Chapter 6.7, Problem 350E

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

350. Use Stokes’ theorem and let C be the boundary of surface $z=x^2+y^2$ with $0\le x\le 2$ and $0\le y\le 1$ , oriented with upward facing normal. Define $F\left(x,y,z\right)=\left[\sin \left(x^3\right)+xz\right]i+\left(x-yz\right)j+\cos \left(z^4\right)k$ and evaluate ${\displaystyle {\int }_{c}F\cdot dS}$ .