Question $6$

Consider a vector field vec$(W)$ representing a linear flow where vec$(W)$ points in the same direction and has the same length

at every point in space. Find a vector field vec$(V)$ such that vec$(W)$ equals the curl of vec$(V)($you may assume that your $x-$axis

and vec$(W)$ point in the same direction$).$ Draw pictures representing vec$(V)$ and vec$(W).$ Also draw two closed curves $C_1$ and $C_2$

such that $o{\displaystyle \underset{C_1}{\overset{\phantom{}}{}}}$dvec$(r)*vec(V)=0$ and $o{\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}$dvec$(r)*vec(V)>0.$ Now draw two surfaces $$ and ${}^{\text{'}}$ such that

${\displaystyle \underset{}{\overset{\phantom{}}{}}}$dAhat$(N)*vec(W)={\displaystyle \underset{{}^{\text{'}}}{\overset{\phantom{}}{}}}$dAhat$(N)*vec(W)=o{\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}$dvec$(r)*vec(V)$

Finally draw a surface $S$ for which ${\displaystyle \underset{S}{\overset{\phantom{}}{}}}$dAhat$(N)*vec(W)=0.$ Indicate on every surface in your pictures the direction of the

unit normal vector hat$(N),$ and on each closed curve the direction of integration.