Certain orthogonal coordinate systems naturally correspond to basis vectors which are both curl-free and divergence-free, and hence solve Maxwell’s equations. After first comparing several different traditional approaches to computing div, grad, and curl in curvilinear coordinates, we present a new approach, based on these “electromagnetic” basis vectors, which combines geometry and physics. Not only is our approach tied to a physical interpretation in terms of the electromagnetic field, it is also a useful way to remember the formulas themselves. We give several important examples of coordinate systems in which this approach is valid, in each case discussing the electromagnetic interpretation of the basis. We also give a general condition for when an electromagnetic interpretation is possible.

Topics
Electromagnetism, Maxwell equations, Euclidean geometries, Numerical linear algebra
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3.
For the record, while a traditional course in multivariable or vector calculus will certainly discuss polar, cylindrical, and spherical coordinates, vectors will most likely be expressed exclusively in terms of their rectangular components.
4.
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5.
H. M. Schey, div, grad, curl, and all that, 3rd ed. (Norton, New York, 1997).
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David J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, New York, 1999).
7.
Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd ed. (Wiley, New York, 1983).
8.
One plate is in fact sufficient. The advantage of two plates is that the field vanishes outside the capacitor.
9.
We use φ rather than θ for compatibility with our later examples, and srather than r to avoid confusion with spherical coordinates.
10.
A spheroid is an ellipsoid with two axes of the same length.
11.
It is instructive to consider this latter example as a “stretched out” point charge.
12.
Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 5.
13.
A similar statement can be made in three dimensions, using quaternions in place of the complex numbers.
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© 2002 American Association of Physics Teachers.
2002
American Association of Physics Teachers
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