Question

Question partpointssubmissions usedverify that the divergence theorem is true for the vector field f on the region

Answer 1

$\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(3x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(5xz)}{\partial z}=3+x+5x=3+6x$

By the divergence theorem, the flux across the boundary of the given region $\mathcal E$ is

$\displaystyle\iint_{\partial\mathcal E}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal E}\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$
$=\displaystyle\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(3+6x)\,\mathrm dx\,\mathrm dy\,\mathrm dz=72$