Consider the vector field, $F(x,y)=(6+4x{y}^2)i+4x^{2}yj,$ and the curve $C$ that is the arc of the hyperbola $y=\frac{1}{x}$ from $(1,1)$ to $(2,\frac{1}{2}).$

$($a$)(12$ pts$.)$ Find a potential function $f$ such that $F=$gradf.

$($b$)(2$ pts$.)$ Use part $($a$)$ to evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r),$ i$.$e$.$ use the fundamental theorem for line integrals.