Apply Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) by finding the flux of \(\operatorname{curl}(\mathbf{F})\) across an appropriate surface.

\(\mathbf{F}=\langle x z, x y, y zangle, \quad C\) is the rectangle with vertices \((0,0,0),(0,0,2),(3,0,2)\), and \((3,0,0)\), oriented counterclockwise as viewed from the positive \(y\)-axis.

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THEOREM 1 Stokes' Theorem Let S be a surface as described earlier, and let F be a vector field whose components have continuous partial derivatives on an open region containing S. fos r=ff curl(F F.dr = curl(F). dS The integral on the left is defined relative to the boundary orientation of 8s. If S is a closed surface, then Js curl(F). dS=0 1