Chapter 6.7, Problem 361E

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

361. [T] Use a CAS and Stokes’ theorem to evaluate ${\displaystyle {\oint }_{c}F\cdot dS}$ , if $F\left(x,y,z\right)=\left(3z-\sin x\right)i+\left(x^2+e^y\right)j+\left(y^3-\cos z\right)k$ , where C is the curve given by $x=\cos t$ , $y=\sin t$ , $z=1$ ; $0\le t\le 2\pi$ .