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.

\(\mathbf{F}=\langle y,-2 z, 4 xangle, \quad C\) is the boundary of that portion of the plane \(x+2 y+3 z=1\) that is in the first octant of space, oriented counterclockwise as viewed from above.

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