The Net Advance of Physics: SPECIAL
BIBLIOGRAPHIES, No. 3
Algebras of Electromagnetics
by Perttu P. Puska (Helsinki University of Technology)
First Edition, 2000 November 28.
Copyright © 2000 by
Perttu P. Puska:
P. Puska, Net Adv. Phys. Spec. Bibliog. 3:1 (2000).
Algebras of Electromagnetics
I have collected here links and references
about algebras used in electromagnetics. The contents of the list
of course reflect my preferences. However, taste has not been the
only criterion, for I have
included only those references that I
have had possibility to evaluate, which explains the fact that so many
of the classical papers are missing.
Contents
Note that classification is quite artificial, for
it is moreoftenthannot difficult to label piece of writing as
belonging to 'exterior algebras' or 'clifford algebras'. So reader is
advised to consult all categories in order to find suitable texts.
Vector and dyadic algebra
General
Did you know that
the traditional vector algebra is included in Clifford algebra
Cl(3,0)? See my notes on that.
J. W. Gibbs and E.B. Wilson,Vector Analysis,
2. ed., Scribner, New York, 1909, Dover reprint, New York, 1960.

J. W. Gibbs is the father of vector and dyadic algebra.
His works are still some of the best expositions on
dyadic algebra.
O. Heaviside,Electromagnetic Theory, Ernest Benn,
London, 1925.

Heaviside is the alternative father of vector algebra.
Oliver Heaviside is credited (among many other things)
for formulating Maxwell's equations in
the language of vector algebra. Bytheway, it is shame that so many
of the modern textbooks still fail to pay homage to a man
who wrote the equations in a form that made it possible to
teach the electromagnetic theory to generations of
engineers and physicists. His own writing is a bit heavi read, though (pun
intended).
Introductory texts and tutorials
Well, it is almost hopeless to try to recommend anything, there
are so many texts available.
Intermediate to advanced
H. C. Chen, Theory of electromagnetic waves, McGrawHill,
New York, 1983.
 Intermediate level dyadics. Out of print for years.
I. Lindell,
Methods for Electromagnetic Field Analysis, Oxford: Clarendon
Press, 1992. Republished by IEEE 1995.
Clifford algebras
General
Clifford algebra researchers have an own society
which publishes a journal called
Advances in Applied Clifford Algebras twice a year.
The Geometric Algebra Research Group at Cavendish Laboratory
has
an online intro and several downloadable psformat research papers
ranging from introductory texts to advanced. Several very nice papers.
It is probably so that
this Cavendish group has become a forerunner in utilisation of Clifford's
geometric algebra in physics.
Tutorials and introductory texts
D. Hestenes: "Vectors, Spinors, and Complex Numbers in Classical and
Quantum Physics, " Am. J. Phys. vol. 39, no. 9, pp. 10131027, 1971.
W. E. Baylis,
J. Huschilt, Jiansu Wei: "Why i?," Am. J. Phys. vol. 60, no. 9,
pp. 788797, 1992.
E. F. Bolinder: "Clifford Algebra, What is it?,"
IEEE Antennas and Propagation
Society Newsletter, August, pp. 1823, 1987.
 Bolinder's brief historical review and discussion of Clifford
algebra (I thank dr. S.Sensiper for sending a copy of
this article).
Did you know that M. Riesz' wrote
his famous lecture notes originally
for Bolinder?
T. G. Vold: "An introduction to geometric algebra with an application to
rigid body mechanics" and "An introduction to geometric
calculus and its application to electrodynamics"
Am. J. Phys. vol. 61, no. 6, pp. 491513, 1993.
A. Lewis, a psformat intro can be downloaded from
his webpage. Also
links to other intros, such as the one written by
R. Harke.
C. Rodriguez has a compact
online intro about Clifford algebra.
T. Smith,
an online intro
not entirely restricted to the subject of Clifford algebra.
Intermediate to advanced
W. E. Baylis,
and G. Jones, ``The Pauli algebra
approach to special relativity,'' J. Phys. A, vol. 22, no. 1,
pp. 115, 1989.

