Section 8.4 Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates
In the activities below, you will construct infinitesimal distance elements (sometimes called line elements) in rectangular, cylindrical, and spherical coordinates. These infinitesimal distance elements are building blocks used to construct multi-dimensional integrals, including surface and volume integrals. To construct flux integrals, you will need vector differentials, which are a slight variant of the infinitesimal distance elements. See Section 8.5.
Activity 8.2. The Distance Element in Rectangular Coordinates.
Given the cube shown below, find \(ds\) on each of the three paths, leading from \(a\) to \(b\text{.}\)
Path 1: \(ds=\)
Path 2: \(ds=\)
Path 3: \(ds=\)
The infinitesimal distance element \(ds\) is an infinitesimal length. Find the appropriate expression for \(ds\) for the path which goes directly from \(a\) to \(c\) as drawn below.
Path 4: \(ds=\)
An infinitesimal element of length in the \(z\)-direction is simply \(dz\)
Cartesian coordinates would be a poor choice to describe a path on a cylindrically or spherically shaped surface. Next we will find appropriate expressions in these cases.
Activity 8.3. The Distance Element in Cylindrical Coordinates.
You will now use geometry to determine the general form for \(ds\) in cylindrical coordinates by determining \(ds\) along the specific paths below. See Figure 8.4.
Geometrically determine the length of the three paths leading from \(a\) to \(b\text{.}\) Notice that, along any of these three paths, only one coordinate \(s\text{,}\) \(\phi\text{,}\) or \(z\) is changing at a time (i.e. along path 1, \(dz\ne0\text{,}\) but \(d\phi=0\) and \(dr=0\)).
Path 1: \(ds=\)
Path 2: \(ds=\)
Path 3: \(ds=\)
If all three coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(ds\) for any path as:
\(ds\)=
This is the general distance element in cylindrical coordinates.
Analogously to rectangular coordinates, an infinitesimal element of length in the \(r\) direction is simply \(dr\text{.}\) But an infinitesimal element of length in the \(\phi\) direction in cylindrical coordinates is not just \(d\phi\text{,}\) since this would be an angle and does not even have the units of length.
Using the relationship between angles (in radians!) and radius, the infinitesimal element of length in the \(\phi\) direction in cylindrical coordinates is \(s\,d\phi\text{.}\)
Activity 8.4. The Distance Element in Spherical Coordinates.
You will now use geometry to determine the general form for \(ds\) in spherical coordinates by determining \(ds\) along the specific paths below. See Figure 8.5. As in the cylindrical case, note that an infinitesimal element of length in the \(\that\) or \(\phat\) direction is not just \(d\theta\) or \(d\phi\text{.}\) You will need to be more careful.
Geometrically determine the length of the three paths leading from \(a\) to \(b\text{.}\) Notice that, along any of these three paths, only one coordinate \(r\text{,}\) \(\theta\text{,}\) or \(\phi\) is changing at a time (i.e. along path 1, \(d\theta\ne0\text{,}\) but \(dr=0\) and \(d\phi=0\)).
Path 1: \(ds=\)
Path 2: \(ds=\qquad\) (Be careful, this is the tricky one.)
Path 3: \(ds=\)
If all 3 coordinates are allowed to change simultaneously, by an infinitesimal amount, we could write this \(ds\) for any path as:
\(ds\)=
This is the general line element in spherical coordinates.
A similar argument to the one used above for cylindrical coordinates, shows that the infinitesimal element of length in the \(\theta\) direction in spherical coordinates is \(r\,d\theta\text{.}\)
What about the infinitesimal element of length in the \(\phi\) direction in spherical coordinates? Make sure to study the diagram carefully. Where is the center of a circle of constant latitude? It is not at the center of the sphere, but rather along the \(z\)-axis. The radius of this circle is not \(r\text{,}\) but rather \(r\,\sin\theta\text{,}\) so the infinitesimal element of length in the \(\phi\) direction in spherical coordinates is \(r\,\sin\theta\,d\phi\text{.}\)