Consider the vector field $\mathbf{F} \left( x, y, z \right) = \left( 3 z+ y\right) \mathbf{i} + \left( 2 z+ x \right) \mathbf{j} + \left( 2 y+ 3 x \right) \mathbf{k}$.

a) Find a function $f$ such that $\mathbf{F} = \nabla f$ and $f(0,0,0) = 0$.
$f(x,y,z) =$

b) Suppose C is any curve from $\left( 0, 0, 0 \right)$ to $\left( 1, 1, 1 \right).$ Use part a) to compute the line integral $\int_{C} \mathbf{F} \cdot d\mathbf{r}$.