Syllabus for learning advanced mathematics and physics for a highschool student.

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brunstheorem
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Syllabus for learning advanced mathematics and physics for a highschool student.

Postby brunstheorem » Wed Aug 08, 2018 4:14 pm UTC

Hello. I am currently a 16 year old high school student preparing for the iPhO. I am really interested in physics and mathematics. I know single variable calculus, classical mechanics and electromagnetism but in an undergraduate level. What kind of a syllabus should I follow for learning advanced mathematics and physics from the ground up?

My plan for the future partially consists of double majoring in physics and mathematics and as a graduate student working on mathematical physics (Quantum Field Theory, Gravitation) and computational neuroscience. What I want is to better spend my time in highschool so I can get a headstart in my future studies as an undergraduate student in the fields mentioned above.

Thus I would really appreciate it if you guys could give an even vague pathway in approaching these fields, since probably some of you are working on these subjects.

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doogly
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Re: Syllabus for learning advanced mathematics and physics for a highschool student.

Postby doogly » Wed Aug 08, 2018 5:35 pm UTC

OK I would roll up the sleeves and get started on some Solving Difficult Problems. I highly recommend
Mechanics - Kleppner and Kolenkow
Electricity and Magnetism - Purcell
Calculus - Spivak

I spent quality time with the first two when actually learning them. Spivak's Calculus I was too late to the party to actually use. I hepped to him as of Comprehensive Introduction to Differential Geometry, which I also recommend, but that is later.

You will very soon get used to the answer to "What next" being "More of the last thing you did, but harder."

If you're also interested in computational neuro, I can't say too much at all. Computational physics though, I think this generalizes pretty easily to other scientific computing, and I think you'd be well served by numerical methods in python. There seem to be a bunch of good books on these which focus more on the computational recipes, and then you might have some more specialized things if you are then focusing on computational neuro, or plasma physics, or numerical general relativity, or what have you. But python is a good way to go.

Oh more disclosure, I also never learned any Python.
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Qaanol
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Re: Syllabus for learning advanced mathematics and physics for a highschool student.

Postby Qaanol » Thu Aug 09, 2018 12:04 am UTC

For mathematics, you will definitely want a solid grasp of linear algebra and differential equations. A significant part of physics comes down to partial differential equations, which can get…difficult.

On the more theoretical side, you may want to learn about topology and manifolds. And, in light of Noether’s theorem, a solid foundation in abstract algebra wouldn’t hurt either.

Regarding classical physics, are you comfortable with the Lagrangian and Hamiltonian formulations? They are often easier to work with and more illuminating than a direct application of Newton’s laws.

You should also try learning how to program, preferably in a language that’s strong on numerics such as Python, Matlab, or R. Being able to create and run your own computer simulations is a useful skill.
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Re: Syllabus for learning advanced mathematics and physics for a highschool student.

Postby Link » Sat Aug 11, 2018 12:06 am UTC

David Griffiths has excellent introductory textbooks on electrodynamics, quantum mechanics and particle physics; those are definitely worth checking out.

Lagrangian formalism and the variational principle are eventually going to be the go-to tools for any new problem, so it's definitely a good idea to get a solid understanding of those. Of course you should have a basic grasp of partial differential equations to even get started with that, but that shouldn't be difficult to pick up. Unfortunately I don't know of any particularly outstanding entry-level texts that really work directly towards understanding those concepts from the ground up, but it's at least a goalpost to try to move towards. Maybe have a look at David Tong's lecture notes and videos. I found his notes on quantum field theory quite insightful (although they're definitely not early-undergraduate level).

Linear algebra is indeed another big one. Quantum mechanics basically is linear algebra, so just working through any LA textbook will probably be useful later on. Pay extra attention to sections on Hilbert spaces and unitary transformations.

For the computational part, first of all, learn Python, and in particular Numpy and Scipy. They're very very easy-to-use and surprisingly fast numerical libraries for mathematical and scientific computation of various kinds, ranging from basic vectorised maths to curve fitting, root finding, and even solving of differential equations. Some of the key computational techniques you'll want to learn are Monte Carlo methods for statistical physics and Runge-Kutta methods for the time-evolution of differential equations; those are always good skills to have. In your case I would imagine neural nets and Deep Learning are also immensely useful to learn, but I can't really help you there since I've never dabbled with either.

Also, get yourself a copy of Wolfram Mathematica and get acquainted with it. It has a slight learning curve due its somewhat awkward syntax and the fact that it almost always has like 37 different built-in ways to do the same thing, but it's an incredibly useful tool for symbolic computation. As a real-life example: I've recently used it to solve a boundary value problem where the final solution easily fills my whole screen. The problem itself wasn't very difficult per se, but the resulting expressions are so horrifyingly long that it would probably have taken weeks to get (and verify) the solution by hand. Mathematica does it in minutes. It won't help you with abstract mathematical physics, but you're almost certainly going to want to actually compute something concrete at some point, and then it can be a real life-saver. Also I vehemently maintain that Wolfram integration is just as valid a technique as, say, contour integration or integration by parts.

brunstheorem
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Re: Syllabus for learning advanced mathematics and physics for a highschool student.

Postby brunstheorem » Sat Aug 11, 2018 10:59 am UTC

Qaanol wrote:You should also try learning how to program, preferably in a language that’s strong on numerics such as Python, Matlab, or R. Being able to create and run your own computer simulations is a useful skill.


I was planning on learning python but I don't know where to start. Can you reccomend any resources for getting started with it?

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Re: Syllabus for learning advanced mathematics and physics for a highschool student.

Postby Link » Sat Aug 11, 2018 3:36 pm UTC

Introductions to Python can be found all over the web, really. You could of course use the official tutorial. Alternatively something like https://www.learnpython.org/ will let you jump right in, and it even embeds an interpreter on its pages so you can play with it as you read. If you have no prior programming experience at all, see e.g. https://en.wikibooks.org/wiki/Non-Programmer%27s_Tutorial_for_Python_3. The Standard Library reference will be your best friend once you get the basic syntax down.


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