Preview Activity 12.7.1.
In this Preview Activity, we will be reviewing the meaning of each of the cylindrical and spherical coordinates by looking at a description of a common surface in either cylindrical or spherical coordinates. For each task, you should draw a plot of the surface described by hand and write a few sentences describing how your plot relates to the cylindrical/spherical coordinates.
(a)
What familiar surface is described by the points in cylindrical coordinates with \(r=2\text{,}\) \(0 \leq \theta \leq 2\pi\text{,}\) and \(0 \leq z \leq 2\text{?}\) How does this example suggest that we call these coordinates cylindrical coordinates? How does the surface change if we restrict \(\theta\) to \(0 \leq \theta \leq \pi\text{?}\)
(b)
What familiar surface is described by the points in cylindrical coordinates with \(\theta=2\text{,}\) \(0 \leq r \leq 2\text{,}\) and \(0 \leq z \leq 2\text{?}\)
(c)
Plot the graph of the cylindrical equation \(z=r\text{,}\) where \(0 \leq \theta \leq 2\pi\) and \(0 \leq r \leq 2\text{.}\) What familiar surface is this a plot of?
(d)
What familiar surface is described by the points in spherical coordinates with \(\rho = 1\text{,}\) \(0 \leq \theta \leq 2\pi\text{,}\) and \(0 \leq \phi \leq \pi\text{?}\) How does this particular example demonstrate the reason for the name of this coordinate system? What if we restrict \(\phi\) to \(0 \leq \phi \leq \frac{\pi}{2}\text{?}\)
(e)
What familiar surface is described by the points in spherical coordinates with \(\phi = \frac{\pi}{3}\text{,}\) \(0 \leq \rho \leq 1\text{,}\) and \(0 \leq \theta \leq 2\pi\text{?}\)
