What to learn next?

1. Apr 30, 2012

stefan10

I'm quite overwhelmed by how many options I have to self-study over the summer.

I want to improve my abilities in mathematics and physics before I start my freshman year of college, with plans to major in both. This will be done by self-studying a book or two over the summer. I'll give a basic outline of my abilities.

I started Algebra I in 8th grade, which upon reflection seems to have been more like a pre-algebra class that touched upon algebra. I took geometry in 9th, and Algebra II in 10th(this class omitted many topics/didn't go into the detail necessary. In 11th I took a trigonometry-focused class, with some review of concepts touched upon in Algebra II: functions, logarithms, etc. I am currently in AP Calculus AB and Probability & Statistics(algebra-based; although I can tell when Calculus can be implemented.) I feel as if I am doing very well. I am likely to get a 5 on my AP exam, and I have an A in the class with much less trouble than I had in Trigonometry. In fact, Calculus is the first math course that really has got me interested in math. Before I saw it as a tool, and played with it superficially, but now it seems much more interesting than that. This year, I've been mostly touching upon some pre-calculus topics not covered in my curriculum at my school, due to the poor organization of the courses: Series/Sequences, Matrices(only learned how to add, subtract and multiply in Algebra II), Polar Coordinates, Vectors(first learned about them in algebra-based physics), Conic sections, Parametric Equations, Solving Polynomials(in detail, never even heard of synthetic division until this year),Complex Numbers(only touched upon imaginary numbers) and the Binomial Thereom.

Now that I've finished up all of that, I feel as if there is too much to choose from, and I don't know what would be best for developing my abilities. I know what interests me, mostly Calculus/Analysis, but I don't know what would be useful at which point, nor do I know what level of depth I should focus on. Since Calculus AB only covers 2/3rds of a Single-Variable Calculus course, I was going to self-study the last few chapters in my not-so great book, in regards to self-learning. Then, I considered to instead learn from the beginning using a different book, particularly the highly regarded Apostol's or Spivak's. This is so that I developed 'mathematical maturity.' Although, I don't know how useful that would be if I am to take non-analysis mathematics courses my first semester of college. The two courses being a course that works as an introduction to proofs(focused on sets I think), and a proof-based matrix theory/linear algebra course.

Basically, I'm interested in which order and with which books I should follow for a comprehensive understanding of mathematics, by self-studying. That is the only way I really do learn, and I'm mostly going to use my courses throughout college as a review. Please consider that I want the best order to maximize my potential in physics courses as well. Physics will be my primary major, and mathematics is more of a special interest at this point. Basically, in which order should I learn undergraduate level mathematics, for the most efficient use of my time?

As for physics, this is the book used in the introductory courses offered at the college I will be attending. I understand that it isn't the conventional introductory course according to the reviews. Will there be anything I will miss out on if I learn solely from this? As of now, my entire experience with physics has been Algebra/Trig-based, although I do understand when Calculus can be implemented, mostly within the context of rate of change type of relations, such as position - velocity - acceleration, or work - power. I sat in on a second semester lecture of this course during my visit of the university, and I could grasp things here and there(they were talking about emf), the math involved wasn't a problem, but some of the concepts were the major obstacle. If I were to familiarize myself over the summer, I think the transition will be much easier.

Also, my ultimate interests lie in astrophysics/cosmology and particle physics. At this time, particularly the latter.

2. May 1, 2012

homeomorphic

No one can have a "comprehensive understanding of mathematics". There are some subjects everyone should learn, but past a certain point, you have to choose which subjects are going to be the most useful/interesting to you.

I'm not sure the order in which you learn subjects matters that much, provided you understand the prerequisites to what you are learning. Also, it may be different for different people. The only thing I would say is that studying linear algebra early on can pay off. Seems like you are already following that advice before I said it. I took ODE, then linear algebra, and it was the wrong order, I think. Also, both classes were too plug-n-chug.

I think choosing the right books and the right approach matters a lot more than order.

Just try to avoid studying a bunch of random topics. Try to understand the relationships between different subjects and maintain some kind of continuity between different subjects. You remember things better if they are connected to other things.