Let $\vec{F}=(6 z + 2)\vec i + 8 z\vec j + (9 z + 5)\vec k$, and let the point $P = (a,b,c)$, where $a$, $b$ and $c$ are constants. In this problem we will calculate $\mbox{div }\vec F$ in two different ways, first by using the geometric definition and second by using partial derivatives.

(a) Consider a (three-dimensional) box with four of its corners at $(a,b,c)$, $(a+w,b,c)$, $(a,b+w,c)$ and $(a,b,c+w)$, where $w$ is a constant edge length. Find the flux through the box.
flux =
Thus, we have
$\mbox{div }\vec F(x,y,z) = \lim\limits_{w\to 0} \left(\right.$ / $\left.\right) =$

(b) Next, find the divergence using partial derivatives:
$\mbox{div }\vec F(x,y,z) =$