## Small Group Activity: 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.
• Media

##### 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 $d\boldsymbol{\vec r}$.

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