Question

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

e. give the flux. f(x, y, z) = 4xi + xyj + 5xzk, e is the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, and z = 2.

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
`\\abla\cdot\mathbf f` over
`E`:

`\displaystyle\iint_(\partial E)\mathbf f\cdot\mathrm d\mathbf S=\iiint_E\\abla\cdot\mathbf f\,\mathrm dV`

We have divergence

`\\abla\cdot\mathbf f=(\partial(4x))/(\partial x)+(\partial(xy))/(\partial y)+(\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`