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:

$\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(x^2)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial z}{\partial z}=2x+x+1=3x+1$

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

$\displaystyle\iint_{\partial\mathcal E}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal E}\nabla\cdot\mathbf f(x,y,z)\,\mathrm dV$

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

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