Chapter 6.7, Problem 364E

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

364. Let S be paraboloid $z=a\left(1-x^2-y^2\right)$ , for $z\ge 0$ , where $a>0$ is a real number. Let $F=\langle x-y,y+z,z-x\rangle$ . For what value(s) of a (if any) does ${\displaystyle {\iint }_s\left(\nabla \times F\right)\cdot ndS}$ have its maximum value?