This is the last page of my Vector Calculus lecture notes:
I've understood most of it I guess. The last equation basically is very similar to that of a conservative vector field, indeed we see that the integral around a closed surface (instead of a closed path) is $0$, and thus the integral is "surface-independent". However I don't understand if there is a mistake when it says
where $s_1$ and $S_2$ are two open surfaces with the same boundary $C=\partial S_i$.
How come they are open? And also, shouldn't they be closed? It says that they have the same boundary, so I guess the boundary should indicate that they are closed curves?
Can you please help me clarify that last step/equation?
Thank you