Suppose $\vec{F}(x,y,z) = 2 \boldsymbol{i} + 4 y \boldsymbol{j} + 2 z^2 \boldsymbol{k}.$

(a) Find the flux of $\vec{F}$ through the cube in the first octant with edge length $c$, one corner at the origin, and edges along the axes. The cube is oriented outward.

Flux through the front face (the plane $x = c$) is
Flux through the back face (the plane $x = 0$) is
Flux through the right face (the plane $y = c$) is
Flux through the left face (the plane $y = 0$) is
Flux through the top face (the plane $z = c$) is
Flux through the bottom face (the plane $z = 0$) is

Total flux through the cube =

(b) Use the geometric definition of divergence and the total flux to find $\mathrm{div}(\vec{F})$ at the origin.
$\mathrm{div}(\vec{F}(0,0,0)) = \displaystyle \lim_{c \to 0}$ =