©2002 American Association of Physics Teachers. All rights reserved.

^{ }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. © *2002
American Association of Physics ^{ }Teachers.*

- I. INTRODUCTION
- II. CALCULATING DIV, GRAD, AND CURL
- III. ELECTROMAGNETIC CONIC SECTIONS
- IV. DISCUSSION
- ACKNOWLEDGMENTS
- REFERENCES
- FIGURES
- FOOTNOTES

^{ }are the divergence and
curl of a vector field? Students^{ }in mathematics courses often learn
algebraic formulas for these derivatives,^{ }without learning the
geometry behind them. Those students who go^{ }on to take physics or
engineering courses which use these^{ }concepts often have trouble
"bridging the gap" between the way^{ }vector calculus is taught by
mathematicians and the way it^{ }is used in applications.^{1}^{,}^{2} One indication of the extent of^{ }this
problem is the fact that few mathematicians have seen^{ }the notation
{,,} for
the unit vectors for spherical coordinates,^{ }yet most physicists
assume that their students learn this in^{ }vector calculus.^{3} This problem is exacerbated by the different
conventions^{ }for spherical coordinates used by physicists and
mathematicians. ^{}

^{ }that the use of
nonrectangular coordinate bases is just one^{ }small step; the gap must
be bridged at a more^{ }fundamental level.^{4} And a few simple examples, such as
spherical^{ }coordinates, are sufficient for, say undergraduate physics.
The goal of^{ }this paper is rather to consider several more
sophisticated examples,^{ }showing explicitly how they relate to
electromagnetism. ^{}

^{ }formulas for
divergence and curl so much harder in curvilinear^{ }coordinates than in
rectangular coordinates? Because the basis vectors are^{ }not constant.
However, if the basis vectors were both curl-free^{ }and
divergence-free, they would pull through the computation of the^{ }curl
and divergence as though they were constant, dramatically simplifying^{
}things. This is the basic idea we will develop here.^{
}Furthermore, a (time-independent) vector field which is both curl-free
and^{ }divergence-free solves Maxwell's equations; such vector fields
correspond to an^{ }electromagnetic field in vacuum. ^{}

^{ }ways of computing div, grad, and curl in curvilinear
coordinates,^{ }and in particular contrasting the approaches used by
mathematicians and^{ }physicists. We then consider several important
examples which naturally correspond^{ }to an "electromagnetic basis," in
each case discussing the electromagnetic^{ }interpretation. Most of the
examples we give are special cases^{ }of ellipsoidal coordinates; we
call them *electromagnetic conic sections*. We^{ }then give a
necessary and sufficient condition for a given^{ }vector field to be
electromagnetic, that is, to admit a^{ }rescaling which is both
curl-free and divergence-free. Finally, we show^{ }how to use these
basis vectors to simplify the computation^{ }of div, grad, and curl.
^{}

^{ }typically works in Cartesian coordinates, using the basis
{**î**,,}.
Given^{ }any vector field

the divergence and curl of
are^{ }defined by the formulas

and

respectively. It is an instructive^{ }exercise to try to use these
formulas to verify that^{ }

(the charge density for a point charge, away from the^{ }source) and
that

(the current density of a line charge,^{ }away from the source).
These are perhaps the two most^{ }important elementary physical examples
of divergence and curl. ^{}

^{ }the symmetry of the problem.
Introducing the spherical basis vectors^{ }

it is straightforward but messy to use (2)
and (3)^{
}to calculate, for instance, that

either using the chain rule,^{ }or by rewriting (*r*,,) in terms
of (*x*,*y*,*z*). If one^{ }now recalls that vector
differentiation satisfies the product rules

then^{ }it is an easy matter to use (9)
and (10)^{
}to show that

for instance by using *r*^{2} = *x*^{2} +
*y*^{2} + *z*^{2} and *r* sin = .^{ }But these are precisely (4)
and (5);
the hard work^{ }here is in deriving the initial formulas (9)
and (10).^{
}This approach is closely related to the concept of covariant^{
}differentiation in differential geometry. ^{}

^{
}curl in terms of a Cartesian basis, an introductory mathematics^{
}course typically goes on to prove the divergence theorem and^{
}Stokes' theorem. If there is time—there often is not—a geometric^{
}interpretation is then provided through the formulas

which relate divergence^{ }and curl to flux and circulation,
respectively. ^{}

^{ }this
around, and use these formulas to *define* the divergence^{ }and
curl, thus turning the divergence theorem and Stokes' theorem^{ }into
tautologies. These formulas are then used to *compute* the^{
}formulas for the divergence and curl in various coordinate systems.^{
}In spherical coordinates, for instance, this leads to formulas such^{
}as

from which (13)
and (14)
follow immediately. For a^{ }good, informal description of this
approach, see Schey.^{5} ^{}

