in physics, and b) won't be in conflict to the usual canon at university. And you can see how you get along with it, i.e. with the language it's written in.

 

[barek]

Here's another good article: https://arxiv.org/pdf/1205.5935.pdf which you can study, which a) will be very helpful for your geometric understanding, which is needed in physics, and b) won't be in conflict to the usual canon at university. And you can see how you get along with it, i.e. with the language it's written in.

Thanks! That looks really interesting