What Advanced Physics or Math Resources Can I Explore to Avoid Boredom?

[Fisicks]

Question 1: Over the summer i briefly went through a introductory college physics class in Classical Mechanics, i covered all of http://www.etsu.edu/physics/lutter/courses/phys2010/index.htm" for the most part. Now in school I'm taking Physics B which is basically the same stuff, just probably not as in depth. So basically I'm bored, and want to take it to the next level. I have a really good understanding of calculus up to multi-variable. What kinds of options are open to me if I am eager to learn?

Question 2: Over the summer i went through all of http://press.princeton.edu/TOCs/c8351.html" and in class I'm taking Calculus BC so I'm basically going through the whole book again, which is boring. Now with all these things in mind, will Apostols Calculus be boring to me or exciting. I don't to spend the money on the textbook if I'm going to be going over the small old stuff.

Thanks in advance,
Fisicks

 

[aPhilosopher]

If you want to take the math further, I highly recommend "Undergraduate Analysis" by Serge Lang. It's terse but it has a full solution manual (you have to buy it separately) and it is a good introduction to the field. It will put what you know about calculus on a rigorous footing and extend it to other metric spaces. He covers Fourier series and introduces differential forms.

Some linear algebra might help with the above book, for which I recommend "Linear Algebra: An Introduction to Abstract Mathematics" by Robert Valenza. It's never too early to find out what a group is ;) The one problem is that it doesn't have any solutions but the problems are all proof oriented and very well graded in terms of difficulty. If you toss a few of them up on the homework board, I'm sure you'll get feedback.

If you only go with one of the above, I'd say linear algebra. It provides a foundation for a lot of physics (QM for example) and a ton of math as well. The author does an excellent job of showing you how to understand it well enough to write proofs.

I don't do a ton of physics (although I'm starting) so I won't make a recommendation for that.

 

[n!kofeyn]

Question 1: Usually the next step after the two basic course that cover Newtonian mechanics, electricity and magnetism, and optics is a course in modern physics. The book I used in school was Modern Physics by Krane. To do much more in physics, you need to know a bit more math including multivariable and vector calculus, differential equations, and some linear algebra, but this book requires none of that. Since you're still in high school (I'm guessing because of the Physics B and Calculus BC), you might want to read Understanding Physics by Isaac Asimov. I say this because it doesn't require math (even though you know singe-variable calculus) and will give you a great introduction to and understanding of the stuff you will learn more thoroughly later. It goes from mechanics all the way to particle physics. A fantastic book.

Question 2: It depends on what you want to do. If you want to learn to do proofs, my suggestion would be to pick up Analysis: With an Introduction to Proof by Steven Lay. It has an excellent buildup that teaches you logic and techniques of proofs and then leads you into introductory analysis, which is the theoretical foundation of calculus. Calculus by Michael Spivak is also very good and slightly similar to Apostol, teaches you basic analysis and more theoretical calculus, and has a solutions manual as well. Apostol's writing is dry, and his books are expensive. Although I've never looked at it, I do not recommend you get the Lange book.

More advanced introductory books to analysis are Mathematical Analysis by Tom Apostol, Advanced Calculus by Creighton Buck, and Understanding Analysis by Stephen Abbott, which are alternates to the Lang book. Apostol's book is by far the more advanced and difficult, but very complete. I'd recommend the others before you go to it.

If you are wanting to just learn more math, then maybe read through the differential equations and linear algebra sections in Calculus by Tom Apostol (the book you mentioned) or in Richard Courant's Differential and Integral Calculus (this book has more physical intuition built in than Apostol and probably more fun to read). Apostol and Courant also have multivariable calculus material. You need to check though as both of the authors' books have two volumes, and I don't remember what material is in which volume.

A book that would be great to go through is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard. I've only read the excerpts and table of contents available online, so I don't know its exact contents, but it seems like the best, although challenging, way to learn multivariable and vector calculus. I don't recommend learning linear by itself as suggested above, as it is extremely boring, and it really helps to see how it's used as you learn it, which is how the above book will treat linear algebra.

 

[Fisicks]

jeez you guys are like science guidance counselors, I wasn't expecting all that! I have some research to do, thank you so much for all the considerations. I've decided to leave the physics to whatever college I'm going to but I am defiantly picking up one of those math books.

 

[n!kofeyn]

jeez you guys are like science guidance counselors, I wasn't expecting all that! I have some research to do, thank you so much for all the considerations. I've decided to leave the physics to whatever college I'm going to but I am defiantly picking up one of those math books.

Cool! No problem. I really recommend the Isaac Asimov book. It's the type of book I wish I would have read before I started, because now I'm so busy with everything else, I don't have time for it. It will read more like a novel or book than a textbook, because it definitely is not a textbook. By the way, this is the same author that wrote I, Robot and other famous science fictions books.

Also, a few more recommendations I always try give out. Take a look at George Gamow's books. The only one I've read is Gravity, which gives a discussion of the different theories of gravity, from Galileo to Einstein. Since you know calculus, it will be a good read. All his other books are classics as well. And of course, there are Richard Feynman's books. These are more fun to read than a textbook, as they are the masters having fun explaining the things they love. The last is Isaac Newton by James Gleick. It's a short, but clear, and fun biography. These are all books that I found out about after undergraduate school. I wish I would have known about them before!