Activity 13.13.2.
In this activity, we will look at calculating both sides of a non-trivial example of the Divergence Theorem. We will look at the region inside the right circular cylinder shown in Figure 13.13.10. Let \(S\) be the closed surface formed by combining \(S_{\text{top}}\) (in yellow), \(S_{\text{sides}}\) (in blue), and \(S_{\text{bottom}}\) (in magenta). The solid volume \(Q\) is the volume bounded by \(S\text{.}\) The region shown has radius \(2\text{,}\) and its height is \(1\text{.}\) The vector field we consider in this activity is given by
\begin{equation*}
\vF=\langle xy-2z,y^2-yz,3x+z^2\rangle\text{.}
\end{equation*}
(a)
Figure 13.13.11 shows the vector field \(\vF\) on a region around \(S\text{.}\) Without doing any computations, write a couple of sentences to explain if you think the flux of \(\vF\) through \(S\) will be positive, negative, or zero.
(b)
Parametrize each of the surfaces \(S_{\text{top}}\text{,}\) \(S_{\text{sides}}\text{,}\) and \(S_{\text{bottom}}\text{.}\) Be sure to give bounds for each of your parametrization.
(c)
Give inequalities in terms of cylindrical coordinates to describe \(Q\text{.}\)
(d)
Set up and evaluate double integrals to calculate the flux of \(\vF\) through \(S_{\text{top}}\text{,}\) \(S_{\text{sides}}\text{,}\) and \(S_{\text{bottom}}\text{.}\)
(e)
What is the net flux through the closed surface? Be sure to state if the net flux is in or out.
(f)
Compute the divergence of \(\vF\) and use this to explain whether you think \(\iiint_Q \divg(\vF)\, dV\) will be positive, negative, or zero.
(g)
Set up and compute the triple integral for \(\iiint_Q \divg(\vF)\, dV\text{.}\)
(h)
(i)
Was the left-hand side or right-hand side of the equation in the Divergence Theorem more tedious to calculate in this example? Do you think this will be true for most other cases where the Divergence Theorem applies?
