Verify that the divergence theorem is true for the vector field $F$ on the region $E.$

$F(x,y,z)=($:$z,y,x$:$)$;$,Eis$ the solid ball $x^2+y^2+z^{2}9$

First compute the divergence of $F$:divF$=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(z)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(y)+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(x)=$

$,$ so

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dV=V(E).$ This calculates to the exact value:

$S$ is a sphere of radius $3$ centered at the origin which can be parametrized by $r(,)=($:$3sin()cos(),3sin()sin(),3cos()$:$),0,02($similar to this example$).$ Then

and

${}_{S}F*dS={}_{D}F*(r_{}x{r}_{})dA$

$={\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{}{}}}(54cos()si{n}^2()cos()+(,)si{n}^3()si{n}^2())dd$

$={\displaystyle \underset{0}{\overset{2}{}}}[18si{n}^3()cos()+()-cos())si{n}^2()$