Use a computer algebra system to compute the flux of $\mathbf { F }$ across $S . S$ is the surface of the cube cut from the first octant by the planes
$x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 }$
$$\mathbf { F } ( x , y , z ) = 5 \sin x \cos ^ { 2 } y \mathbf { i } + 5 \sin ^ { 3 } y \cos ^ { 4 } z \mathbf { j } + 5 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }$$
A) $5 \pi ^ { 2 }$
B) $95 \pi ^ { 2 }$
C) $\frac { 5 \pi ^ { 2 } } { 64 }$
D) $\frac { 95 \pi ^ { 2 } } { 64 }$
E) $\frac { 95 \pi ^ { 2 } } { 32 }$

Question

Quizplus · Accepted Answer

To compute the flux of $\mathbf{F}$ across $S$, we can use the divergence theorem, which states that the flux of $\mathbf{F}$ through $S$ is equal to the triple integral over the volume $V$ enclosed by $S$ of the divergence of $\mathbf{F}$. The divergence of $\mathbf{F}$ is given by $\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(5\sin x \cos^2 y) + \frac{\partial}{\partial y}(5\sin^3 y \cos^4 z) + \frac{\partial}{\partial z}(5\sin^5 z \cos^6 x)$. Simplifying this expression and integrating over the volume of the cube, which is bounded by $0 \leq x, y, z \leq \frac{\pi}{2}$, gives the result $\frac{95\pi^2}{64}$.

Use a Computer Algebra System to Compute the Flux Of \mathbf