Hello,
plz try to do these questions.
Let me try...
First we need to find,
$\displaystyle \mbox{div} \b{F}=2x-1$
And $\displaystyle S$ is a box.
Thus,
$\displaystyle \int_0^3 \int_0^1 \int_0^2 2x-1 dz\, dy\, dx$
Thus,
$\displaystyle \int_0^3 \int_0^1 4x-2 dy\, dx$
Thus,
$\displaystyle \int_0^3 4x-2 dx$
Thus,
$\displaystyle 2(3)^2=18$
Question 2]
You are given,
$\displaystyle \b{V}=<e^x\cos y,-e^x\sin y,0>$
To determine if it is conservative for well-behaved vector fields the necessary and sufficient conditions is to check if $\displaystyle \mbox{curl}\b{F}=\b{0}$
Thus, the we have,
$\displaystyle \left| \begin{array}{ccc}\bold{i}&\bold{j}&\bold{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ e^x\cos y&-e^x\sin y&0 \end{array} \right|=<0,0,-e^x\sin y+e^x\sin y>=\b{0}$
Thus if is.
It turns out that there is an easier way in the special case for two dimension vector fields.
$\displaystyle \b{F}(x,y)=u(x,y)\b{i}+v(x,y)\b{j}$
Necessary and suffient for well-behaves fields,
$\displaystyle u_y(x,y)=v_x(x,y)$
Called the "cross-partials test"
(If you taken Differencial Equations you will note it is the same test to determine if a differencial equation is exact or not).
In this case,
$\displaystyle u_y=-e^x\sin y$
$\displaystyle v_x=-e^x\sin y$
Thus, it is conservative.