Need advice on which math classes to take

[HeLiXe]

This is a direct spinoff to the thread I started about which class general relativity is offered in. That question was answered and now I would like to align my math classes to prepare for tensor calculus. Here are the classes I am thinking of taking

Linear Algebra
Abstract Algebra I and II
Vector Analysis
Number Theory
Numerical Analysis
Probability and Statistics

I have already taken Calc I-III and Differential Equations. Because of the scheduling of courses at my school, I will probably have to take the most of these after I graduate. My question is, which classes are most essential for tensor analysis, and which of these would you recommend for self study (as they probably would not be a prerequisite for graduate courses)? I plan to go to graduate school for astrophysics and am currently majoring in physics and chemistry. Thanks!

 

[WannabeNewton]

One usually encounters tensor analysis either in a GR course or a riemannian / smooth manifolds class. Typically you need to know your calc 3 and your LA very well if you want to get started on an introduction that isn't heavy on theory but is more bent towards computation (I'm not sure what your vector analysis class entails but for example at my uni there's an undergrad class called vector and tensor analysis so is it a tensor analysis class for you as well?).

 

[HeLiXe]

I don't think so...from what I have read in the catalog, it covers vector fields, divergence theorem, Greene's theorem, and stokes theorem. There is no clear tensor analysis course at my school. The only course description that even mentions "tensor" is a graduate course on continuum mechanics.

 

[WannabeNewton]

Yeah tensor analysis is big in fluid dynamics. Well if the majority if the stuff in the vector analysis class are things you haven't already learned in calc 3 then it would probably help to take it (I'm not sure how theoretical that class gets but if you know the textbook you can probably gauge it from that - the one at my uni uses Geometry of Physics - Frankel). Anyways, if you are right now looking to get used to the computational / classical aspects of it all (basically what you would see in a first GR course) then you need to have calc 3 and LA under your belt but not much else really. Note that if your goal is to eventually get to a good learning of GR then tensor calculus isn't really the major thing to focus on as its own separate entity because most GR books will teach that along the way. It is really differential geometry that forms the core of GR. So yeah put your efforts on LA and, if needed, vector analysis. If you have the time and interest then take Abstract Algebra 1 but for a first GR course you won't see the stuff in that show up much at all in terms of tensor calculus.

 

[Matrix0]

I would recommend taking Linear Algebra, Vector Analysis, and Numerical Analysis as essential classes for tensor calculus. These classes will provide a strong foundation in the mathematical concepts necessary for understanding tensors and their applications in physics. Abstract Algebra may also be useful, but it may not be as directly applicable to tensor calculus as the other classes. Number Theory and Probability and Statistics may also be helpful, but they may be more useful for specific applications in physics rather than for tensor calculus itself.

If you are unable to take all of these classes before you graduate, I would suggest prioritizing Linear Algebra and Vector Analysis as they are the most directly related to tensors. You could also consider taking a course in Differential Geometry, as it is closely related to tensor calculus and is often used in physics and astrophysics.

For self-study, I would recommend starting with Linear Algebra and Vector Analysis, as they are the most fundamental for understanding tensors. Numerical Analysis would also be a good choice for self-study, as it will give you practical skills in solving problems involving tensors. Abstract Algebra and Number Theory may not be necessary for self-study unless you have a specific interest in those areas.

Overall, it is important to have a strong foundation in mathematics for a career in astrophysics, and taking these classes will certainly be beneficial. However, keep in mind that graduate courses will also cover these topics and may provide a more in-depth understanding of them in the context of astrophysics. Good luck with your studies!