Question

Question part points submissions used verify that the divergence theorem is true for the vector field f on the region

e. give the flux. f(x, y, z) = x2i + xyj + zk, e is the solid bounded by the paraboloid z = 4 − x2 − y2 and the xy-plane.

Answer 1

Compute the divergence:


`\\abla\cdot\mathbf f(x,y,z)=(\partial(x^2))/(\partial x)+(\partial(xy))/(\partial y)+(\partial z)/(\partial z)=2x+x+1=3x+1`

By the divergence theorem, the flux is of
`\mathbf f` across
`\partial\mathcal E` (the boundary of the 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`

We set up and compute the volume integral with respect to cylindrical coordinates.


`\displaystyle\iiint_(\mathcal E)(3x+1)\,\mathrm dV=\int_(\theta=0)^(\theta=2\pi)\int_(r=0)^(r=2)\int_(z=0)^(z=4-r^2)(3r\cos\theta+1)r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\frac{7\pi}2`