Here S is the surface of the sphere lying above the xy-plane.Let C be the boundary PQK of the surface S.Then the curve C in the xy-plane and equations of C are
The parametric equations of C are --(1)
We need to verify Stokes'theorm that is --(2)
LHS of (2)=
= as on C,z=0 and dz=0.
Let R be the plane circular region PQK bounded by C.Let S1 be the surface consisting of the surfaces S and R. Then S1 is a closed surface enclosing volume V(say). By Gauss Divergence theorm,we have
Since S1 consists of S and R,we have
--(6),noting that n denotes unit vector to S1.
From (5) and (6),we have --(7)
Therefore,RHS of (2)= by using (7)
From (3) and (8),we obtain (2) and this verifies Stokes'theorm.