Is it worth it to take multivariable calculus and linear algebra over the summer through harvard extension? I am a sophomore in high school that is finishing BC calculus. What kinds of math would I take if I did that? Will it help me when applying for colleges? Any tips on how to succeed in those classes?
I suggest you take the class for your own interest in math, but don’t do it for college admission because it won’t help. If you have done well in BC calculus, you should be fine with the relatively elementary versions of multivariable calculus and linear algebra courses offered by Harvard Extension School.
What would the plan be for the fall - would you take any more math junior and senior years of high school? It may make more sense to spread those out by taking them during the school year, to avoid either getting too far into college math (such that it becomes difficult to place out of those courses in college), or alternatively, having too much time off from math before college.
For summer, consider some math off the regular sequence, such as one of the interesting courses at https://artofproblemsolving.com/. Perhaps Harvard Extension has courses on alternate topics as well.
Are you interested in taking more math, or interested in a math-heavy major?
The usual courses after single variable calculus (calculus BC) are multivariable calculus, linear algebra, and differential equations (the latter two are often combined). These are typical math courses for math, statistics, physical science, engineering, pre-PhD or quantitative economics majors. Computer science majors generally will take linear algebra and discrete math; those in engineering-based programs will also take multivariable calculus and differential equations.
Beyond that, math majors will take upper level math courses with a much greater emphasis on proofs (e.g. real analysis, abstract algebra); some statistics, physics, and economics majors will take some of those courses as well. Computer science majors typically take proof-heavy upper level computer science theory courses.
DS took BC in 9th and then was able to audit Linear Algebra at a local college during sophomore year. In his junior year he has taken Graph Theory and Multivariable at his high school. He should take Differential Equations next year. The issue we are seeing is that he will probably have to repeat these courses in college since the courses were not taken at a college.
Depending on which college it might not be an issue as I am sure some of the tippy tops will challenge him no matter if he has taken it before, but this is not necessarily true at other colleges. As a white male with no hooks, a tippy top is far from a probability although his high school does have a good success rate. He enjoyed all of the courses and is glad he took them but does not look forward to repeating material.
He is aware that some colleges will allow him to test out of certain classes but it seems this is not always a very easy process and varies quite a bit. He would do the same sequence if he were to do it over but you should be aware that it might cause boredom in the future.
He recently read an article where an individual that had taken many upper level math courses in high school indicated that he felt that many top mathletes drift away from math as they are oftentimes repeating material in college in math but physics or CS is more interesting as it is new.
It seems you are interested in math, which is great. However you should think about why you are taking such advanced math? For me, I am doing AP calc BC as a sophomore, because it is almost mandatory for learning physics olympiads at a deeper level.
Why do you want to learn multi? I don’t know how much it would help for college admissions, though of course that shouldn’t be your only motivation. You can read the famous calculus trap https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap here.
If you want, try math competitions! It really helps with problem solving in many fields later on in life, and good for college admissions.