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 + 2xzk, e is the cube bounded by the planes x = 0, x = 3, y = 0, y = 3, z = 0, and z = 3.

Answer 1

We have

`\mathrm{div}(\mathbf f)=\\abla\cdot\mathbf f=(\partial(2x))/(\partial x)+(\partial(xy))/(\partial y)+(\partial(2xz))/(\partial z)=2+3x`

By the divergence theorem, the flux of
`\mathbf f` over the region
`\mathcal S` that bounds the region
`\mathcal E` is

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

`=\displaystyle\int_(x=0)^(x=3)\int_(y=0)^(y=3)\int_(z=0)^(z=3)(3+2x)\,\mathrm dz\,\mathrm dy\,\mathrm dx`

`=\displaystyle3^2\int_(x=0)^(x=3)(3+2x)\,\mathrm dx=162`