Shows that there is enough structure in Cl(3,0) ( i.e. Pauli algebra )
to represent spacetime.
See also the book by W. E. Baylis.
W. E. Baylis,
Electrodynamics, A Modern Geometric Approach
Birkhduser, Boston 1999.
I. M. Benn and R. W. Tucker,An introduction to Spinors and
Geometry, Adam Hilger, London 1987.
 Discusses tensors, Clifford algebras ( spinors are
elements of minimal left or right
ideals of Clifford algebras,
which explains why the word spinor appears so often in Clifford
algebra literature )
and applications.
Chapter on electromagnetism. Modern differential geometry
well represented, therefore
this book could also be in the exterior algebra
section of this list.
E. F. Bolinder: ``Unified microwave network theory
based on Clifford algebra in Lorentz space,'' 12th
European Microwave Conference, Helsinki, 2535, 1982.
Microwave Exhibitions and Publishers, Turnbridge Wells, Kent, 1982.
 Circuit engineers will find this interesting.
F. Brackx,
R. Delanghe,
F. Sommen,Clifford analysis,
Pitman, London 1982.
 Analysis branch of the research. Hypercomplex analysis
explained. Also from the same Gent research group is
R. Delanghe,
F. Sommen and V. Soucek,
Clifford Algebra and SpinorValued Functions,
Kluwer Academic Publisher, Dordrecht/Boston, 1992.

Analysis branch of the research. More into analysis is also
K. Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus
for Physicists and Engineers, Wiley, Chichester, 1997.

PDEs and Clifford algebra. Contains
also a short section on electromagnetics.
D. Hestenes, Spacetime Algebra, Gordon & Breach,
New York, 1966.
D. Hestenes, and
G. Sobczyk,
Clifford Algebra to Geometric Calculus,
Reidel, Dordrecht, 1984, reprint with corrections 1992.
 Expands the work started in Spacetime Algebra.
Advocates the use of geometric algebras instead
of differential forms.
P. Hillion, ``Constitutive relations and Clifford
algebra in electromagnetism,'' Adv. in Appl. Cliff. Alg.
vol. 5, no. 2,
pp. 141158, 1995.

One of the few to discuss constitutive relations in Clifford algebra context.
Note that the author uses a 'wedge' usually reserved for the exterior
products in place of the 'cross' in
cross products.
D. A. Hurley, M. A. Vandyck, Geometry Spinors and Applications,
Springer and Praxis Publishing, Chichester, 2000.
 Modern mathematical apparatus of physics well presented.
Contains an especially interesting section on electromagnetism.
One very nice thing about the text is that the formulas
in the text are written out fully, e.g. vectors are
written with their components and basis vectors.
B. Jancewicz, Multivectors and Clifford Algebra
in Electrodynamics, World Scientific, Singapore, 1988.
 New insights abound in this 1988 classic. Both
electromagnetism and algebra are carefully developed
sidebyside, the latter in intuitive manner that will
not scare away electrical engineers. This work and
the recent book by Baylis are the most
mature texts available for those who wish to use
Clifford algebra in electromagnetics.
B. Jancewicz: "A Hilbert space for the classical electromagnetic
field," (now where was that conference where this appeared, let me see..)
 Hilbert space of the vacuum electromagnetic field
rendered in Cl(3,0).
P. Lounesto,
Clifford Algebras and Spinors,
Cambridge University Press, Cambridge, 1997.
 A review of
the current state of research. Prof. Lounesto's
webpage has a
nice collection of links related to Clifford algebras. Prof. Lounesto is also
one the authors of CLICAL,
a Clifford calculus program written for
DOS that is small enough to run in an emulator (Yours truly runs it
in dosemu.)
J. Kot, G. C. James:
"Clifford algebra in electromagnetics", Proceedings of the
International Symposium on
Electromagnetic Theory, URSI International Union of Radio Science,
Aristotle University of Thessaloniki, 2528 May 1998, Thessaloniki, Greece.
pp. 822824.
M. Riesz, Marcel Riesz: Clifford Numbers and Spinors,
Kluwer Academic Publisher, Dordrecht/Boston, 1993.

