Let $S$ be the lower hemisphere $x^2+y^2+z^2=25,z0,$ and let $F(x,y,z)=($:$y-z,-x,x-z$:$).$ Evaluate the line integral ${\displaystyle \underset{\text{d}\text{e}\text{l}\text{S}}{\overset{\phantom{}}{}}}F*Tds,$ where delS is the boundary of this hemisphere. According to Stokes' Theorem, your answer should be the same as in the previous exercise.

Let $C$ be the curve $x^2+y^2=2,z=1,$ and let $F(x,y,z)=($:$y,x,x+z$:$).$ Find ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*Tds$ by using Stokes' Theorem to rewrite it as a surface integral over some surface whose boundary is $C.$