Chapter 6.7, Problem 373E

For the following exercises, let $S$ he the disk enclosed by curve $C:\text{r}(t)=\langle \cos \phi \cos t,\sin t\sin \phi \cos t\rangle$ , for $0\le t\le 2\pi$ , where $0\le \phi \le \frac{\pi }{2}$ is a fixed angle.

373. Evaluate integral ${\displaystyle {\iint }_s(\nabla \times \text{F})\cdot \text{n}dS}$ , where $\text{F}=-xz\text{i}+yz\text{j}+xy{e}^{{}^z}\text{k}$ and $S$ is the cap of paraboloid $z=5=x^2-y^2$ above plane $z=3$ , and $\text{n}$ points in the positive $z$ -direction on $S$ .