Learn Deeper Math for Physics & Chemistry Undergrads

[DavidGuimont]

Hey there, I've got a really pretty good comprehension of calculus, I've even had an Advanced Calculus class and a Linear Algebra class, so this is pretty much my knowledge. I'll be going in Chemistry in University so I'll be doing much less math, but I'm pretty interested in physics and math, so I was wondering what steps one took to go maybe a little deeper. What part of mathematics is next? What books could I buy (even if they're expensive textbooks)? What classes would be next for a math undergrad let's say? I may be going a little too far too quick, but I want to learn about Fourier series, Lorentz transformations, just be able to understand Maxwell's equations mathematically, I know I might not be at that point right now, but if you could point me toward what parts of mathematics are next would be great and give me references to things I could read, etc. Thanks a lot and see you on this board, if I'm learning I might keep coming back and interacting! Cya!

 

[micromass]

I can only comment on the math. But if you already finished calculus, then there are a lot of options open to you. So "what's next" is actually entirely up to you.

You could study linear algebra. I know you already took a course in linear algebra, but it was probably quite computational and not very theoretical. You might want to learn linear algebra from a more theoretical point of view. Books like Axler, Friedberg or Lax should be fine for you.
https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20
https://www.amazon.com/dp/0130084514/?tag=pfamazon01-20
https://www.amazon.com/dp/0471751561/?tag=pfamazon01-20

You could also do abstract algebra. This studies structures such as groups, rings, fields, etc. Normally, people study linear algebra first, but it is not really necessary. A good first book is Pinter.
https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20

You could also do analysis. If you want to study Fourier series, then this is the way to go. I think it's best to study a book like Spivak, Lang or Abbott first, since they are quite gentle. Don't start of with Rudin.
https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20
https://www.amazon.com/dp/1441928537/?tag=pfamazon01-20
https://www.amazon.com/dp/1441928669/?tag=pfamazon01-20

Then there's also discrete mathematics. Here you study combinatorics, graph theory, designs, generating functions, etc. Books like Grimaldi and Knuth are good.
https://www.amazon.com/dp/B008NO91I2/?tag=pfamazon01-20
https://www.amazon.com/dp/0201558025/?tag=pfamazon01-20

Now, all the math courses I listed are proofy. If you are not comfortable with proofs, then it might help to go through a proof book first. The obvious choice here is Velleman and Houston.
https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20
https://www.amazon.com/dp/052171978X/?tag=pfamazon01-20

Good luck!