Facsimile of Riesz' lectures to
E. Folke Bolinder and also a review of Riesz' work by Pertti Lounesto.
A classic. Riesz treats isometries in a considerable detail,
and later discusses very briefly electromagnetism (in vacuo). Rather
expensive book, I must add.
M. Riesz, "Sur certaines notions fondamentales en thiorie quantique
relativiste," C.R. 10^{e} Congrhs Math. Scandinaves, Copenhagen,
1946. Jul. Gjellerups Forlag, Copenhagen 1947, pp. 123148 (You can find the
article in the collected papers of M.Riesz, pp. 545570 ).
J. Snygg,Clifford Algebra, A Computational Tool for Physicists,
Oxford University Press, New York, 1997.
 Use of Clifford algebra in flat and curved spaces. General relativists
are targeted but the clear and detailed
discussion will appeal to electrical engineer as well. The
author makes a nice point in the preface about the use
of full differential geometry apparatus when the metric is present.
His point essentially is that the use of differential forms is
an overkill when the metric has been introduced (I agree with him.)
Therefore, enter the Clifford algebra.
Hypercomplex numbers:
the Quaternions ( a Cl(3,0) subalgebra ) and
their bigger cousins, the Octonions.
General
Timeline of
hypercomplex numbers (i.e. quaternions and octonions ).
Ftpsite
by H. Baker about historical quaternion papers.
Introductory texts and tutorials
D. Sweetser has several
online tutorials concerning applications of quaternions.
Geoffrey Dixon's
site about octonions and related physical
applications/implications. Conference announcements.
A concise tutorial
also available.
Intermediate to advanced
more to come..
Exterior algebras; Differential forms
General
An excellent
list of references has been collected by
Richard H. Selfridge, David V. Arnold and Karl F. Warnick at
Brigham Young University. Their site is very much worth checking
out, since they have several papers online. The level of exposition
in these papers varies from introductory level texts to
advanced research papers. Especially recommendable is
the one entitled "Teaching Electromagnetic Field Theory Using
Differential Forms," available in psformat.
Many of the references listed in
the link above are of intro level, so we do not
print them here again, with the exception of
G. A. Deschamps: "Electromagnetics and differential forms", Proc. IEEE,
vol. 69, pp. 676696, June 1981.

An article which started the differential form boom.
And while we are at it, Deschamps' earlier text
G. A. Deschamps: "Exterior differential forms," pp. 112161,
in
E. Roubine (ed): Mathematics Applied to Physics,
SpringerVerlag, Berlin and UNESCO, Paris, 1970
 is not
an older version of the article that appeared in Proc. IEEE,
but a more detailed excursion in the area of forms and
manifolds. Discusses electromagnetism, too.
It might come as a surprise to a modern reader that
J. W. Gibbs wrote an article about Grassmann's algebra:
"On multiple algebra", Proc. Am. Ass. Adv. of Science, vol. XXXV.
pp. 3766, 1886.
His approach looks very 'dyadescian', an interesting
and relevant critique of his
approach is recorded in the pages 635652 of Bull. A.M.S.,
vol. 78, 1972,
in F.Dyson's "Missed Opportunities,". See also pp. 1415 of
preprint
by J.Parra.
Introductory texts and tutorials
T. Frankel: "Maxwell's equations", Am. Math. Monthly, vol. ?,
pp. 343349, April 1974.
 Mathematician introduces salient features in a very concise
manner.
Intermediate to advanced
J. Baez, J. P. Muniain, Gauge Fields, Knots and Gravity,
World Scientific, Singapore, 1994.
 Comprehensive and lucid review of the present day
theoretical physics in modern differential geometry
notation. Electromagnetism included, of course.
As the title says, readers interested in
modern theories of gravity might be the targeted audience,
but nevertheless recommendable also for those readers who want
to put their understanding of electromagnetism in to a
broader context.
D. Baldomir, P. Hammond Geometry of Electromagnetic Systems,
Clarendon Press, Oxford 1996.
 If you are to own one
'differential forms in electromagnetics'book,
this is the one  not quite, but close. There are
mainly two reasons why not: Typos and omission of
pseudoforms. Otherwise I would
recommend this book without reservations.
Nevertheless, any lab
or faculty library should own a copy of this one.
B. Jancewicz: ``A Variable Metric Electrodynamics.
The Coulomb and BiotSavart Laws in Anisotropic Media,''
Ann. Phys. (N.Y.), vol. 245, 227274, 1996.

