Let S be the closed surface which is the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ with $- 2 \leq \boldsymbol { z } \leq 1$ topped by the disk in the plane $\bar { z } = 1$ , oriented outward.Then the flux of $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ through S is:
A) $9 \pi$
B) $24 \pi$
C) $27 \pi$
D) $18 \pi$
E) $36 \pi$

Question

Quizplus · Accepted Answer

The flux of $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ through a closed surface S can be calculated using the divergence theorem. The divergence of $\vec{r}$ is $\nabla \cdot \vec{r} = 3$, since the partial derivative of $x$ with respect to $x$, $y$ with respect to $y$, and $z$ with respect to $z$ each contribute 1. The volume enclosed by S is the volume of the portion of the sphere from $z = -2$ to $z = 1$ plus the volume of the cylinder capping it. The volume of the sphere segment is $\frac{4}{3}\pi(2^3) - \frac{1}{3}\pi(1^3) = \frac{31}{3}\pi$, and the volume of the cylinder is $\pi(2^2)(1) = 4\pi$. Adding these gives $\frac{43}{3}\pi$. Multiplying this volume by the divergence, $3$, gives the total flux: $3 \times \frac{43}{3}\pi = 43\pi$. However, this calculation mistakenly includes the volume of the cylinder, which should not be added separately as it's part of the spherical cap calculation. The correct approach is to directly apply the divergence theorem to the volume of the sphere segment, which yields $3 \times \frac{31}{3}\pi = 31\pi$, indicating a mistake in the initial explanation. The correct flux calculation should directly follow from applying the divergence theorem to the given vector field and the specified volume, without incorrectly adding volumes. The correct answer, considering the divergence theorem and the specified limits for z, should reflect the flux through the described surface, which is not directly given by the initial incorrect calculation.

Let S Be the Closed Surface Which Is the Portion x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4

Let S Be the Closed Surface Which Is the Portion $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$