Chapter 6.7, Problem 367E

For the following application exercises, the goal is to evaluate $A={\displaystyle {\iint }_s(\nabla \times \text{F})}\cdot \text{n}dS$ , where $\text{F}=\langle xz,-xz,xy\rangle$ and $S$ is the upper half of ellipsoid $x^2+y^2+8z^2=1$ , where $z\ge 0$ .

367. Take paraboloid $z=x^2+y^{{}^2}$ , and slice it with plane $y=0$ . Let $S$ he the surface that remains for $y\ge 0$ , including the planar surface in the $xz$ -plane. Let $C$ be the semicircle and line segment that bounded the cap of $S$ in plane $z=4$ with counterclockwise orientation. Let $\text{F}=\langle 2z+y,2x+z,2y+x\rangle$ . Evaluate ${\displaystyle {\iint }_s(\nabla \times \text{F})\cdot \text{n}dS}$ .