Mathematical frameworks for representing fields and waves and expressing Maxwell's equations of electromagnetism include vector calculus, differential forms, dyadics, bivectors, tensors, quaternions, and Clifford algebras. Vector notation is by far the most widely used, particularly in applications. Of the more sophisticated notations, differential forms stand out as being close enough to vectors that most practitioners can readily understand the notation, yet at the same time offering unique visualization tools and graphical insight into the behavior of fields and waves. We survey recent papers and book on differential forms and review the basic concepts, notation, graphical representations, and key applications of the differential forms notation to Maxwell's equations and electromagnetic field theory.
2. Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 2, Oxford University Press, New York, 1998.
3. Grifths, H., "Oliver heaviside," History of Wireless, 1st Edition, 229-246, T. K. Sarkar, R. Mailloux, and A. A. Oliner, Eds., Wiley & Sons, Hoboken, New Jersey, 2006.
4. Grassmann, H. and L. Kannenberg, A New Branch of Mathematics: The ``Ausdehnungslehre" of 1844 and Other Works, Open Court Publishing, Chicago, 1995.
5. Cartan, E., Les Systemes Differentielles Exterieurs, Hermann, Paris, 1945.
6. Miller, A. I., Imagery in Scientic Thought, Birkhauser, Boston, 1984.
7. Flanders, H., Differential Forms with Applications to the Physical Sciences, Dover, New York, 1963.
8. Misner, C., K. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisc, 1973.
9. Thirring, W., Classical Field Theory, Vol. 2, 2nd Edition, Springer-Verlag, New York, 1978.
10. Deschamps, G. A., "Electromagnetics and differential forms," IEEE Proc., Vol. 69, 676-696, June 1981.
doi:10.1109/PROC.1981.12048
11. Burke, W. L., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985.
doi:10.1017/CBO9781139171786
12. Weck, N., "Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries," J. Math. Anal. Appl., Vol. 46, 410-437, 1974.
doi:10.1016/0022-247X(74)90250-9
12. Sasaki, I. and T. Kasai, "Algebraic-topological interpretations for basic equations of electromagnetic fields," Bull. Univ. Osaka Prefecture A, Vol. 25, No. 1-2, 49-57, 1976.
14. Schleifer, N., "Differential forms as a basis for vector analysis --- With applications to electrodynamics," Am. J. Phys., Vol. 51, 1139-1145, December 1983.
doi:10.1119/1.13325
15. Burke, W. L., "Manifestly parity invariant electromagnetic theory and twisted tensors ," J. Math. Phys., Vol. 24, 65-69, January 1983.
doi:10.1063/1.525603
16. Engl, W. L., "Topology and geometry of the electromagnetic field," Radio Sci., Vol. 19, 1131-1138, September-October 1984.
17. Baldomir, D., "Differential forms and electromagnetism in 3-dimensional Euclidean space R3," IEE Proc., Vol. 133, 139-143, May 1986.
18. Karloukovski, V. I., "On the formulation of electrodynamics in terms of differential forms," Annuaire de l'Universitede SoaFacultede Physique, Vol. 79, 3-12, 1986.
19. Baldomir, D. and P. Hammond, "Global geometry of electromagnetic systems," IEE Proc., Vol. 140, 142-150, March 1992.
20. Ingarden, R. S. and A. Jamiokowksi, Classical Electrodynamics, Elsevier, Amsterdam, The Netherlands, 1985.
21. Bamberg, P. and S. Sternberg, A Course in Mathematics for Students of Physics, Vol. 2, Cambridge University Press, Cambridge, 1988.
doi:10.1017/CBO9781139171670
22. Parrott, S., Relativistic Electrodynamics and Differential Geometry, Springer-Verlag, New York, 1987.
doi:10.1007/978-1-4612-4684-8
23. Frankel, T., The Geometry of Physics, Cambridge University Press, Cambridge, 1997.
24. Weintraub, S., Differential Forms --- A Complement to Vector Calculus, Academic Press, New York, 1997.
25. Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, Artech House, Boston, 2003.
26. Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering , 2nd edition, Artech House, Boston, 2006.
27. Warnick, K. F. and P. Russer, Problem Solving in Electromagnetics, Microwave Circuit, and Antenna Design for Communications Engineering, Artech House, Norwood, MA, 2006.
28. Hehl, F. W. and Y. N. Obukov, Foundations of Classical Electrodynamics, Birkhauser, Boston, Basel, Berlin, 2003.
doi:10.1007/978-1-4612-0051-2
29. Lindell, I. V., Differential Forms in Electromagnetics, IEEE Press, New York, 2004.
doi:10.1002/0471723096
30. Warnick, K. F., R. H. Selfridge, and D. V. Arnold, "Electromagnetic boundary conditions using differential forms," IEE Proc., Vol. 142, No. 4, 326-332, 1995.
