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

• Vector and dyadic algebra
• Clifford algebras and
• related algebras: Quaternions and Octonions
• Exterior algebras; Differential forms

Vector and dyadic algebra

General

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.

• 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. By-the-way, 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

• Intermediate level dyadics. Out of print for years.

I. Lindell, Methods for Electromagnetic Field Analysis, Oxford: Clarendon Press, 1992. Republished by IEEE 1995.

• Dyadics galore.

Clifford algebras

General

The Geometric Algebra Research Group at Cavendish Laboratory has an on-line intro and several down-loadable ps-format 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

W. E. Baylis, J. Huschilt, Jiansu Wei: "Why i?," Am. J. Phys. vol. 60, no. 9, pp. 788-797, 1992.

E. F. Bolinder: "Clifford Algebra, What is it?," IEEE Antennas and Propagation Society Newsletter, August, pp. 18-23, 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. 491-513, 1993.

A. Lewis, a ps-format intro can be down-loaded from his webpage. Also links to other intros, such as the one written by R. Harke.

C. Rodriguez has a compact on-line intro about Clifford algebra.

T. Smith, an on-line intro not entirely restricted to the subject of Clifford algebra.

Intermediate to advanced

• 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.

• The book covers typical material of intermediate electromagnetics (or electrodynamics) course and some advanced topics, but all these are developed in Cl(3,0)! This is possible because Cl(3,0) is rich enough in structure: For instance, a space-time point (t,x,y,z) is represented as a paravector:

        x = t + x e1 + y e2 + z e3. (1)
  

  Note that time is just a scalar parameter and many of the familiar Gibbs-Heaviside vector 'algebra' ideas are carried over to paravector algebra. This approach Baylis believes to be paedagogically better than the more usual Cl(1,3) or Cl(3,1) formulation of space-time. Now how does the metric of space-time arise from (1)? Consider the (Clifford) product

      _              
    x x =
  (t+xe1+ye2+ze3)(t-xe1-ye2-ze3) =
  t^2 - x^2 - y^2 -z^2,    (2)
  
  

  where 'bar' operation - denotes Clifford conjugation, that is, combination of reversion and grade involution (we are using Lounesto's notation here). Thus (2) gives the correct signature and is a valid candidate for a scalar product in space-time.

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.

• 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

• Analysis branch of the research. More into analysis is also

• PDEs and Clifford algebra. Contains also a short section on electromagnetics.

D. Hestenes, Space-time Algebra, Gordon & Breach, New York, 1966.

• Standard reference. Especially noteworthy for its treatment on energy-momentum (Maxwell's stress-energy) tensor: It is a pleasant fact that in Clifford algebra (vacuum) energy-momentum tensor can be simply written as

           T_ij =  scalar part of (F ei F ej),
       

  where ek are the basis vectors of Cl(1,3) (or Cl(3,1)) (This great discovery was first made by M. Riesz in 1946) These as well as some other EM-related points are discussed in this work. An update of the book is coming?

D. Hestenes, and G. Sobczyk, Clifford Algebra to Geometric Calculus, Reidel, Dordrecht, 1984, reprint with corrections 1992.

• Expands the work started in Space-time 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. 141-158, 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 side-by-side, 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, 25-28 May 1998, Thessaloniki, Greece. pp. 822-824.

• Hypercomplex analysis.

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.

• Cf. discussion earlier.

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.

Ftp-site by H. Baker about historical quaternion papers.

Introductory texts and tutorials

Geoffrey Dixon's site about octonions and related physical applications/implications. Conference announcements. A concise tutorial also available.

Intermediate to advanced

Exterior algebras; Differential forms

General

G. A. Deschamps: "Electromagnetics and differential forms", Proc. IEEE, vol. 69, pp. 676-696, June 1981.

• An article which started the differential form boom.

G. A. Deschamps: "Exterior differential forms," pp. 112-161, in E. Roubine (ed): Mathematics Applied to Physics, Springer-Verlag, 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. 37-66, 1886. His approach looks very 'dyadescian', an interesting and relevant critique of his approach is recorded in the pages 635-652 of Bull. A.M.S., vol. 78, 1972, in F.Dyson's "Missed Opportunities,". See also pp. 14-15 of preprint by J.Parra.

Introductory texts and tutorials

• Mathematician introduces salient features in a very concise manner.

Intermediate to advanced

• 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 never-the-less recommendable also for those readers who want to put their understanding of electromagnetism in to a broader context.

• 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 pseudo-forms. 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 Biot-Savart Laws in Anisotropic Media,'' Ann. Phys. (N.Y.), vol. 245, 227-274, 1996.

• Author introduces pseudo-multivectors 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. 366-400, is self-contained 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, Springer-Verlag, 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.

• The first one that I have seen to utilize differential forms in FDTD. Interested? Then you need also K.F. Warnick, D.V. Arnold:"Green forms for anisotropic inhomogenous media", J. Electrom. Waves Appls., vol. 12, pp. 1145-1164, 1997.

J. W. Wheeler, Geometrodynamics, Academic Press, London 1962.

• Physics as geometry.

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.

• Whitney elements for FDTD. An interesting paper by A. Bossavit, and L. Kettunen, director of the Tampere U. Tech. (TUT) computational electromagnetics group. From this TUT group comes also the following paper,

• Discrete Hodge operator is constructed and its significance in finite-element schemes is discussed. Finite-element 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. 289-309.

• 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. 1217-1233.

• Numerical methods, PDEs (not just Maxwell).

F. L. Teixeira,W. C. Chew: "Lattice electromagnetic theory from a topological viewpoint," J. Math. Phys., vol.40, no.1, Jan. 1999, pp.169-187.

is with HUT Electromagnetics Laboratory