Author introduces pseudomultivectors and multiforms
(some authors call these 'twisted','impair').
Anisotropy embedded in the metric tensor is discussed in a static case.
D. G. B. Edelen, Applied Exterior Calculus, Wiley, New York 1985.
 A good dose of topology and differential geometry.
A nice electromagnetics chapter (ch. 9). One of the few
works to discuss the significance of constitutive relations.
If your library
does not carry this particular book, Edelen's article
"A Metric Free Electrodynamics with Electric and Magnetic Charges,"
in Ann. Phys. (N.Y.), vol. 112, 1978, pp. 366400, is
selfcontained and has essentially all the information
that later appeared in chapter 9.
I.
Lindell and P. Lounesto,
Differentiaalimuodot sdhkvmagnetiikassa (Differential forms in
electromagnetics),
Helsinki University of Technology,
Electromagnetics laboratory report, Espoo 1995.

In Finnish. Formulates constitutive relations of bianisotropic media nicely
and some other attractive developments.
K. Meetz and W. L. Engl, Elektromagnetische Felder,
SpringerVerlag, Berlin, 1980.
 Develops electromagnetics entirely in terms of differential
forms. Electrical engineers will like this one, as the
spirit of Meetz's and Engl's text is clearly
'engineering' minded. This does not prevent them from
giving an account on special relativity,
a topic which usually is omitted in
el. eng. textbooks.
C. Misner, K. Thorne and J. Wheeler, Gravitation, Freeman,
San Francisco 1973.
 Cosmology bible. Advocates the idea of physics as geometry.
First chapters explain and discuss differential forms
and their usage in electromagnetics. An excellent book.
See also
J. Wheeler's Geometrodynamics.
F. L. Teixeira
and W. C. Chew: "Unified
analysis of perfectly
matched layers using differential forms," Microwave and Optical
Technology Letters, vol. 20, no. 2, pp. 124126, 1999.
J. W. Wheeler, Geometrodynamics, Academic Press,
London 1962.
An astute observer has probably noticed that
a renaissance of exterior algebras in
electromagnetics is already here.
My recent INSPEC database search indicated
that within last year or so, several papers about
differential forms in FEM/FDTD have been published, e.g.
A. Bossavit, L. Kettunen:
"Yeelike schemes on a tetrahedral mesh, with diagonal lumping"
Int. J. of Num. Mod.: ElectronicNetworks,Devices and Fields.
vol.12, no.12, Jan.April 1999, pp.129142.
T. Tarhasaari, L. Kettunen: "Some realizations of discrete Hodge operator:
A reinterpretation of finite element techniques," IEEE
Tr. Magn., vol. 35, no. 3, 1999, pp. 14941497.
 Discrete Hodge operator is constructed and its significance
in finiteelement schemes is discussed. Finiteelement schemes can
be nicely formulated in terms of discretizations of differential forms
i.e. Whitney elements (this partly explains
the popularity of differential forms among numerical simulation
scientists).
C. Mattiussi: "An Analysis of Finite Volume, Finite Element, and
Difference Methods Using Some Concepts from Algebraic Topology,"
J. Comp. Phys., vol. 133, 1997, pp. 289309.
 Algebra of forms with its de Rham operator is
a standard example of cohomology in practice.
W. Schwalm, B. Moritz, M. Giona, M. Schwalm:
"Vector difference calculus for physical lattice models,"
Phys. Rev. E, vol. 59, no. 1, 1999, pp. 12171233.

Numerical methods, PDEs (not just Maxwell).
F. L. Teixeira, W. C. Chew: "Differential forms,
metrics,
and the reflectionless absorption of electromagnetic waves,"
J. Electrom. Waves Appls., vol.13,
no.5, 1999, pp .665686.
F. L. Teixeira,W. C. Chew:
"Lattice electromagnetic theory from
a topological viewpoint,"
J. Math. Phys., vol.40, no.1, Jan. 1999, pp.169187.
Perttu Puska
is with HUT
Electromagnetics Laboratory
HUT mirror site
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