^{ }that is, one in which
the three coordinate directions are^{ }everywhere orthogonal. Typical
examples are rectangular, cylindrical, and spherical coordinates,^{ }but
there are many more. ^{}

^{ }(*u*,*v*,*w*) will have a line element of the
form

If^{ }we denote the unit vector fields in the coordinate
directions^{ }by {**û**,,}, then
we can expand any vector field ^{
}as

It is then a fairly simple computation^{6} to derive^{ }the general formulas

using the formulas (15)
and (16).
^{}

^{ }formulas can hardly be
called obvious. The corresponding formula for^{ }the gradient is much
more natural. Starting from the chain^{ }rule, in the form

the all-important directional derivative, in the^{ }form

together with the "square root" of the line element^{ }(in the sense
*d*·*d* =
*d**s*^{2}), given by

we obtain

^{}

^{
}and (22),
we see that there are special vector fields^{ }which are divergence or
curl free, since

and similarly for^{ } and
. These formulas can also be derived from^{ }the
identities

when one realizes that in orthogonal coordinates one^{ }has

^{}

^{7} the existence of a natural^{
}divergence-free basis along the lines of (27)
can be used^{ }to reduce the computation of the divergence to the
much^{ }simpler computation of the gradient. Similarly, the existence of
a^{ }natural curl-free basis along the lines of (28)
can be^{ }used to simplify the computation of the curl. In each^{
}case, this is accomplished using the appropriate product rule, (11)^{
}or (12),
respectively. However, it is noteworthy that these two^{ }natural bases
only agree in rectangular coordinates. ^{}

^{ }could find a basis
which was *both* divergence- and curl-free?^{ }In that case, one
would never need to remember the^{ }formulas for the divergence and
curl; all computations would reduce^{ }to the much simpler formula for
the gradient. ^{}

^{ }basis would also be of
physical interest. A vector field^{ }which is both divergence- and
curl-free solves Maxwell's vacuum equations,^{ }and can hence be
interpreted as an electric or magnetic^{ }field. We are thus led to ask
whether we can^{ }find a basis of electromagnetic fields.
^{}

^{
}several examples. ^{}

**î**,,}^{
}is constant, and therefore, of course, both divergence- and
curl-free.^{ }Each basis vector field must therefore describe an
electromagnetic field.^{ }Which one? Consider an infinite parallel-plate
capacitor,^{8} with infinite separation^{ }between the
plates. If the plates have equal but opposite^{ }(uniform) charge
densities, then there is a constant electric field^{ }orthogonal to the
plates. If, instead, the plates have equal^{ }but opposite (uniform)
current densities, then there is a constant^{ }magnetic field parallel
to the plates (but orthogonal to the^{ }currents). ^{}

^{9}^{ }

Horizontal (*z* = constant) and vertical (tan = constant) slices
through this coordinate system^{ }are shown in Fig. 1. Denoting the orthonormal basis for^{ }cylindrical
coordinates as usual by {,,}, we
have ,
and^{ }thus this basis vector field is both divergence- and
curl-free.^{ }But what about the other basis vectors? ^{}

^{
}electromagnetic fields correspond to an infinite straight wire carrying
either^{ }a uniform charge density or a uniform current density. It^{
}is straightforward to work out the corresponding fields: Up to^{
}scale factors, the electric field of the (positively) charged
*z*^{ }axis is

and the magnetic field of the (upward) current-carrying^{ }*z*
axis is

Thus, an "electromagnetic" basis in this case^{ }is given by {,,}.
^{}

^{
}be axially symmetric, and will thus have as a^{
}coordinate, as a
basis vector field, and
as^{ }an electromagnetic basis vector field (although *s* = will need to^{ }be expressed in
terms of the given coordinates). We will^{ }omit further discussion of
this case in (most of) the^{ }subsequent examples, and we will have no
further use for^{ }horizontal slices analogous to (a) in Fig. 1.
^{}

^{ }about the other standard
coordinate system, namely spherical coordinates, defined^{ }implicitly
by

