i thought stokes theorem (green's thm) was hard after reading it in spivak, who calls it trivial nonetheless. however lang showed it is indeed trivial in his analysis I. the same proof occurs in courant. I.e. the point is that the theorem is easy for a rectangle, where it follows immediately from fubini and the fundamental theorem of calculus. the same argument works on a circle. then one discusses the concept of a "parametrized" plane region, i.e. the image of a rectangle or circle under a mapping. there is a concept also of "pulling back" a vector field or covector filed, under the mapping. Then the pull back of the curl is the curl of the pullback, the pull back of the integral is the integral of the pullback, etc... the upshot is that every term in the theorem pulls back faithfully under the parameter map so that once the theorem is proved for a rectangle or circle it is also true for every region which can be parametrized by a rectangle or circle, i.e. essentially any convex region or deformation of one. that does it. i apologize for posting a new thread on this question asked elswhere but I could not find that thread again.