Calculate the circulation, $\int_C\vec F\cdot d\vec r$, in two ways, directly and using Stokes' Theorem. The vector field $\vec F = (2x-4y+2z)(\vec i+\vec j)$ and $C$ is the triangle with vertices $(0,0,0)$, $(7,0,0)$, $(7,8,0)$, 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 $0\le t\le1$.
On $C_1$ from $(0,0,0)$ to $(7,0,0)$, $\vec r(t) =$
On $C_2$ from $(7,0,0)$ to $(7,8,0)$, $\vec r(t) =$
On $C_3$ from $(7,8,0)$ to $(0,0,0)$, $\vec r(t) =$

So that, integrating, we have $\int_{C_1}\,\vec F\cdot d\vec r =$
$\int_{C_2}\,\vec F\cdot d\vec r =$
$\int_{C_3}\,\vec F\cdot d\vec r =$
and so $\int_C\vec F\cdot d\vec r =$ .

Using Stokes' Theorem, we have
$\mbox{curl }\vec F =$
So that the surface integral on $S$, the triangular region on the plane enclosed by the indicated triangle, is
$\int_S\mbox{curl }\vec F\cdot d\vec A = \int_a^b\int_c^d$ $dy\,dx$,
where $a =$ , $b =$ , $c =$ , and $d =$ .
Integrating, we get $\int_C\vec F\cdot d\vec r = \int_S\mbox{curl }\vec F\cdot d\vec A =$