Chapter 6.7, Problem 352E

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

352. Let $F\left(x,y,z\right)=xyi+\left(e^{z^2}+y\right)j+\left(x+y\right)k$ and let S be the graph of function $y=\frac{x^2}{9}+\frac{z^2}{9}-1$ with $z\le 0$ oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral ${\displaystyle {\iint }_{s}curlF\cdot dS}$ .