Calculate the circulation, $\_$C vec$($F$)*$dvec$($r$),$ in two ways, directly and using Stokes' Theorem. The vector field vec$($F$)=(2$x$-7$y$+7$z$)($vec$($i$)+$vec$($j$))$ and C is the triangle with vertices $(0,0,0),(3,0,0),(3,6,0),$ traversed in that order. Calculating directly, we break C into three paths. For each, give a parameterization vec$($r$)($t$)$ that traverses the path from start to end for $0<=$t$<=1.$ On C$\_(1)$ from $(0,0,0)$ to $(3,0,0),$vec$($r$)($t$)=$ On C$\_(2)$ from $(3,0,0)$ to $(3,6,0),$vec$($r$)($t$)=$ On C$\_(3)$ from $(3,6,0)$ to $(0,0,0),$vec$($r$)($t$)=$ So that, integrating, we have $\_($C$\_(1))$ vec$($F$)*$dvec$($r$)=\backslash$int$\_($C$\_(2))$ vec$($F$)*$dvec$($r$)=\backslash$int$\_($C$\_(3))$ vec$($F$)*$dvec$($r$)=$ and so $\backslash$int$\_$C vec$($F$)*$dvec$($r$)=$ Using Stokes' Theorem, we have curlvec$($F$)=$ So that the surface integral on S $,$ the triangular region on the plane enclosed by the indicated triangle, is $\_$S curlvec$($F$)*$dvec$($A$)=\_$a$^$b $\_$c$^$d $,$dydx where a$=,$b$=,$c$=,,$ and d$=$ Integrating, we get $\_$C vec$($F$)*$dvec$($r$)=\_$S curlvec$($F$)*$vec$($d$($vec$($A$)))=$