Question

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

Answer 1

By the divergence theorem, the flux of $\mathbf f(x,y,z)=4x\,\mathbf i+xy\,\mathbf j+5xz\,\mathbf k$ across the boundary $\partial E$ of the cube $E$ is equal to the integral of $\nabla\cdot\mathbf f$ over $E$:

$\displaystyle\iint_{\partial E}\mathbf f\cdot\mathrm d\mathbf S=\iiint_E\nabla\cdot\mathbf f\,\mathrm dV$

We have divergence

$\nabla\cdot\mathbf f=\dfrac{\partial(4x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(5xz)}{\partial z}=4+6x$

and so the flux is

$\displaystyle\iiint_E(4+6x)\,\mathrm dV=\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(4+6x)\,\mathrm dx\,\mathrm dy\,\mathrm dz$

$=4(4x+3x^2)\bigg|_{x=0}^{x=2}=80$