The discussion revolves around selecting appropriate math classes for a high school student entering 11th grade, who has already completed AP Calculus BC and an introductory course in Linear Algebra. The focus is on balancing advanced math coursework with preparation for math competitions and future studies in physics, applied math, or electrical engineering.
Participants express a variety of opinions on the best math courses for the student, with no clear consensus on a single path. Some advocate for traditional courses while others suggest alternative or additional subjects based on the student's interests and competition goals.
Participants note that the student is advanced for his age and may not need to follow the typical sequence of intermediate university math courses. There are also considerations regarding the balance between formal coursework and self-study, particularly in relation to math competitions.
Parents and educators seeking guidance on advanced math course selection for high school students interested in STEM fields, particularly those preparing for math competitions or pursuing physics and engineering.
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.Muu9 said: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?
Thanks a lot for sharing! That's very kind of you! I will print it and he will go through it.mathwonk said: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).