Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x , y , z ^ { 2 } \right\rangle$
A) $\frac { 12 } { 3 } \pi$

B) $\frac { 8 } { 3 } \pi$

C) $\frac { 4 } { 3 } \pi$

D) $\frac { 1 } { 3 } \pi$

Question

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x , y , z ^ { 2 } \right\rangle$
A) $\frac { 12 } { 3 } \pi$

B) $\frac { 8 } { 3 } \pi$

C) $\frac { 4 } { 3 } \pi$

D) $\frac { 1 } { 3 } \pi$

Quizplus · Accepted Answer

By the Divergence Theorem, $\iint _ { S } \vec { F } \cdot \vec { n } d S = \iiint _ { V } \nabla \cdot \vec { F } \, dV$, where $V$ is the volume enclosed by $S$. The divergence $\nabla \cdot \vec { F } = 1 + 1 + 2z = 2$. The volume of the unit sphere is $\frac{4}{3}\pi$, so the integral is $2 \times \frac{4}{3}\pi = \frac{8}{3}\pi$.

Compute ∬SF⃗⋅n⃗dS\iint _ { S } \vec { F } \cdot \vec { n } d S∬S​F⋅ndS

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$