a$)$ Consider the vector field $F(x,y)=(4xy{y}^2-3y)i(2x^{2}2xy-3xy)j.$

i$)$ Show that this field is conservative.

ii$)$ If it is$,$ find all functions $f$ such that $F=$gradf.

iii$)$ Find the work done by the force field $F$ on a particle moving from the point $(0,0)$ to the point $(2,48)$ along the curve $y=3x^4.$

b$)$ Evaluate the integral

$o{\displaystyle \underset{c}{\overset{\phantom{}}{}}}(sin(y)$ exp$(x^2)4xy-y^6)dx(xcos(y)2sin(y^5)2x^2)dy,$

where C is the circle $x^2{y}^2=4$ using Green's theorem.