(with as before), and shown in Fig.^{ }2. The orthonormal basis for spherical coordinates is
{,,},
and^{ }we already know that

is both divergence- and curl-free. ^{}

^{ }only obvious spherical
electromagnetic field is the electric field of^{ }a point charge, which
is, up to a scale factor^{ }

This solves part of the problem. But what electromagnetic field,^{
}if any, looks like ?
Somewhat surprisingly, it turns out^{ }there is one, namely the electric
field of two half-infinite^{ }uniform line charges, with equal but
opposite charge densities, as^{ }shown in Fig. 3. Up to a scale factor, the^{ }resulting
divergence-free and curl-free basis vector field is

and an^{ }electromagnetic basis is given by {,,}. ^{}

^{ }about other, less common,
orthogonal coordinate systems? Consider first prolate^{ }spheroidal^{10} coordinates, defined by

as shown in Fig. 4. The^{ }relevant orthonormal basis vectors are
**û** and ; our goal^{ }is to find multiples of
these which are both divergence-^{ }and curl-free, if possible.
^{}

^{
}is, after having first computed the answer by brute force,^{ }it
is clear that such vector fields do indeed exist.^{ }Consider the
spherical model above, in which a multiple of^{ } was produced by two half-infinite line charges which
were^{ }joined at the origin. Separate the two instead by a^{
}finite distance, as shown in Fig. 5. The resulting electric^{ }field is just
(proportional to)

and is therefore spheroidal. Similarly,^{ }the electric field of the
"missing" finite line segment is^{ }just (proportional to)

which is hyperboloidal, as shown in Fig.^{ }6.^{11} An electromagnetic basis in this case is
therefore given^{ }by {,,}. ^{}

^{ }defined by

and shown in Fig. 7. Do there exist^{ }multiples of **û** and
which are both divergence- and^{ }curl-free?
^{}

^{ }clearly yes. The electric field of a half-infinite, uniform
line^{ }charge is shown in Fig. 8, corresponding to

respectively. ^{}

^{ }(inverse paraboloidal) coordinates, defined by

and shown in Fig. 9.^{ }We have

and we seek multiples of **û** and ^{
}which are both divergence- and curl-free. ^{}

^{}

^{}

^{ }, we ask whether there exists a function such^{
}that is both divergence- and curl-free, that is,
such^{ }that

Using the product rules (11)
and (12),
we can^{ }rewrite these conditions as

On the other hand, the identity^{ }

leads to

Rearranging terms and using (56)
and (57)
then^{ }yields

^{}

^{ }both sides yields

since the left-hand side is the curl^{ }of ln . The necessary
and sufficient condition that a suitable^{ } exist is
therefore (61);
if exists, then (61)^{
}is satisfied due to the identity (30),
whereas if (61)^{
}is satisfied, then there exists a (local) potential function, which^{
}is ln . ^{}

^{
}possible alternative way to compute the divergence and curl in^{
}certain standard cases. For instance, in spherical coordinates, one
really^{ }need only remember that {,,}
is an electromagnetic basis—ideally by^{ }recalling the corresponding
electromagnetic fields. The divergence and curl are^{ }then easily
computed from formulas like

Yes, this requires knowing^{ }how to compute the gradient in
spherical coordinates, but this^{ }can easily be rederived as needed
from the geometrically obvious^{ }formula

^{}

^{ }admit an "electromagnetic basis." All of
these examples are *separable ^{ }coordinates* in the sense of Morse
and Feschbach,

^{ }that
separable coordinates are the only ones which admit an^{
}electromagnetic basis. However, there are also nonseparable coordinates
which admit^{ }an electromagnetic basis, an example being
"logcoshcylindrical" coordinates, defined by^{ }

^{}

^{ }characterizes the vector fields
which can be rescaled so^{ }as to be both divergence- and curl-free, it
also provides^{ }an explicit algorithm for determining . There is
another, simpler^{ }characterization, but without this property.
^{}

^{ }means that (locally)

In particular, since we are assuming =
,^{ }this forces the original vector
field to be orthogonal^{ }to the surfaces
{*f* = constant}. Thus, a necessary condition on^{ } is that it be *hypersurface orthogonal*. This
condition is^{ }always satisfied for the examples considered here,
constructed from a^{ }coordinate system. ^{}

^{ }further
condition that

so that must be the gradient^{ }of a
*harmonic* function. Thus, the question of which coordinate^{
}systems admit basis vectors which can (all) be rescaled so^{ }as
to be divergence- and curl-free is equivalent to the^{ }question of
which coordinate systems can themselves be rescaled so^{ }as to be
harmonic coordinates. ^{}

