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.

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*coordinates*, vectors will most likely be expressed exclusively in terms of their rectangular components.

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*Mathematical Methods in the Physical Sciences*, 2nd ed. (Wiley, New York, 1983).

*Methods of Theoretical Physics*(McGraw-Hill, New York, 1953), Chap. 5.

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