I am looking for a good textbook/online notes that treat vector analysis theoretically, but more from a analysis/pde approach (emphasis on divergence theorem and symbolically computing integrals), and less from a differentiable manifolds approach (heavy emphasis on generalized Stoke's theorem).
Ideally it would give a good sense about how to compute in hyperspherical coordinates (example: spherical averages from Evan's pde chapter 2) but I will definitely settle for not going full $\mathbb{R}^n$ with $n>3$. I ask because I am having some difficulties with some of the vector calculus in PDE and haven't been able to find anything suitable online or in the library.
Hopefully this isn't asking too much. Thank you very much!
Have you seen Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger? Is this the kind of textbook you are looking for?
Perhaps this book does not cover all the topics you might be interested in, but at least you should be all set for when it comes to Elliptic PDEs.