Students construct the volume element in cylindrical and spherical coordinates.
Cylindrical Coordinates
Using the first figure below, determine the length \(d\ell\) of each of the three segments shown (the three thick lines). Notice that, along any of these three segments, only one coordinate \(s\), \(\phi\), or \(z\) is changing at a time (i.e. along segment 1, \(dz\ne0\), but \(d\phi=0\) and \(ds=0\)). Use your results to determine the volume of the region.
Segment 1: \(d\ell=\)
Segment 2: \(d\ell=\)
Segment 3: \(d\ell=\)
\(dV=\)
Spherical Coordinates
Using the second figure below, determine the length \(d\ell\) of each of the three segments shown (the three thick lines). Notice that, along any of these three segments, only one coordinate \(r\), \(\theta\), or \(\phi\) is changing at a time (i.e. along segment 1, \(d\theta\ne0\), but \(dr=0\) and \(d\phi=0\)). Use your results to determine the volume of the region.
(Be careful: One segment is trickier than the others.)
Segment 1: \(d\ell=\)
Segment 2: \(d\ell=\)
Segment 3: \(d\ell=\)
\(dV=\)
This is one of several similar activities using infinitesimal reasoning in curvilinear activities. Unlike most of the others, this one does not use \(d\boldsymbol{\vec r}\).