How should I proceed in my study of physics / mathematics?

[Arcthor]

I recently graduated from High School (or Sweden's equivalent). I am entering quite a difficult university program studying physics / math in a couple of months.

I am interested in self-studying this summer simply because I find the subjects fascinating, especially the mathematical side of it. However, I don't know how to proceed. What should I learn with knowledge of high school physics and maths? What is the typical roadmap to understanding quantum-mechanics and other high-level physics or developing extraordinary math skills? I have all the time in the world right this summer.

Are there any good books / websites etc? Thank you.

 

[Amrator]

Worry about quantum mechanics and other high-level physics later, man. First, learn calculus and calculus-based mechanics.

Edit: Also, you did not say what exactly your math background is?

 

[Arcthor]

Worry about quantum mechanics and other high-level physics later, man. First, learn calculus and calculus-based mechanics.

Edit: Also, you did not say what exactly your math background is?

Yeah, I am nowhere near learning quantum mechanics :) Just wanted to know how to get there.

I don't really know how Sweden's education system compares to USA etc, but I've definitely learned basic calculus (derivation, integration, different integration techniques etc and how they relate to physics) and I have learned some basic matrix transformations.

I have not, however, gone too far into general relativity and special relativity. Could this be a good starting point and if so, where do I start?

 

[Amrator]

Yeah, I am nowhere near learning quantum mechanics :) Just wanted to know how to get there.

I don't really know how Sweden's education system compares to USA etc, but I've definitely learned basic calculus (derivation, integration, different integration techniques etc and how they relate to physics) and I have learned some basic matrix transformations.

I have not, however, gone too far into general relativity and special relativity. Could this be a good starting point and if so, where do I start?

Relativity is still way too high right now. What is the first physics and mathematics course you will be enrolled in once your school starts? I suggest you just worry about those courses for now. If you major in physics, you're going to learn quantum mechanics and some special relativity either way. Other than that, I'm going to let the older guys take in, since I haven't even completed undergraduate yet :P

 

[Arcthor]

Relativity is still way too high right now. What is the first physics and mathematics course you will be enrolled in once your school starts? I suggest you just worry about those courses for now. If you major in physics, you're going to learn quantum mechanics and some special relativity either way. Other than that, I'm going to let the older guys take in, since I haven't even completed undergraduate yet :P

Multivariable calculus is one of the first courses I am going to learn, so I don't know.

 

[Amrator]

Then prepare for multi-variable calculus. There you go. Also, if you are at that level in math, then you should be able to study for whatever physics course you're going to take.

For resources: Check out both the STEM Learning Materials and Science and Math Textbooks sub-forums.
Resources off of the top of my head: MIT OpenCourseWare, Physics Classroom, and Wikipedia.
A great resource for math is Khan Academy.

Have you read any of ZapperZ's So You Want To Be a Physicist? Be sure to check that out. Just check out the whole Physics Forums Insights while you're at it. Great stuff.

 

[JorisL]

I'd say first make you single-variable calculus more rigorous.
In high school I learned about derivatives, integrals etc. as well. However that was a more mechanistic approach, basically shut up and calculate after we had 'derived' integrals from Riemann sums and the same for derivatives from the Newton quotient.

In college you will learn how they actually work. If it is anything like the Belgian system (high school seems similar in my case) you'll be able to go through the first part rather quick but then you learn some new things (approximation etc.)

But for well-tailored mechanics courses you should have enough background I believe. (well-tailored in the sense that you won't need the Taylor series approximations to get a meaningful result)

 

[Amrator]

I agree with JorisL. Make sure you understand the material intuitively. Do NOT try to memorize.

 

[pmr]

I'd study the following things, and in this order:

1) Vector calculus
2) Basic linear algebra, of the non-abstract sort (especially eigenvalue problems)
4) Classical Mechanics, on a moderately advanced level)
5) Special Relativity
6) Electrostatics
7) Electrodynamics

 

[TubbaBlubba]

I just completed my freshman year of physics at Göteborgs Universitet, so here are my basic recommendations:
1. Make sure you are very confident in single variable calculus. Understand and learn to do epsilon-delta proofs, utilize the mean value theorem, learn precise definitions, characteristics of the real numbers (such as Cauchy sequences), etc. I studied under the old Gymnasium program, but in my experience the curriculum was far from sufficient for this; we simply learned _how_ to differentiate and integrate and solve very basic differential equations. It is important to understand why things are as they are, how things are defined and why, and so on, especially since you will go on to learn more advanced mathematics. Personally I recommend Spivak's Calclulus due to its mathematical rigor, although other texts may be more focused on problem solving and such.
2. Basic linear algebra (Gaussian elimiantion, Eigenvalue problems, matrix multiplications, Gram-Schmidt, linear differential equations, etc - the stuff you will find in an introductory textbook) is very important, and it is a prerequisite to understanding both multivariable calculus and much of physics. Linear Algebra Done Right is a good book.
3. Classical mechanics; basic problem-solving, finding equations, transforms, etc. Nicholas Apazidis' Mekanik I contains great exercises, although it is weak on the theory side. ETA: Also, you should probably look at some exercises for numerical solutions of ordinary differential equations, e.g. in MATLAB, since generally Newton's equations cannot be solved analytically.
4. Multivariable calculus, up to and including line integrals, Kelvin-Stokes' Theorem, and so on. You may also be interested in Spivak's pamphlet, Calculus on Manifolds, which goes a bit further but contains many useful results, theorems and explanations.

This should keep you busy for a while, and prepare you well!

Edit to add: One more important thing, in both single- and multivariable calculus, that you probably know already: While learning to differentiate is mostly about understanding and applying a set of rules (in any number of dimensions), remember that integrals are significantly more difficult and generally cannot be evaluated in a single methodical step-by-step fashion. Therefore, I strongly recommend you to evaluate lots, and lots, and lots of integrals, and learn all the tips and tricks well, recognize patterns and develop an intuition for how to evaluate or otherwise handle them. Learning the methods of numerical quadrature will also be essential eventually, although you might wish to wait a bit with that.