# VC5.21

From Example Problems

Let the given surface S is cube (OCBA,GDEF) which is bounded by planes x=0,y=0,z=0,x=2,y=2,z=2. The given cube meets the xy-plane in the square of OABC where the coordinates of A,B,C are (2,0),(2,2) and (0,2) respectively. Thus the curve C bounding S is square OABC.

Hence,by Stokes'theorm we have

=

=

= as on C,z=0 and dz=0.

= --(1)

Along OA,y=0,dy=0 and x varies from 0 to 2. Along AB,x=2,dx=0 and y varies from 0 to 2. Along BC,y=2,dy=0 and x varies from 2 to 0. and along CO,x=0,dx=0 and y varies from 2 to 0.

Therefore,from(1),

=

=