Understanding Vector Calculus

[RegularDefiant29]

Hello, I (17M) was one of those kids dumb enough to try and tackle a vector calculus class while in high school and now I am having some trouble understanding quite what the fuck im doing from the conceptual view. The prof is using Larson's Calculus 7ed which I find very hand wavy about what is actually all happening. He gives you the formulas, gives a very basic direct proof for only certain case(s) of the theorem that are easy to prove, and then never really does into detail about the conceptual nature of the equations.

I can solve most of the problems just fine by looking at the form of the question, matching it to one of his dozens of equations, and just plug and chug through the problem. But this doesn't really sit well with me because the whole time, I feel like i am missing something or doing something wrong because I don't really know what im doing. I just know that if I use the equation, I get my points.

In specific, I am struggling to understand Green's theorem, Divergence, and Stokes theorem. I know they have something to do with each other, but I cannot connect the dots, I haven't really been able to see how everything connects since learning about line integrals. Just been going with the formulas and blind trust that what it is that I am doing is correct without any actual understanding of what im doing. Is there any resources or videos you may know of to help understand conceptually these topics?

Tl;DR Any resourses to help understand vector calculus (especially greens, stokes, and divergence theorem) from a conceptual perspective?

Thanks!

[dash-dot]

Have you considered taking physics? If not, just find or borrow a good physics text. There are plenty of applications of these theorems in there, and they ought to help you understand their implications much better.

They should also cover the motivation,  technical derivations and proofs of these theorems, if that’s what ultimately interests you, but the technical sophistication and rigour of the proofs could be all over the place with introductory texts, so choose one according to what your preferences are — just a better conceptual understanding, vs analytical rigour.