Find the flux of the vector field F across the surface S in the indicated direction.
-$\mathbf { F } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = 12 \mathrm { x } + 12 \mathrm { y } \mathbf { j } + 12 \mathrm { z } \mathbf { k } , \mathrm { S }$ is the surface of the sphere $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 1$ in the first octant, direction away from the origin
A) $4 \pi$
B) $6 \pi$
C) $3 \pi$
D) 0

Question

Quizplus · Accepted Answer

The flux of the vector field $\mathbf{F} = 12x\mathbf{i} + 12y\mathbf{j} + 12z\mathbf{k}$ across the surface of the sphere $x^2 + y^2 + z^2 = 1$ in the first octant can be calculated using the divergence theorem. The divergence of $\mathbf{F}$ is $\nabla \cdot \mathbf{F} = 12 + 12 + 12 = 36$. The volume of the sphere in the first octant is $\frac{1}{8} \cdot \frac{4}{3}\pi r^3 = \frac{1}{8} \cdot \frac{4}{3}\pi (1)^3 = \frac{\pi}{6}$. The flux is then the product of the divergence and the volume, which is $36 \cdot \frac{\pi}{6} = 6\pi$.

Find the Flux of the Vector Field F Across the Surface