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\\abla\cdot\mathbf f(x,y,z)\,\mathrm dV`

`=\displaystyle\iiint_V\left((\mathrm d\mathbf f)/(\mathrm dx)+(\mathrm d\mathbf f)/(\mathrm dy)+(\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`