Question

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

Answer 1

$\mathbf f(x,y,z)=3x\,\mathbf i+xy\,\mathbf j+4xz\,\mathbf k$
$\implies\nabla\cdot\mathbf f(x,y,z)=3+x+4x=3+5x$

$\displaystyle\iint_{\partial E}\mathbf f(x,y,z)\,\mathrm dS=\iiint_E\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$
$=\displaystyle\int_{z=0}^{z=3}\int_{y=0}^{y=3}\int_{x=0}^{x=3}(3+5x)\,\mathrm dx\,\mathrm dy\,\mathrm dz$
$=\displaystyle9\int_0^3(3+5x)\,\mathrm dx$
$=\dfrac{567}2$