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.

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 length elements \(d\ell\) to
determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

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.