You will now derive expressions for infintesimal distances in coordinate directions in cylindrical coordinates.

Geometrically determine the length $ds$ of each of the three paths shown below. Notice that, along any of these three paths, only one coordinate $r$, $\phi$, or $z$ is changing at a time (i.e. along path 1, $dz\ne0$, but $d\phi=0$ and $dr=0$).

Path 1: $ds=$

Path 2: $ds=$

Path 3: $ds=$

You will now derive expressions for infintesimal distances in coordinate directions in spherical coordinates.

Geometrically determine the length $ds$ of each of the three paths shown above. Notice that, along any of these three paths, only one coordinate $r$, $\theta$, or $\phi$ is changing at a time (i.e. along path 1, $d\theta\ne0$, but $dr=0$ and $d\phi=0$).

Path 1: $ds=$

Path 2: $ds=$
(Be careful, this is the tricky one.)

Path 3: $ds=$