Question

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

e. give the flux. f(x, y, z) = 2xi + xyj + 3xzk, e is the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, and z = 2.

Answer 1

`\\abla\cdot\mathbf f(x,y,z)=(\partial(2x))/(\partial x)+(\partial(xy))/(\partial y)+(\partial(3xz))/(\partial z)=2+x+3x=4x+2`

The flux across
`\partial\mathcal E` (boundary of
`\mathcal E`) is


`\displaystyle\iint_(\partial\mathcal E)\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_(\mathcal E)(4x+2)\,\mathrm dV`

`=\displaystyle\int_(z=0)^(z=2)\int_(y=0)^(y=2)\int_(x=0)^(x=2)(4x+2)\,\mathrm dx\,\mathrm dy\,\mathrm dz=48`