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Section 1.2 Curvilinear Coordinates

Choosing an appropriate coordinate system for a given problem is an important skill. The most frequently used coordinate system is rectangular coordinates, also known as Cartesian coordinates, after René Déscartes. One of the great advantages of rectangular coordinates is that they can be used in any number of dimensions.

In three dimensions, holding any one coordinate fixed yields a surface, which is a plane in the case of rectangular coordinates. A coordinate system can be thought of as a collection of such “constant coordinate” surfaces, and the coordinates of a given point are just the values of those constants on all the surfaces which intersect at the point. These planes are illustrated for rectangular coordinates in Figure 1.2.1.

Figure 1.2.1. The coordinate planes in rectangular coordinates. On each of these planes, one of the rectangular coordinates is constant.

It is often useful, however, to use a coordinate system which shares the symmetry of a given problem — round problems should be done in round coordinates. The two standard “round” coordinate systems are cylindrical coordinates (\(s\text{,}\)\(\phi\text{,}\)\(z\)), shown in Figure 1.2.2, and spherical coordinates (\(r\text{,}\)\(\theta\text{,}\)\(\phi\)), shown in Figure 1.2.3. Either of these coordinate systems can also be restricted to the \(x\text{,}\) \(y\)-plane, where they both reduce to polar coordinates. You should be aware that the standard physics conventions for spherical coordinates, used here, differ from the standard (American) math conventions; the roles of \(\theta\) and \(\phi\) are reversed. We also reserve \(r\) for the radial coordinate \(r\) in spherical coordinates, using \(s\) instead in cylindrical (and occasionally also in polar) coordinates.

Figure 1.2.2. The geometric definition of cylindrical coordinates.
Figure 1.2.3. The geometric definition of spherical coordinates.
Exploration 1.2.1. Coordinate planes in curvilinear coordinates.

Find the “constant coordinate” surfaces (analogous to those in Figure 1.2.4) for cylindrical and spherical coordinates.

Solution
Figure 1.2.4. The coordinate planes in cylindrical and spherical coordinates.
Notation.

Both of these coordinate systems reduce to polar coordinates in the \(x\text{,}\) \(y\)-plane, where \(z=0\) and \(\theta=\pi/2\) if, in the cylindrical case you relabel \(s\) to the more standard \(r\text{.}\) In both cases, \(\phi\) rather than \(\theta\) is the label for the angle around the \(z\)-axis. Make sure you know which geometric angles \(\theta\) and \(\phi\) represent, rather than just memorizing their names. Whether or not you adopt the conventions used here, you should be aware that many different labels are in common use for both of these angles. In particular, you will often see the roles of \(\theta\) and \(\phi\) interchanged, particularly in mathematics texts.

Another common convention for curvilinear coordinates is to use \(\rho\) for the spherical coordinate \(r\text{.}\) We will not use \(\rho\) for the radial coordinate in spherical coordinates because we want to reserve it to represent charge or mass density. Some sources use \(r\) for both the axial distance in cylindrical coordinates and the radial distance in spherical coordinates.