Calculate the circulation, $\int_C\vec F\cdot d\vec r$, in two ways, directly and using Stokes' Theorem. The vector field $\vec F = 3 y\vec i - 3 x\vec j$ and $C$ is the boundary of $S$, the part of the surface $z = 9 -x^2-y^2$ above the $xy$-plane, oriented upward.

Note that $C$ is a circle in the $xy$-plane. Find a $\vec r(t)$ that parameterizes this curve.
$\vec r(t) =$ ,
with $\le t\le$
(Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.)
With this parameterization, the circulation integral is
$\int_C\vec F\cdot d\vec r = \int_a^b$ $dt$, where $a$ and $b$ are the endpoints you gave above.
Evaluate your integral to find the circulation: $\int_C\vec F\cdot d\vec r =$

Using Stokes' Theorem, we equate $\int_C\vec F\cdot d\vec r = \int_S\mbox{curl }\vec F\cdot d\vec A$. Find $\mbox{curl }\vec F =$ .
Noting that the surface is given by $z = 9 - x^2 - y^2$, find
$d\vec A =$ $dy\,dx$.
With $R$ giving the region in the $xy$-plane enclosed by the surface, this gives
$\int_S \mbox{curl }\vec F\cdot d\vec A = \int_{R}$ $dy\,dx$.
Evaluate this integral to find the circulation:
$\int_C \vec F\cdot d\vec r = \int_{S} \mbox{curl }\vec F\cdot d\vec A =$ .