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 + 4xzk, 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,


`\displaystyle\iint_(\partial\mathcal E)\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_(\mathcal E)\\abla\cdot\mathbf f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz`

where
`\partial\mathcal E` is the boundary of
`\mathcal E`. We have


`\\abla\cdot\mathbf f(x,y,z)=(\partial(4x))/(\partial x)+(\partial(xy))/(\partial y)+(\partial(4xz))/(\partial z)=4+x+4x=5x+4`

so the flux is


`\displaystyle\int_(z=0)^(z=2)\int_(y=0)^(y=2)\int_(x=0)^(x=2)(5x+4)\,\mathrm dx\,\mathrm dy\,\mathrm dz=72`