Other Coordinate Systems

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Section 8.3 Other Coordinate Systems

It is important to realize that \(d\rr\) and \(ds\) are defined geometrically, not by the component expressions in (8.1.2) and (8.1.4). Because of this coordinate-independent nature of \(d\rr\text{,}\) it is possible and useful to study \(d\rr\) in another coordinate system, such as polar coordinates (\(r\text{,}\)\(\phi\)) in the plane.¹ It is then natural to use basis vectors \(\{\rhat,\phat\}\) adapted to these coordinates, with \(\rhat\) being the unit vector in the radial direction, and \(\phat\) being the unit vector in the direction of increasing \(\phi\text{;}\) see Section 1.11. ²

Figure 8.3. The infinitesimal vector version of the Pythagorean Theorem, in both rectangular and polar coordinates.

Figure 8.3. The infinitesimal vector version of the Pythagorean Theorem, in both rectangular and polar coordinates.

By determining the lengths of the sides of the infinitesimal polar “rectangle” shown in the last drawing of Figure 8.3, one obtains

\begin{equation} d\rr = dr\,\rhat + r\,d\phi\,\phat\tag{8.3.1} \end{equation}

Notice the factor of \(r\) in the \(\phat\) term; \(d\phi\) by itself is not a length. The length of an infinitesimal arc is \(r\,d\phi\text{.}\) Using (8.1.3) to find the length of \(d\rr\) as before leads to

\begin{equation} ds^2 = dr^2 + r^2 \, d\phi^2\tag{8.3.2} \end{equation}

which is the infinitesimal Pythagorean Theorem in polar coordinates.

We choose \(\phi\) for the polar angle in order to agree with the standard conventions for spherical coordinates used by everyone but (American) mathematicians.

One can of course relate \(\rhat\) and \(\phat\) to \(\xhat\) and \(\yhat\text{,}\) for instance using the second figure in Figure 1.25. However, in most physical applications (as opposed to problems in calculus textbooks) this step can be avoided by making an appropriate initial choice of coordinates.