Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary of the region, where $A$ can in some sense be thought of as a "curvature at one higher derivative" of $B$ or a closely related quantity. Is either of these theorems a special case of the other? If not, is there a more general theorem of which they are both special cases (which isn't too many levels higher up in abstraction)?
Edit: the answers to this follow-up question provide derivations of the Gauss-Bonnet theorem from Stokes's theorem in this paper, on pg. 105 of this textbook, and in Chapter 6 Section 1 of this textbook. Unfortunately, the derivations are too advanced for me to understand, as I haven't formally studied graduate-level differential geometry. I would appreciate any answer that summarizes the basic idea of the derivation.