# Let F(x,y)=<<xy^2+8x, x^2y-8y>>. Compute the flux oint F * nds of F across a simple closed curve that is the boundary of the half-disk given by x^2+y^2<=7, y>= 0 using the vector form Green's Theorem. oint F * nds = ?

Let $F\left(x,y\right)=⟨x{y}^{2}+8x,{x}^{2}y-8y⟩$. Compute the flux $\oint F\cdot nds$ of F across a simple closed curve that is the boundary of the half-disk given by ${x}^{2}+{y}^{2}\le 7,y\ge 0$ using the vector form Greens
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Step 1
Consider the equation
Step 2
$F\left(x,y\right)=\left(x{y}^{2}+8x,{x}^{2}y-8y\right)$
$\oint F.nds==\int \int \mathrm{\nabla }×DdA$
$\oint F.nds=\int {\int }_{S}\left(8xdy+y\right)=3\pi$
$\int \left(8x,y\right).\left(x,y\right)ds=\int 8xdy+y\left(-dx\right)$
$x=\mathrm{cos}0y=\mathrm{sin}0ds=rd0⇒ds=d0$ because the radius is 1
$dx=-yd0dy=\mathrm{cos}0d0$
$\oint \left(-y,8x\right).dx$
$\int {\int }_{S}\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}=3$
(the area) $=3\left(\pi {\left(1\right)}^{2}\right)=3\pi$