Chapter 6.7, Problem 347E

For the following exercises, use Stokes’ theorem to evaluate ${\displaystyle {\iint }_s\left(curlF\cdot N\right)dS}$ for the vector fields and surface.

347. Use Stokes’ theorem for vector field $F\left(x,y,z\right)=zi+3xj+2zk$ where S is surface $z=1-x^2-2y^2$ , $z\ge 0$ , C is boundary circle $x^2+y^2=1$ , and S is oriented in the positive z-direction.