Courses Need help in choosing math classes for a HS student

[Homelilly]

For next year, check out https://loveofmath.com/course?courseid=multivariable for MVC - it's run by a former AoPS teacher. What's his physics background?

I love Heather with all my mom's heart - she is doing an incredible job! Our two kids have a lot of fun learning with with her - and the kid is taking MVC there next year, and other two are having Geometry and Algebra. He is doing HRK now and taking AP Physics C in May.

 

[Homelilly]

After vol.1 Apostol, he should use vol.2 Apostol. In particular, multivariable differential calculus is conceptually much more natural using linear algebra, which Courant does not do.

By the way, here is a free, short (about 125 pages), book on linear algebra that I wrote for fun.
https://www.math.uga.edu/sites/default/files/laprimexp.pdf

It is an expansion of a much shorter version (15 pages) available on the same website, which I wrote just as an exercise to see how short I could make a book on linear algebra, leaving many easier facts as exercises.

By the way, the book by Richard Penney looks quite good, but there are many ways to present the subject, and even though his book is almost 500 pages, and covers many, many things mine does not, he still does not quite cover all the same bases. E.g. he presents the Jordan form of a matrix, but not the rational canonical form. One difference between them I like to point out, is that one cannot usually actually compute the Jordan form in practice, since it requires factoring the characteristic polynomial, and there is no practical way to factor polynomials. There is however an algorithm for computing the rational canonical form by hand, so one can actually find this form in practice. My little book covers both topics, and even proves the (less important) uniqueness of the reduced echelon form of a matrix, which Penney omits. I also at least sketch a more general form of the spectral theorem than he does, for "normal" operators. He also seems to omit any applications of linear algebra to differential equations, which my book includes. I think this is curious, although common, since differentiation is arguably the most important example of a linear operator. So it never hurts to have more than one book around.

Another glance at Penney reveals that he also treats numerical methods. In the computer age, this is another response to the fact that polynomials are hard to factor. i.e. since one cannot always compute
"eigenvalues" exactly by factoring, one can use numerical methods to at least compute them approximately. I do not treat this (out of ignorance).

Thanks a lot for sharing! That's very kind of you! I will print it and he will go through it.

 

[mathwonk]

If he finds my book impenetrable, I will be happy to answer questions on it. If he has the appetite, here is another short (67 pages) free linear algebra book I wrote. This one includes a complete treatment of determinants. These are based on lectures from a course I taught.
https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

Warning: It is possible these short "books" of mine may not be so easy to read, even though he has completed the course of linear algebra from Penney.

I am interested in his reaction to them however, if he does look at them. It is quite possible they are too terse, or too abstract, but I hope they may be useful at least for a different perspective. And perhaps the fact they are brief may make them more useful for review, highlighting and recalling the main ideas.

As long as I am at it, here are the notes from my course on rigorous proof - based geometry, from Euclid, taught in the first year (2011) of epsilon camp, a 2 week summer experience for brilliant kids aged roughly 10-12. These notes start from scratch and go far enough to deduce the volume of a 4 dimensional ball by techniques Archimedes would have understood. They also discuss how Euclid's and Archimedes' ideas pave the way for, and relate to, those of Newton in calculus.
https://www.math.uga.edu/sites/default/files/inline-files/10.pdf