Sounds like fun! I’m going to bed soonish but I’m willing to answer questions about multivariable calculus probably when I wake up.

When I took multivariable calculus, the two books that really helped me “get the picture” were Multivariable Calculus with Linear Algebra and Series by Trench and Kolman, and Calculus of Vector Functions by Williamson, Crowell, and Trotter. Both are on LibGen and both are cheap because they’re old books. But their real strength lies in the fact that both books start with basic matrix algebra, and the interplay between calculus and linear algebra is stressed throughout, unlike a lot of the books I looked at (and frankly the class I took) which tried to hide the underlying linear algebra.