Question

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

Answer 1

By the divergence theorem,

$\displaystyle\iint_{\partial\mathcal E}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal E}\nabla\cdot\mathbf f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz$

where $\partial\mathcal E$ is the boundary of $\mathcal E$. We have

$\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(4x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(4xz)}{\partial z}=4+x+4x=5x+4$

so the flux is

$\displaystyle\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(5x+4)\,\mathrm dx\,\mathrm dy\,\mathrm dz=72$