Advanced Calculus fi..

324 Advanced Calculus, Fifth EditionFigure 5.34Proof of Stokes's theorem.1'4If the normal n chosen on S is the upper normal, the direction on Cxy is the positivedirection, and by Green's theorem,/L(~.Y.Z)~~ = /LIX,Y, f(x, y)]dx = -11(5 + "%) dx dy.Caz ayC x ~ R ~ Y 4Under the same assumption about the normal,by (5.80). It at once follows thatIf the direction of n is reversed, both sides change sign, so that (5.97) holds ingeneral. By the reasoning described above, one then finds that (5.97) holds for ageneral orientable S. In the same way, equations analogous to (5.97) are establishedfor M and N for a general S. Upon adding the equations for L, M. and N, oneobtains Stokes's theorem in full generality.Just as the Divergence theorem gives a new interpretation for the divergenceof a vector, so does the Stokes's theorem give a new interpretation for the curl of avector. To obtain this, we take S, to be a circular disk in space of radius r and center(x,, y ~, 21) bounded by the circle C, (see Fig 5.35). By the Stokes's theorem andMean Value theorem for integrals,