Preview Activity 13.13.1. Locating sources of a poisonous gas.
We will use three different surfaces to examine the flux through closed surfaces. Let \(S_{\text{top}}\) be the top half of the unit sphere centered at the origin (graphed in magenta in the figures below). Let \(S_{\text{bottom}}\) be the bottom half of the unit sphere centered at the origin (graphed in yellow). Finally, let \(S_{\text{mid}}\) be the unit disk centered at the origin in the \(xy\)-plane (graphed in blue). With these definitions, \(S_{\text{top}}\) and \(S_{\text{bottom}}\) will make a closed surface given by the unit sphere. The surfaces \(S_{\text{top}}\) and \(S_{\text{mid}}\) will enclose the top half of the unit ball, while \(S_{\text{bottom}}\) and \(S_{\text{mid}}\) will enclose the bottom half of the unit ball.
In this problem we will be using the surfaces defined above and the flux integrals of a poisonous gas through these surfaces to try to determine whether different regions of space are emitting or absorbing the poisonous gas.
(a)
In this part, we will consider the unit ball shown in Figure 13.13.4, with boundary and normal vectors as shown in the plot. If the flux integral of a poisonous gas through \(S_{\text{top}}\) is \(15\) and the flux integral of the poison gas through \(S_{\text{bottom}}\) is \(-3\text{,}\) is the interior of the sphere emitting or absorbing poisonous gas? Explain your reasoning.
(b)
In this part, we will consider the top half of the unit ball shown in Figure 13.13.5, with boundary and normal vectors as shown in the plot. If the flux integral of a poisonous gas through \(S_{\text{top}}\) is \(15\) and the flux integral of the poison gas through \(S_{\text{mid}}\) is \(-20\text{,}\) is the top half of the unit ball emitting or absorbing poisonous gas? Explain your reasoning.
(c)
In this part, we will consider the bottom half of the unit ball shown in Figure 13.13.6, with boundary and normal vectors as shown in the plot. Based on the information given in the previous two parts, what will the flux integrals of the poison gas be for \(S_{\text{bottom}}\) and \(S_{\text{mid}}\) be in this case? Be sure to pay attention to the orientation of what we consider positive flow. Explain your reasoning.
(d)
Using your answer from the previous part, is the bottom half of the unit ball emitting or absorbing poisonous gas? Explain your reasoning.
