Question

Question partpointssubmissions usedverify that the divergence theorem is true for the vector field f on the region

e. give the flux.f(x, y, z) = 3xi + 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

`\\abla\cdot\mathbf f(x,y,z)=(\partial(3x))/(\partial x)+(\partial(xy))/(\partial y)+(\partial(5xz))/(\partial z)=3+x+5x=3+6x`

By the divergence theorem, the flux across the boundary of the given region
`\mathcal E` is


`\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 dV`

`=\displaystyle\int_(z=0)^(z=2)\int_(y=0)^(y=2)\int_(x=0)^(x=2)(3+6x)\,\mathrm dx\,\mathrm dy\,\mathrm dz=72`