$8$ marks The following vector field $is$ conservative $in{R}^3$

For vec$(F)(x,y,z)=(3x^{2}y+z^3)hat(i)+(x^3-Ay{z}^2)hat(j)+(-6y^{2}z+Bx{z}^2+2z)hat(k).$

$(a)$ Find the values $of$ the constants A and $B.$

$(b)$ Find a potential function $such$ that vec$(F)(x,y,z)=$grad$in{R}^3$ and $(1,1,1)=5.$

$(c)$ For the curve $C$ parametrized $by{\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)vec(r)(t)=(5cost,4sin3t,t),0,$ compute the

line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r).$