^{
}harmonic functions in two dimensions are closely related to analytic^{
}functions. A vector field =
*P***î** + *Q* is divergence- and curl-free if,^{ }and
only if, *P*–*i**Q* is analytic, since^{13}

^{}

^{
}College for a colloquium invitation which got this project started.^{
}This material is based upon work supported by the National^{
}Science Foundation under Grants Nos. DUE-9653250 (Paradigms Project) and
DUE-0088901^{ }(Vector Calculus Bridging Project). This work has also
been supported^{ }by the Oregon Collaborative for Excellence in the
Preparation of^{ }Teachers (OCEPT) and by an L L Stewart Faculty
Development^{ }Award from Oregon State University. ^{}

- Jason W. Dunn and Julius Barbane, "One model for an integrated
math/physics course focusing on electricity and magnetism and related calculus
topics," Am. J. Phys.
**68**, 749–757 (2000). first citation in article

- Tevian Dray and Corinne A. Manogue, "The vector calculus gap:
mathematicsphysics," PRIMUS
**9**, 21–28 (1999). first citation in article

- 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. first citation in article

- Tevian Dray and Corinne A. Manogue, "Using differentials to
bridge the vector calculus gap," College Math. J. (to appear). first
citation in article

- H. M. Schey,
*div, grad, curl, and all that*, 3rd ed. (Norton, New York, 1997). first citation in article

- David J. Griffiths,
*Introduction to Electrodynamics*, 3rd ed. (Prentice-Hall, New York, 1999). first citation in article

- Mary L. Boas,
*Mathematical Methods in the Physical Sciences*, 2nd ed. (Wiley, New York, 1983). first citation in article

- One plate is in fact sufficient. The advantage of two plates is
that the field vanishes outside the capacitor. first
citation in article

- We use rather than for compatibility
with our later examples, and
*s*rather than*r*to avoid confusion with spherical coordinates. first citation in article

- A spheroid is an ellipsoid with two axes of the same length.
first
citation in article

- It is instructive to consider this latter example as a
"stretched out" point charge. first
citation in article

- Philip M. Morse and Herman Feshbach,
*Methods of Theoretical Physics*(McGraw-Hill, New York, 1953), Chap. 5. first citation in article

- A similar statement can be made in three dimensions, using
quaternions in place of the complex numbers. first
citation in article

Full figure (11 kB)

Fig. 1. (a) A horizontal^{ }slice of cylindrical coordinates,
resulting in the usual polar coordinate^{ }grid. (b) A vertical slice of
cylindrical coordinates, through the^{ }*z* axis (shown as a heavy
line). First
citation in article

Full figure (8 kB)

Fig. 2. A vertical slice of^{ }spherical coordinates, showing the
*r* coordinate grid. First
citation in article

Full figure (5 kB)

Fig. 3. A spherical electric field.^{ }If the positive *z* axis
is given a uniform positive^{ }charge density, and the negative *z*
axis is given an^{ }equal and opposite charge density, the resulting
field lines are^{ }spherical, that is, in the direction. First
citation in article

Full figure (10 kB)

Fig. 4. A vertical slice of^{ }prolate spheroidal coordinates. The
curves orthogonal to the ellipses are^{ }hyperbolas. First
citation in article

Full figure (7 kB)

Fig. 5. A spheroidal electric field. If the oppositely charged half-lines
in^{ }the spherical example are separated by a finite gap, the^{
}resulting field lines are spheroidal. First
citation in article

Full figure (3 kB)

Fig. 6. A hyperboloidal electric field, the electric^{ }field of a
uniformly charged line segment. First
citation in article

Full figure (5 kB)

Fig. 7. A vertical slice of^{ }parabolic coordinates. Both families
of orthogonal curves are parabolas. First
citation in article

Full figure (8 kB)

Fig. 8. Two paraboloidal^{ }electrical fields, namely the electric
field of a half-infinite uniform^{ }line charge, along (a) the negative
*z* axis and (b)^{ }the positive *z* axis. First
citation in article

Full figure (7 kB)

Fig. 9. A vertical slice of hyperboloidal coordinates. Both^{
}families of orthogonal curves are hyperbolas. First
citation in article

^{a}Electronic mail: tevian@math.orst.edu

^{b}Electronic mail:^{ }corinne@physics.orst.edu