Last update: 30th March 2006,

Keywords: Electromagnetics, electromagnetism, algebra, algebras, vector, vectors, dyad, dyads, dyadic, dyadics, polyadic, multivector, multivectors, bivector, bivectors, Clifford algebra, geometric algebra, geometric calculus, quaternion, quaternions, hypercomplex, exterior calculus, exterior algebra, differential forms, exterior forms, spinor, spinors, tensor, tensors, Maxwell, Grassmann, Hamilton, Cayley, Clifford, Gibbs, Heaviside, Cartan, de Rham

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.


Note that classification is quite artificial, for it is more-often-than-not difficult to label piece of writing as belonging to 'exterior algebras' or 'clifford algebras'. So the reader is advised to consult all categories in order to find suitable texts.

Vector and dyadic 'algebra'


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.

O. Heaviside,Electromagnetic Theory, Ernest Benn, London, 1925.

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, McGraw-Hill, New York, 1983.

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

Clifford algebras


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 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

D. Hestenes: "Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics, " Am. J. Phys. vol. 39, no. 9, pp. 1013-1027, 1971.

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.

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

W. E. Baylis, and G. Jones, ``The Pauli algebra approach to special relativity,'' J. Phys. A, vol. 22, no. 1, pp. 1-15, 1989.

W. E. Baylis, Electrodynamics, A Modern Geometric Approach Birkhäuser, Boston 1999.

I. M. Benn and R. W. Tucker,An introduction to Spinors and Geometry, Adam Hilger, London 1987.

E. F. Bolinder: ``Unified microwave network theory based on Clifford algebra in Lorentz space,'' 12th European Microwave Conference, Helsinki, 25-35, 1982. Microwave Exhibitions and Publishers, Turnbridge Wells, Kent, 1982.

F. Brackx, R. Delanghe, F. Sommen,Clifford analysis, Pitman, London 1982.

R. Delanghe, F. Sommen and V. Soucek, Clifford Algebra and Spinor-Valued Functions, Kluwer Academic Publisher, Dordrecht/Boston, 1992. K. Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley, Chichester, 1997.

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

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

P. Hillion, ``Constitutive relations and Clifford algebra in electromagnetism,'' Adv. in Appl. Cliff. Alg. vol. 5, no. 2, pp. 141-158, 1995.

D. A. Hurley, M. A. Vandyck, Geometry Spinors and Applications, Springer and Praxis Publishing, Chichester, 2000.

B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics, World Scientific, Singapore, 1988.

B. Jancewicz: "A Hilbert space for the classical electromagnetic field," Found. Phys. vol. 23, no. 11, 1405-1421, 1993.

G. Juvet, A. Schidlof, Sur les nombres hypercomplexes de Clifford et leurs applications à l'analyse vectorielle ordinaire, à l'électromagnetisme de Minkowski et à la théorie de Dirac. Bull. Soc. Neuchat. Sci. Nat., vol. 57, pp. 127-147, 1932.

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.

P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 1997.

P. Puska "Covariant Isotropic Constitutive Relations in Clifford's Geometric Algebra," PIER 32, EMW Publishing, Cambridge, 2001.

M. Riesz, Marcel Riesz: Clifford Numbers and Spinors, Kluwer Academic Publisher, Dordrecht/Boston, 1993.

M. Riesz, "Sur certaines notions fondamentales en théorie quantique relativiste," C.R. 10e Congrès Math. Scandinaves, Copenhagen, 1946. Jul. Gjellerups Forlag, Copenhagen 1947, pp. 123-148 (You can find the article in the collected papers of M.Riesz, pp. 545-570 ).

J. Snygg,Clifford Algebra, A Computational Tool for Physicists, Oxford University Press, New York, 1997.

Hypercomplex numbers: the Quaternions ( a Cl(3,0) subalgebra ) and their bigger cousins, the Octonions.


Timeline of
hypercomplex numbers (i.e. quaternions and octonions ).

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

Clyde Davenport's pages discuss the use and application of quaternion algebra and commutative hypercomplex algebra in physics. These pages include sections on electromagnetism and special relativity.

Introductory texts and tutorials

D. Sweetser has several on-line tutorials concerning applications of quaternions.

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

Alessandro Rosa's website "Complex field & quaternion space," displays many nice 3D Mandelbrot and Julia sets computed with quaternions. (not exactly electromagnetics, but the pictures are worth looking. For a similar picure, see also the cover of "Clifford Algebras with Numeric and Symbolic Computations", R. Ablamowicz, P. Lounesto, Josep M. Parra (Eds.), Springer, New York, 1996)

Intermediate to advanced

S. Cornbleet, "An electromagnetic theory of rays in a uniform medium," in Huygens' Principle 1690-1990: Theory and Applications, H. Blok, H.A. Ferwerda, H.K. Kuiken (editors), North-Holland, Amsterdam, 1992.

Exterior algebras; Differential forms


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 on-line. 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 ps-format. 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. 676-696, June 1981.

And while we are at it, Deschamps' earlier text

G. A. Deschamps: "Exterior differential forms," pp. 112-161, in E. Roubine (ed): Mathematics Applied to Physics, Springer-Verlag, Berlin and UNESCO, Paris, 1970

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

T. Frankel: "Maxwell's equations", Am. Math. Monthly, vol. ?, pp. 343-349, April 1974.

Intermediate to advanced

J. Baez, J. P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, Singapore, 1994. D. Baldomir, P. Hammond Geometry of Electromagnetic Systems, Clarendon Press, Oxford 1996.

B. Jancewicz: ``A Variable Metric Electrodynamics. The Coulomb and Biot-Savart Laws in Anisotropic Media,'' Ann. Phys. (N.Y.), vol. 245, 227-274, 1996.

D. G. B. Edelen, Applied Exterior Calculus, Wiley, New York 1985.

I. Lindell and P. Lounesto, Differentiaalimuodot sähkömagnetiikassa (Differential forms in electromagnetics), Helsinki University of Technology, Electromagnetics laboratory report, Espoo 1995.

K. Meetz and W. L. Engl, Elektromagnetische Felder, Springer-Verlag, Berlin, 1980.

C. Misner, K. Thorne and J. Wheeler, Gravitation, Freeman, San Francisco 1973.

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. 124-126, 1999.

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

Exterior algebras in computational electromagnetics

PIER 32 concentrated on Geometrical Methods for Computational Electromagnetics. Geometrical methods in this context means algebraic topology with its many incarnations (Tonti diagrams, de Rham cohomology...) The material is available on the Progress In Electromagnetics Research book series website, with the special focus articles under PIER 32 heading. Several good papers (you might even spot the contribution of the yours truly in the last section, which is tentatively entitled as Geometric Algebra for Electromagnetics.*). Stanly Steinberg's website provides a convenient starting point for those readers who find the topics in PIER32 interesting.

A. Bossavit, L. Kettunen: "Yee-like schemes on a tetrahedral mesh, with diagonal lumping" Int. J. of Num. Mod.: ElectronicNetworks,Devices and Fields. vol.12, no.1-2, Jan.-April 1999, pp.129-142.

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. 1494-1497.

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.

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.

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 .665-686.

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.

[*]This one is mostly about old-fashioned, pre-computer electromagnetics in Clifford algebra. But at least the "geometrical methods" is in common with the other articles. By-the-way, check the errata.

Perttu Puska
HUT Electromagnetics Laboratory