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 RhamI 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 the reader is advised to consult all categories in order to find suitable texts.
 Vector and dyadic 'algebra'
 Clifford algebras and
 related algebras: Quaternions and Octonions
 Exterior algebras; Differential forms
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.
 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.
 Dyadics galore.
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 Birkhäuser, 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 spacetime 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 GibbsHeaviside 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 spacetime. Now how does the Minkowski metric arise from (1)? Consider the (Clifford) product_ x x = (t+xe1+ye2+ze3)(txe1ye2ze3) = 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 spacetime.
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, Spacetime Algebra, Gordon & Breach, New York, 1966.
 Standard reference. Especially noteworthy for its treatment
on energymomentum (Maxwell's stressenergy) tensor:
It is a pleasant fact that in Clifford algebra
(vacuum) energymomentum 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 EMrelated 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 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," Found. Phys. vol. 23, no. 11, 14051421, 1993.
 Hilbert space of the vacuum electromagnetic field rendered in Cl(3,0).
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. 127147, 1932.
 To the best of my knowledge, Juvet and Schidlof were the first authors to write Maxwell equations as a single equation in Clifford algebra. However, they worked with C x Cl(4,0) (Minkowski space with x_4=i ct, where i is the imaginary unit of the field C) and thus did not fully utilize the potential of Clifford algebra (a more natural choice would have been Cl(3,1)/Cl(1,3) or Cl(3,0) with paravectors, cf. Baylis).
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.
 Hypercomplex analysis and electromagnetics.
P. Lounesto,
Clifford Algebras and Spinors,
Cambridge University Press, Cambridge, 1997.
 Pertti Lounesto died on Friday 2002 June 21 in Greece. His webpages are no longer available on Helsinki Polytechnic Stadia's website. Here you can find locally mirrored copies of Pertti's webpages. 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.)
P. Puska "Covariant Isotropic Constitutive Relations in Clifford's Geometric Algebra," PIER 32, EMW Publishing, Cambridge, 2001.
 My own contribution (Some silly typos always slip in, here is an errata.)
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.
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.
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 online 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 16901990: Theory and Applications, H. Blok, H.A. Ferwerda, H.K. Kuiken (editors), NorthHolland, Amsterdam, 1992. An excursion into biquaternions ( i.e. C x H, isomorphic to Cl(3,0) ) by the microwave optics master.
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 ofG. A. Deschamps: "Electromagnetics and differential forms", Proc. IEEE, vol. 69, pp. 676696, June 1981.
 An article which started the differential form boom.
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.
 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 sähkömagnetiikassa (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, and very likely you have heard of this one. 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.
 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. 11451164, 1997.
J. W. Wheeler, Geometrodynamics, Academic Press, London 1962.
 Physics as geometry.
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: "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.
 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,
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.
[*]This one is mostly about oldfashioned, precomputer electromagnetics in Clifford algebra. But at least the "geometrical methods" is in common with the other articles. Bytheway, check the errata.