Take the vector field $F$ to be: $$F(x,\,y,\,z) = \left( f_1(x,y,z), \ f_2(x,y,z), \ f_3(x,y,z)\right).$$
Then, $$\text{div} \ F = \left( \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z} \right).$$ If divergence is the flow through an infinitesimal cube, then couldn't the $x$-component of $F$ be changing, but not contributing to the overall flow through the cube with this definition of flux?
Here's a case that confuses me: $$F(x,\,y,\,z) = (z,\ x,\ y).$$ So, $$\text{div} \ F = 0 + 0 + 0 \text{,} $$ but the vector field is nonzero everywhere besides the origin.
I understand that the $x$-component is not changing as $x$ is varied. But I'm imagining that there is flow through a tiny axis-aligned cube in the $x$-direction (since $z$ is nonzero).
Any tips on where I'm going wrong?