Stoke’s theorem is that the total change over the boundary is equal to the sum/integral of the derivatives over the region. Written out like this: $$ \int_D df = \int_{\partial D}f$$ If D is the region and $\partial D$ is the boundary of that region.
I understand what the integral of the derivatives over the entire region means. What I don’t understand is how the total change over the boundary on the right is calculated when thinking in terms of abstract spaces like manifolds or even 2D space. Conceptually I’m also having trouble understanding what I’m supposed to look for along the boundary or what is even changing: direction, position or something else. It would be nice if I could get some help on this. Thanks!