Find the flux of $\vec { F }$ over the closed surface let $\vec{N}$ be the outward unit normal vector of the surface).
$\overrightarrow { \mathbf { F } } ( x , y , z ) = 49 x y \hat { \mathbf { i } } + z ^ { 2 } \tilde { \mathbf { j } } + y z \tilde { \mathbf { k } }$
$S :$ cube bounded by $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$
A) 200
B) 800
C) 400
D) 1,000
E) 600

Question

Find the flux of $\vec { F }$ over the closed surface let  $\vec{N}$ be the outward unit normal vector of the surface). 
$\overrightarrow { \mathbf { F } } ( x , y , z ) = 49 x y \hat { \mathbf { i } } + z ^ { 2 } \tilde { \mathbf { j } } + y z \tilde { \mathbf { k } }$
$S :$ cube bounded by $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$
A) 200
B) 800
C) 400
D) 1,000
E) 600

Quizplus · Accepted Answer

The flux of $\vec{F}$ through the surface can be calculated using the divergence theorem, which states that the flux through a closed surface is equal to the triple integral of the divergence of $\vec{F}$ over the volume enclosed by the surface. The divergence of $\vec{F}$ is $\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(49xy) + \frac{\partial}{\partial y}(z^2) + \frac{\partial}{\partial z}(yz) = 49y + 0 + y = 50y$. The volume integral of $50y$ over the cube from $x=0$ to $x=2$, $y=0$ to $y=2$, and $z=0$ to $z=2$ is $\int_{0}^{2}\int_{0}^{2}\int_{0}^{2} 50y \,dx\,dy\,dz = 50 \times 2 \times \int_{0}^{2} y \,dy \times 2 = 200 \times \left[\frac{1}{2}y^2\right]_0^2 = 200 \times 2 = 400$.

Find the Flux Of F⃗\vec { F }F Over the Closed Surface Let

Find the Flux Of $\vec { F }$ Over the Closed Surface Let