EXAMPLE $1$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}F*dr,$ where $F(x,y,z)=-y^{2}i3xj{z}^{2}k$ and $C$ is the curve of the intersection of the plane $yz=4$ and the cylinder $x^2{y}^2=16.($Orient $C$ to be counterclockwise when viewed from above.$)$

SOLUTION The curve $C($an ellipse$)$ is shown in the figure. Although ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}F*dr$ could be evaluated directly, it's easier to use Stokes' Theorem. We first compute

curlF$=|[i,j,k],[\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}},\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}},\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}],[,3x,z^2]|=(\frac{x}{x})k$

Although there are many surfaces with boundary $C,$ the most convenient choice is the elliptical region $S$ in the plane $yz=4$ that is bounded by $C.$ If we orient $S$ upwards, then $C$ has the induced positive orientation. The projection $D$ of $S$ on the $xy-$plane is the disk $x^2{y}^{2}16$ and so with $z=g(x,y)=4-y,$ we have