According to my textbook, Stokes’ theorem differs from the Divergence Theorem because it is applicable to many different surfaces (I.e., as integrals.)
I came across this explanation but am puzzled - all of the unfamiliar symbols in the formula don’t help, either! Here is the quote: “In the divergence theorem, the region bounded by a closed surface is unambiguous; for instance, there is exactly one region bounded by a given sphere. When you apply Stokes’ theorem, however, one curve can be the boundary of many different surfaces” (image attached).
First of all, what relevance does Stokes theorem have if the result can be so many vastly different shapes. And how can the surface integral of the curl of a vector field be equal to the path integral along the closed path forming the perimeter of the surface? Based on the image, it seems like all these shapes have very different surface areas, and I’m quite confused.
Thank you so much.