Use the Divergence Theorem to calculate the surface integral $\iint _ { s } \mathbf { F } \cdot d \mathbf { S }$; that is, calculate the flux of $\mathbf { F }$ across $S$.
$$\mathbf { F } ( x , y , z ) = x y e ^ { z } \mathbf { i } + x y ^ { 2 } z ^ { 3 } \mathbf { j } - y e ^ { z } \mathbf { k }$$
$S$ is the surface of the box bounded by the coordinate planes and the planes $x = 5 , y = 2$ and $z = 1$.
Select the correct answer.
A) $\frac { 11 } { 3 }$
B) $\frac { 25 } { 2 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 2 } { 25 }$
E) $\frac { 7 } { 3 }$

Question

Quizplus · Accepted Answer

Using the Divergence Theorem, we have
$$\iint _ { s } \mathbf { F } \cdot d \mathbf { S } = \iiint _ { V } \operatorname { div } \mathbf { F } d V$$
where $V$ is the region enclosed by the surface $S$.
We can calculate the divergence of $\mathbf{F}$ as follows:
$$\operatorname { div } \mathbf { F } = \frac { \partial } { \partial x } ( x y e ^ { z } ) + \frac { \partial } { \partial y } ( x y ^ { 2 } z ^ { 3 } ) + \frac { \partial } { \partial z } ( - y e ^ { z } ) = y e ^ { z } + 3 x y z ^ { 3 } - y e ^ { z } = 3 x y z ^ { 3 }$$
The volume $V$ can be divided into six congruent rectangular parallelepipeds or boxes, and the value of $div \mathbf{F}$ is constant on each of them. Each of these boxes has volume $V_{0} = (5/3)\cdot (2/3)\cdot (1/3) = 10/81$. Therefore, we have
$$\iiint _ { V } \operatorname { div } \mathbf { F } d V = 6 V_{0}  \cdot \operatorname { div } \mathbf { F } = 6 \cdot \frac { 10 } { 81 } \cdot \frac { 25 } { 2 } = \frac { 125 } { 27 }$$
Thus, the surface integral is
$$\iint _ { s } \mathbf { F } \cdot d \mathbf { S } = \iiint _ { V } \operatorname { div } \mathbf { F } d V = \frac { 125 } { 27 }$$
Therefore, the correct answer is B.

Use the Divergence Theorem to Calculate the Surface Integral ∬sF⋅dS\iint _ { s } \mathbf { F } \cdot d \mathbf { S }∬s​F⋅dS

Use the Divergence Theorem to Calculate the Surface Integral $\iint _ { s } \mathbf { F } \cdot d \mathbf { S }$