Use Stokes' Theorem to evaluate . ; S is the part of the paraboloid lying below the plane and oriented with normal pointing downward.
C)0

Question

Use Stokes' Theorem to evaluate  .  ; S is the part of the paraboloid  lying below the plane  and oriented with normal pointing downward.
C)0

Quizplus · Accepted Answer

Stokes' Theorem relates the surface integral of the curl of a vector field over a surface S to the line integral of the vector field around the boundary curve of S. Since the curl of a gradient is always zero, and the vector field in this problem is the gradient of a scalar function (namely, f(x,y,z) = xyz), the curl of the vector field is zero everywhere. Thus, the surface integral of the curl over S is zero, and Stokes' Theorem tells us that the line integral around the boundary curve of S is also zero. However, there is no boundary curve given for S, so we cannot evaluate this line integral, and the answer must be zero.

Use Stokes' Theorem to Evaluate