Coordinate Systems

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Section B.1 Coordinate Systems

Subsection Cylindrical Coordinates

\begin{align*} x \amp = s\cos\phi \qquad\qquad s^2 = x^2+y^2\\ y \amp = s\sin\phi \qquad\qquad \tan\phi = y/x\\ z \amp = z \qquad\qquad\qquad\> \, z=z \end{align*}

\begin{align*} 0\amp \le s\lt\infty \\ 0\amp \le \phi\lt 2\pi\\ -\infty\amp \lt z\lt\infty \end{align*}

Figure B.1. The definition of cylindrical coordinates, also showing the associated basis vectors.

Subsection Spherical Coordinates

\begin{align} x \amp = r\sin\theta\cos\phi \qquad\qquad r^2 = x^2+y^2+z^2\notag\\ y \amp = r\sin\theta\sin\phi \qquad\qquad \tan\theta = \sqrt{x^2+y^2}/z\tag{B.1.1}\\ z \amp = r\cos\theta \qquad\qquad\qquad \tan\phi = y/x\notag \end{align}

\begin{align*} 0\amp \le r\lt\infty \\ 0\amp \le \theta\lt \pi\\ 0\amp \le \phi\lt 2\pi \end{align*}

Figure B.2. The geometric definition of spherical coordinates, also showing the associated basis vectors.

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.