Compute the flux \(\oint \mathbf{F} \cdot \mathbf{n} d\) s of \(\mathbf{F}\) across the curve \(C\) for the given vector field and curve using the vector form of Green's Theorem.

\(\mathbf{F}(x, y)=\left\langle x y^{2}+2 x, x^{2} y-2 yightangle\) across the simple closed curve that is the boundary of the half-disk given by \(x^{2}+y^{2} \leq 3, y \geq 0\)

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THEOREM 1 Green's Theorem Let D be a domain whose boundary 3D is a simple closed curve, oriented counterclockwise. If F and F2 have continuous partial deriva- tives in an open region containing D, then F aF = - SS (F-F) dA y & Fi dx + F2 dy: 2