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.

\(\mathbf{F}=\langle y, z, xangle, \quad C\) is the triangle with vertices \((0,0,0),(3,0,0)\), and \((0,3,3)\), oriented counterclockwise as viewed from above.

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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. ko F dr= = curl(F). ds The integral on the left is defined relative to the boundary orientation of S. If S is a closed surface, then Is curl(F) dS=0 1