Question

Verify that the divergence theorem is true for the vector field f on the region

Answer 1

$\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(2x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(3xz)}{\partial z}=2+x+3x=4x+2$

The flux across $\partial\mathcal E$ (boundary of $\mathcal E$) is

$\displaystyle\iint_{\partial\mathcal E}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal E}(4x+2)\,\mathrm dV$
$=\displaystyle\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(4x+2)\,\mathrm dx\,\mathrm dy\,\mathrm dz=48$