In $(x, y, z)$ space is considered the vector field

$V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization:

$\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =r(u,v,w)=\left[ \begin{matrix} wu \\ wv \\ 2-2w \end{matrix} \right]$

Where $u\in \left[ -2,2 \right] ,\quad v\in \left[ -2,2 \right] ,\quad w\in \left[ -2,2 \right]$

a) Determine the divergence and the rotation of V.

b) Determine$\int _{ \partial \Omega }^{ }{ V\cdot n }\ d\mu$, $n$ is an outwardly directed on unit-normal vector field $\partial \Omega$.

For any number represents the intersection of points $\Omega$ and the plan has equation $y = c$, a surface termed $F_c$. The surface likely equipped with a unit normal vector field $n$ pointing away from the $z-axis$. The basket rim $\Partial F_c$ likely equipped with a unit tangent vector field $e$ pointing counter-clockwise when viewed from the $y-axis$ positive end.

c) Determine the value for $c$ which determines,

$\int _{ F_{ c } }^{ }{ V\cdot n } d\mu =\int _{ \partial F_{ c } }^{ }{ V\cdot e } d\mu $

$V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization:

$\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =r(u,v,w)=\left[ \begin{matrix} wu \\ wv \\ 2-2w \end{matrix} \right]$

Where $u\in \left[ -2,2 \right] ,\quad v\in \left[ -2,2 \right] ,\quad w\in \left[ -2,2 \right]$

a) Determine the divergence and the rotation of V.

b) Determine$\int _{ \partial \Omega }^{ }{ V\cdot n }\ d\mu$, $n$ is an outwardly directed on unit-normal vector field $\partial \Omega$.

For any number represents the intersection of points $\Omega$ and the plan has equation $y = c$, a surface termed $F_c$. The surface likely equipped with a unit normal vector field $n$ pointing away from the $z-axis$. The basket rim $\Partial F_c$ likely equipped with a unit tangent vector field $e$ pointing counter-clockwise when viewed from the $y-axis$ positive end.

c) Determine the value for $c$ which determines,

$\int _{ F_{ c } }^{ }{ V\cdot n } d\mu =\int _{ \partial F_{ c } }^{ }{ V\cdot e } d\mu $

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