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.

The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists’ and mathematicians’ notational conventions) and the basis vectors adapted to these coordinate systems.

The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point
\(\vec r'\) is a coordinate-independent, physical and geometric quantity. But,
in practice, you will need to know how to express this quantity in different
coordinate systems.

Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point \(\vec
r'\) in rectangular coordinates.

Show that this same distance written in cylindrical coordinates is:
\begin{equation}
\left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi\,{}'-\phi) +(z\,{}'-z)^2}
\end{equation}

Show that this same distance written in spherical coordinates is:
\begin{equation}
\left\vert\vec r\,{}' -\vec r\right\vert
=\sqrt{r\,{}'^2+r^2-2rr\,{}'
\left[\sin\theta\sin\theta\,{}'\cos(\phi\,{}'-\phi)
+\cos\theta\,{}'\cos\theta\right]}
\end{equation}

Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify
the previous two formulas.

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula.
\[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).

Students use known algebraic expressions for vector line elements \(d\vec{r}\) to
determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to
Scalar Surface and Volume Elements except uses a more sophisticated vector approach to find surface, and volume elements.