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.

\(\mathbf{F}=\langle y+2 x, 2 x+5 z, 7 y+8 xangle, \quad C\) is the circle with radius 5 , center at \((2,0,0)\), in the plane \(x=2\), and oriented counterclockwise as viewed from the origin \((0,0,0)\).

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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. fors 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 11 curl(F). dS = 0 1