Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$ in the first octant. Then the flux of $\text { F }$ through S is

A)78?
B)84?
C)99?
D)108?
E)121?

Question

Let

\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }

and S is

x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36

in the first octant. Then the flux of

\text { F }

through S is

A)78?
B)84?
C)99?
D)108?
E)121?

Quizplus · Accepted Answer

The flux of $\mathbf{F}$ through S 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 $\mathbf{F}$ over the volume enclosed by the surface. The divergence of $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ is $\nabla \cdot \mathbf{F} = 1 + 1 + 1 = 3$. The volume of the sphere in the first octant (one-eighth of the full sphere) is $\frac{1}{8} \times \frac{4}{3}\pi r^3 = \frac{1}{8} \times \frac{4}{3}\pi (6)^3 = 36\pi$. Therefore, the flux through S is $3 \times 36\pi = 108\pi$. Since the question seems to be looking for a numerical answer without $\pi$, the correct answer is 108, assuming a misinterpretation in the question's presentation regarding the inclusion of $\pi$.

Let F(x,y,z)=xi+yj+zk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }F(x,y,z)=xi+yj+zk

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$