Let vec$(F)=(x^3-3x{y}^2)vec(i)+(3x^{2}y-y^3)vec(j)$ be a force field.

a$.$ Find the work done by vec$(F)$ moving an object from the point $A(1,0)$ to the point $B(0,1)$ along the unit circle $x^2+y^2=1$ in the $xy-$plane.

b$.$ Let $D$ be the domain in the plane given by:

$D=\{(x,y)in{R}^2,x^2+y^{2}1,x0,y0\}$

$($i$)$ Evaluate the double integral ${}_{(D)}xydxdy.$

$($ii$)$ Deduce the counterclockwise circulation of vec$(F)$ around the boundary of $D.$

c$.$ Deduce from questions $1.$ and $2.$ the value of

${\displaystyle \underset{[OA]}{\overset{\phantom{}}{}}}(x^3-3x{y}^2)dx+(3x^{2}y-y^3)dy+{\displaystyle \underset{[BO]}{\overset{\phantom{}}{}}}(x^3-3x{y}^2)dx+(3x^{2}y-y^3)dy$

where $OA$ is the segment joining the origin to A and $BO$ is the segment joining B to the origin.