Sections $16\mathrm{.}5$ and $17\mathrm{.}3$

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by $F=($:$x-y,z+y+10,z^2$:$)$ and the surface of the net is described by the equation $S_1$:$y=\sqrt[\phantom{2}]{1-x^2-z^2},y0,$ and oriented in the positive $y-$direction.

$($Use symbolic notation and fractions where needed. Hint contains a video.$)$

Correct Answer

We close the surface of net with a disk $S_2$ described by the $y=0$ and $x^2+z^{2}1$ oriented in the negative $y-$direction. Let $S$ be the closed surface of the union of $S_1$ and $S_2$ oriented outward and $W$ be the solid entrapped in this closed surface.

We close the surface of net with a disk $S_2$ described by the $y=0$ and $x^2+z^{2}1$ oriented in the negative $y-$direction. Let $S$ be the closed surface of the union of $S_1$ and $S_2$ oriented outward and $W$ be the solid entrapped in this closed surface.

Div$(F)dV=$

Use symmetry.$)$

Correct Answer

According to Divergence theorem, the outward flux of any closed surface is equal to the triple integral of the divergence of the vector field on the solid entrapped in the surface.

Div$(F)dV={}_{S}F*dS$

According to Divergence theorem, the outward flux of any closed surface is equal to the triple integral of the divergence of the vector field on the solid entrapped in the surface.

${}_w$Div$(F)dV={}_{S}F*dS$

${}_{S}F*dS=\frac{4}{3}$

$$

Correct Answer

$B$

${}_{S_2}F*dS=$

$($Parametrize the disk and compute. Mind the orientation.$)$

incorrect Answer

${}_{S_2}F*dS=-\frac{4}{3}$

$($Parametrize the disk and compute. Mind the orientation.$)$

Incorrect Answer

${}_{S}F*dS={}_{S_1}F*dS+{}_{S_2}F*dS.So$

${}_{S_1}F*$

Incorrect Answer