Gauss and Stokes Theorem Problem!! URGENT HELP PLEASE!!

May 2016
3
0
Spain
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 $
 
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romsek

MHF Helper
Nov 2013
6,746
3,037
California
the translation got garbled or something. I can't make any sense of part b.

You should be able to compute the divergence and rotation of some field.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Since you titled this "Gauss and Stokes theorem", what are those theorems?
What do they have to do with the "divergence" and "rotation" (I would have said "curl") that you found in part (a)?
 
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