Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = zi + xyj + zyk; D: the solid cube cut by the coordinate planes and the planes x = 2, y = 2, and z = 2
A) 32
B) 4
C) 16
D) 2

Question

Quizplus · Accepted Answer

By the Divergence Theorem, the outward flux of $F$ across the boundary of $D$ is equal to the triple integral of the divergence of $F$ over the region $D$.

We have that $
abla \cdot F = \frac{\partial}{\partial z}z + \frac{\partial}{\partial x}(xy) + \frac{\partial}{\partial y}(z) = 1 + y$.

Thus, the outward flux is given by:

$$\iint_S F \cdot \mathbf{n} \ dS = \iiint_D (
abla \cdot F) \ dV = \int_0^2 \int_0^2 \int_0^2 (1+y) \ dx \ dy \ dz = 16.$$

Therefore, the best choice is $\boxed{	ext{(C) 16}}$.

Using the Divergence Theorem, Find the Outward Flux of F