"I'm a fan of higher maths, so I think it's definitely worth it if you have the interest. If your goal is to learn about how the world works, the best thing is probably Applied Math, with Differential Equations as top priority.
The first thing you should do is make sure you have Linear Algebra and Calculus down pat. The heavy theory of LA can wait til later, but I'd recommend learning as much of it as you can right away because I think it's awesome.
At this point you can choose to either grab an elementary book on DiffEQ that uses only Calc or go into more theory: Real Anlysis and Functional Analysis. Both of those are harder but essential for advanced stuff. Real Analysis will teach you how Calculus really works, and at the MINIMUM you should get used to the ideas of epsilon-delta proofs, basic point-set topology, and metric spaces. Once you're comfortable with that you can do measure and integration theory. Functional Analysis is just like linear algebra, and is the math of quantum mechanics.
After that, you'll be ready to understand DiffEQ at a fairly high level (advanced undergrad/beginning grad), and that leads to a lot of physical applications. Heat equation and Laplace's equation are nice and pop up all over. Also, with measure theory you can learn advanced Probability and Statistics which is pretty cool.
After you're comfortable with basic Real Analysis, you can move into Differential Geometry. For that you'll need to make sure your Linear Algebra is up to snuff, as you'll be needing it a lot. Differential Geometry is really awesome. It's a physicist's best friend and is the math of classical mechanics and general relativity. The first thing to do in this area is learn about manifolds and differential forms so you can do the generalized Stokes' Theorem.
Complex Analysis, which you can also try after getting comfortable with Real, is a neat subject. The most basic ideas aren't too hard and also very interesting.
If you want to get more theoretical with less obvious physical applications, you can go the Abstract Algebra route. Abstract Algebra can be learned any time, really. I'd recommend keeping an introductory book (like Dummit and Foote or Artin) around and glancing at it when you're in the mood. As your mathematical maturity builds you will understand more and more of it. Abstract Algebra opens up the fields of Number Theory and anything with Algebraic in the title." ---ee tak hochetsa otimet'!
The first thing you should do is make sure you have Linear Algebra and Calculus down pat. The heavy theory of LA can wait til later, but I'd recommend learning as much of it as you can right away because I think it's awesome.
At this point you can choose to either grab an elementary book on DiffEQ that uses only Calc or go into more theory: Real Anlysis and Functional Analysis. Both of those are harder but essential for advanced stuff. Real Analysis will teach you how Calculus really works, and at the MINIMUM you should get used to the ideas of epsilon-delta proofs, basic point-set topology, and metric spaces. Once you're comfortable with that you can do measure and integration theory. Functional Analysis is just like linear algebra, and is the math of quantum mechanics.
After that, you'll be ready to understand DiffEQ at a fairly high level (advanced undergrad/beginning grad), and that leads to a lot of physical applications. Heat equation and Laplace's equation are nice and pop up all over. Also, with measure theory you can learn advanced Probability and Statistics which is pretty cool.
After you're comfortable with basic Real Analysis, you can move into Differential Geometry. For that you'll need to make sure your Linear Algebra is up to snuff, as you'll be needing it a lot. Differential Geometry is really awesome. It's a physicist's best friend and is the math of classical mechanics and general relativity. The first thing to do in this area is learn about manifolds and differential forms so you can do the generalized Stokes' Theorem.
Complex Analysis, which you can also try after getting comfortable with Real, is a neat subject. The most basic ideas aren't too hard and also very interesting.
If you want to get more theoretical with less obvious physical applications, you can go the Abstract Algebra route. Abstract Algebra can be learned any time, really. I'd recommend keeping an introductory book (like Dummit and Foote or Artin) around and glancing at it when you're in the mood. As your mathematical maturity builds you will understand more and more of it. Abstract Algebra opens up the fields of Number Theory and anything with Algebraic in the title." ---ee tak hochetsa otimet'!
