Activity 12.8.2.
In this activity, we will be looking at what happens to the region \(D\text{,}\) as shown in Figure 12.8.5, under the change of coordinates given by
\begin{equation*}
x=\sqrt{st} \quad \text{ and } \quad y =\sqrt{\frac{s}{t}}
\end{equation*}
(a)
Find the coordinates of the four corner points of the region \(D\text{.}\)
(b)
Given the form of the change of variable equations above, it is not obvious how to change \((x,y)\) points into \((s,t)\) points. Show that the following equations are satisfied by our change of variable equations:
\begin{equation*}
yx=s \quad \text{and} \quad \frac{y}{x}=\frac{1}{t}
\end{equation*}
Use these new relationships between our coordinates to convert each of these four corner points from \(xy\)-coordinates to \(st\)-coordinates and draw these points on Figure 12.8.6.
(c)
We hope that the curved boundaries of the region \(D\) get transformed to horizontal and vertical lines in the \(st\)-plane, as is suggested by the result of the previous task. Let’s look at how each of the boundary curves gets transformed by our change of variables formula.
Use the change of variables equations to write \(y=2x\) in terms of \(s\) and \(t\) coordinates and solve for either \(s\) or \(t\) (as would make sense). Plot the resulting line on Figure 12.8.6.
(d)
Write each of the other boundary curves of \(D\) (\(y=\frac{2}{x}, y=\frac{1}{2x}, y=\frac{x}{2}\)) in terms of \(s\) and \(t\) coordinates, solve for either \(s\) or \(t\) (as would make sense), and plot the resulting lines on Figure 12.8.6.
(e)
We have now confirmed that our irregular region \(D\) gets mapped to a rectangle in the \(st\)-plane by the change of variables formula given. Some areas of the region \(D\) needed to be stretched more than others in order to straighten out our sides into the nice rectangle you have drawn on Figure 12.8.6. Write a few sentences to describe what part of \(D\) you think is stretched the most in the transformation from \(xy\)-coordinates to \(st\)-coordinates.
