Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

A)64 $\pi$
B)72 $\pi$
C)84 $\pi$
D)96 $\pi$
E) $108 \pi$

Question

Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

A)64 $\pi$
B)72 $\pi$
C)84 $\pi$
D)96 $\pi$
E) $108 \pi$

Quizplus · Accepted Answer

The divergence of $\mathbf{F}$ is $\nabla \cdot \mathbf{F} = 3$. The volume of the sphere $x^2 + y^2 + z^2 = 9$ is $\frac{4}{3} \pi (3)^3 = 36\pi$. By the Divergence Theorem, $\iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V (\nabla \cdot \mathbf{F}) \, dV = 3 \times 36\pi = 108\pi$.

Let F(x,y,z)=3zk\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }F(x,y,z)=3zk

Let $\mathbf { F } ( x , y , z ) = 3 z \mathbf { k }$