I'm currently in the middle of a year long vector calc + linear algebra class, and we saw the diagonals procedure. We have not defined n by n determinants yet, though I think our book defines it axiomatically (the unique multilinear map with some list of properties).
To quote Vladimir Arnold,
"The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns. If the students are told this secret (which is carefully hidden in the purified algebraic education), then the whole theory of determinants becomes a clear...
All three books you listed are good books, although Art of Problem Solving is not at all similar to the other two. Art of Problem Solving is aimed specifically at people who are preparing for Middle School and/or High School Math Competitions (especially the AMC/AIME/USAMO series). The other...
Out of curiosity, does anyone know if there are English translations of any of the Indian work on calculus? I read somewhere that in the 12th century they figured out differentiation, Taylor series, and even a bit of analytic number theory.
http://www.gravityfromthegroundup.org/ might be an interesting book for you guys to work through. It uses no calc and covers quite a few physically interesting topics, ranging through various parts of classical mechanics to a bit of general relativity, all with an emphasis on astro and cosmology.
Mathwonk, have you felt that your new insights into classical geometry have helped you understand modern (say differential or algebraic or something) geometry?
For the relativity book, once you've finished AP French's special relativity book, I suggest you read Bernard Schutz's "A First Course in General Relativity" or James Hartle "Gravity: An Introduction to Einstein's General Relativity."
I'm currently a freshman in college (at Yale) and am very interested in a major in math (or math/physics) and quite possibly grad school beyond (though it is very premature for me to make such a decision). The math class I am currently taking (Year-long Vector Calc+Linear Algebra with textbook...
I think its more high school than physics that is causing your lack of interest in physics class. I was the same way with my HS math classes (luckily had a good, interesting physics class tho). Is your physics teacher the type who would teach you fun stuff outside of class if you asked? If...
Yes, that is true. At least mathematically, goldstein or landau is manageable after calc 3, but the physics might be tough going. Personally, I worked through landau with my HS physics teacher with only BC calc as background (plus independant study on a motley array of fun topics), and I found...
Goldstein's Mechanics and Landau/Lif****z are about at the same level. I personally prefer L/L because its much shorter and doesn't feel like it rambles, but many people prefer Goldstein also. If you have library access, you could always just check out a copy of each of these books!
To answer the OP's question, integrals can differ by a constant. You can't just "cancel" your integrals in the equation where you seem to have 1=0. If you differentiate both sides of the equation, everything works out nicely. At least in basic calculus, the indefinite integral is simply...
After calc 3, you can do even harder mechanics books than Kleppner. I highly recommend "Mechanics" by Landau and Lif****z if you want a top notch mechanics book. You will have to learn a little bit of math (rudimentary calculus of variations, and you'll have to look up a theorem here and...
You might want to try Steven Weinberg- "The Quantum Theory of Fields- Volume 3: Supersymmetry"
If you make it through, you should be a physics genius by the end.