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Section 3.2 Change of Coordinates

The whole point of using curvilinear coordinates is that they may be better adapted to the symmetries of a given problem. Ideally, this means that the entire problem should be done in curvilinear coordinates, without converting between coordinate systems (although this is not always possible). From this point of view, while it is certainly worth learning how to convert between, say, rectangular and polar coordinates, it is also worth learning how to avoid doing so as much as possible.

For completeness, the explicit transformations between these coordinate systems are given below. You should be able to use Figure 3.1.2, Figure 3.1.3, and trigonometry to verify these results. We reiterate that the conventions used here are “physicists' conventions”, used by almost everyone except American mathematicians [1].

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*}

Spherical Coordinates:

\begin{align*} x \amp = r\sin\theta\cos\phi \qquad\qquad r^2 = x^2+y^2+z^2\\ y \amp = r\sin\theta\sin\phi \qquad\qquad \tan\theta = \sqrt{x^2+y^2}/z\\ z \amp = r\cos\theta \qquad\qquad\qquad \tan\phi = y/x \end{align*}