Student handout: Curvilinear Volume Elements

Vector Calculus I 2022
Students construct the volume element in cylindrical and spherical coordinates.
What students learn Students practice infinitesimal reasoning 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: \(d\ell=\)
\(dV=\)



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