Question

Use the divergence theorem to calculate the surface integral ??s f · ds; that is, calculate the flux of f across s. f(x,y,z = exsin(y i + excos(y j + yz2 k s is is the surface of the box bounded by the planes x = 0, x = 4, y = 0, y = 3, and z = 0, z = 4.

Answer 1

$\mathbf f(x,y,z)=e^x\sin y\,\mathbf i+e^x\cos y\,\mathbf j+yz^2\,\mathbf k$

By the divergence theorem, the surface integral can be evaluated by computing the triple integral

$\displaystyle\iint_Sf(x,y,z)\,\mathrm dS=\iiint_V\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$
$=\displaystyle\iiint_V\left(\dfrac{\mathrm d\mathbf f}{\mathrm dx}+\dfrac{\mathrm d\mathbf f}{\mathrm dy}+\dfrac{\mathrm d\mathbf f}{\mathrm dz}\right)\,\mathrm dV$
$=\displaystyle\int_{x=0}^{x=4}\int_{y=0}^{y=3}\int_{z=0}^{z=4}2yz\,\mathrm dz\,\mathrm dy\,\mathrm dx$
$=72$