First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles \(\theta\) and \(\phi\). Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
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First, show students diagrams of cylindrical and spherical coordinates. Discuss common notation systems, especially that mathematicians and physicists use opposite notations for the angles \(\theta\) and \(\phi\). Don't forget to discuss the ranges of each of the coordinates.
It can be very helpful to have a set of coordinate axes, perhaps suspended from the ceiling somewhere in the room, to refer to as needed.
Attach a string to the \(z\)-axis at the origin so that it can move freely and demonstrate, by moving the string appropriately, how the various angles in cylindrical and spherical coordinates change.Ask students to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards. Appropriate figures are attached. Beware, this part of the activity can take longer than you expect; you can try to speed things up by limiting the coordinate equals constant sketches to \(s\)=constant and \(\phi\)=constant in cylindrical and \(\theta\)=constant in spherical.
For the cylindrical coordinate system shown below, draw three surfaces: one for constant \(s\), one for constant \(\phi\), and one for constant \(z\).
\begin{align} x&=s\cos\phi\\ y&=s\sin\phi\\ z&=z \end{align} \begin{align} 0\le &s<\infty\\ 0\le &\phi <2\pi\\ -\infty < &z <\infty \end{align}
For the spherical coordinate system shown below, draw three surfaces: one for constant \(r\), one for constant \(\theta\), and one for constant \(\phi\).
\begin{align} x&=r\, \sin\theta\, \cos\phi\\ y&=r\, \sin\theta\, \sin\phi\\ z&=r\, \cos\theta \end{align} \begin{align} 0\le &r<\infty\\ 0\le &\theta<\pi\\ 0\le &\phi <2\pi\\ \end{align}