Expert: Daniel Mazur - 9/15/2008

Question
Hi!
I'm a high school student and I have always had a great interest for maths and, especially, physics. I would like to enlarge my knowledge about it and soon be on a good university, but I get the impression school physics is not getting me anywhere... We repeat subjects, don't look deeper at them, and take only a very little glimpse to what is physics. So, I'm writing to you, hoping you could tell me where I should begin (maybe some good books you know), and what is the best way to study physics by myself once I don't get what I want from school.
Thanks,
Alexandra

Answer
Hello Alexandra,
thank you for the question. What a good point to start is and what would be a good book, that depends very much on how much you already know. Since to study physics seriously one needs a thorough knowledge of certain parts of maths, you need to advance both in parallel. I think this is one of the reasons, why towards the end of high school physics the teachers repeat old stuff rather than teach something new: they would need you first to learn a lot more maths to gain anything from the more advanced subjects.
The mathematics you need to get familiar with (mostly undergraduate college subjects) are:
Linear Algebra : vector and tensor (matrix) calculus, operators, their eigenvectors and eigennumbers,
Mathematical Analysis : derivatives, simple integrals, solving ordinary differential equations, vector and tensor operators (like 3D derivative), integration in a complex plane
both : transformations of systems of coordinates, series expansions of functions
With the above you can go on tackling most of the problems you might encounter as a student of physics and even during your later career. There is a big chunk of theory behind each keyword I wrote and 95% of it is only useful to those intent on going on as physicists, chemists or engineers. In my old high school class of 30 people we were 3 going on to anything alike, and only I went to do physics. Our teachers also did not go any deeper in physics as there would be at most 3 people benefiting from it.
Hence, I think that a good thing to start with is some standard college textbook of maths for physicists. There are many results when searching Amazon.com for "mathematics for physicists". I looked into a cheap paperback by Dennery and Krzywicki and there is more than you need right there. As mathematics never changes, once established, cheap book contains as much information as an expensive one. The difference between them is usually the level of accessibility, "reader-friendliness". The cheaper textbooks are "raw" maths, not colored, intended for users with quite a good understanding of standard mathematical notation of statements. The expensive textbooks have a lot of redundancy, as if "holding your hand" all the way through the new topics, telling a lot of "intuitive" guides, what this or that abstract term means in the real world, giving examples... Most colleges nowadays (all the American ones) use the modern and expensive books, so you can bet on it, that as soon as you enter college, you will have to buy several. Therefore, I recommend something "raw" and cheap like the paperback I mentioned. It is a challenge to read and study from it, but it will give you a good test of character (i.e. "Are you willing to go through all this?") and all the information there will be guaranteed correct (apart from typos). The "modern" books frequently have the flaw of overdoing the "intuitive" approach and "real life examples", so that they sometimes say quite a bit of nonsense.... Put it this way: The "raw" maths textbooks are written by those, who understand maths. The "reader-friendly" textbooks are written by those, who understand writing textbooks, but not necessarily the maths.
A similar thing happens with college physics books, but I have a suggestion. Feynman, Leighton and Sands wrote a series of textbooks for undergraduates. By standards of teaching first-timers they are considered not very good. However, they offer a lot of narrative that does not so much teach, but radiates the authors' *enthusiasm for physics*. The books are very famous and many physicists around the world buy them for the inspiration, for insight. I heartily recommend them to you along with a maths textbook, it is a good reading, reasonably good for first encounter and learning and again, the physics in there is guaranteed correct.
Lastly, I think you might consider finding yourself a tutor. Perhaps if you put up an ad at a physics college nearby your home for a 3rd- or 4th-year physics student (or even graduates, no problem with them) to tutor you the basics of the undergraduate college maths and physics, someone will be willing. With a good coach you could get skills that you will be able to use as a freshman already, while your classmates are still only learning them.
Good luck and don't hesitate to write again, if you have more questions.
Cheers,
Daniel