31. Warnick, K. F. and P. Russer, "Two, three, and four-dimensional electromagnetics using differential forms," Turkish Journal of Electrical Engineering and Computer Sciences, Vol. 14, No. 1, 153-172, 2006.
32. Warnick, K. F. and P. Russer, "Green's theorem in electromagnetic field theory," Proceedings of the European Microwave Association, Vol. 12, 141-146, June 2006.
33. Warnick, K. F. and D. V. Arnold, "Electromagnetic Green functions using differential forms," Journal of Electromagnetic Waves and Applications, Vol. 10, No. 3, 427-438, 1996.
doi:10.1163/156939396X00504
34. Warnick, K. F. and D. V. Arnold, "Green forms for anisotropic, inhomogeneous media," Journal of Electromagnetic Waves and Applications, Vol. 11, No. 8, 1145-1164, 1997.
doi:10.1163/156939397X01061
35. Nguyen, D. B., "Relativistic constitutive relations, differential forms, and the p-compound," Am. J. Phys., Vol. 60, 1137-1147, December 1992.
doi:10.1119/1.16962
36. Warnick, K. F., R. H. Selfridge, and D. V. Arnold, "Teaching electromagnetic field theory using differential forms," IEEE Trans. Educ., Vol. 40, No. 1, 53-68, 1997.
doi:10.1109/13.554670
37. Mingzhong, R., T. Banding, and H. Jian, "Differential forms with applications to description and analysis of electromagnetic problems," Proc. CSEE, Vol. 14, 56-62, September 1994.
38. Picard, R., "Eigensolution expansions for generalized Maxwell fields on C0;1-manifolds with boundary," Applic. Anal., Vol. 21, 261-296, 1986.
doi:10.1080/00036818608839597
39. Bossavit, A., "Differential forms and the computation of fields and forces in electromagnetism," Eur. J. Mech. B, Vol. 10, No. 5, 474-488, 1991.
40. Hammond, P. and D. Baldomir, "Dual energy methods in electromagnetics using tubes and slices," IEE Proc., Vol. 135, 167-172, March 1988.
41. Hiptmair, R., "Multigrid method for Maxwell's equations," SIAM Journal on Numerical Analysis, Vol. 36, No. 1, 204-225, 1998.
doi:10.1137/S0036142997326203
42. Castillo, P., R. Rieben, and D. White, "FEMSTER: An object-oriented class library of high-order discrete differential forms," ACM Transactions on Mathematical Software (TOMS), Vol. 31, No. 4, 425-457, 2005.
doi:10.1145/1114268.1114269
43. Desbrun, M., E. Kanso, and Y. Tong, "Discrete differential forms for computational modeling," Discrete Di®erential Geometry, 287-324, Springer, 2008.
doi:10.1007/978-3-7643-8621-4_16
44. Buffa, A., J. Rivas, G. Sangalli, and R. Vazquez, "Isogeometric discrete differential forms in three dimensions," SIAM Journal on Numerical Analysis, Vol. 49, No. 2, 818-844, 2011.
doi:10.1137/100786708
45. Teixeira, F. and W. Chew, "Differential forms, metrics, and the reflectionless absorption of electromagnetic waves," Journal of Electromagnetic Waves and Applications, Vol. 13, No. 5, 665-686, 1999.
doi:10.1163/156939399X01104
46. Trautman, A., "Deformations of the hodge map and optical geometry," JGP, Vol. 1, No. 2, 85-95, 1984.
47. Warnick, K. F., A differential forms approach to electromagnetics in anisotropic media, Ph.D. Thesis, Brigham Young University, Provo, UT, 1997.
48. Teixeira, F. L., H. Odabasi, and K. F. Warnick, "Anisotropic metamaterial blueprints for cladding control of waveguide modes," JOSAB, Vol. 27, No. 8, 1603-1609, 2010.
doi:10.1364/JOSAB.27.001603
49. Caro, P. and T. Zhou, "On global uniqueness for an IBVP for the time-harmonic Maxwell equations," Mathematical Physics, 1210.7602, 2012.
50. Russer, P., M. Mongiardo, and L. B. Felsen, "Electromagnetic field representations and computations in complex structures III: Network representations of the connection and subdomain circuits," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 15, No. 1, 127-145, 2002.
doi:10.1002/jnm.435
51. Deschamps, G., "Electromagnetics and differential forms," Proceedings of the IEEE, 676-696, June 1981.
doi:10.1109/PROC.1981.12048
52. Tellegen, B., "A general network theorem with applications," Philips Research Reports, Vol. 7, 259-269, 1952.
53. Peneld, P., R. Spence, and S. Duinker, Tellegen's Theorem and Electrical Networks, MIT Press, Cambridge, Massachusetts, 1970.
54. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.
55. Harrington, R. F., Time Harmonic Electromagnetic Fields, McGraw-Hill, New York, 1961.
56. Kong, J. A., Electromagnetic Wave Theory, Wiley-Interscience, 1986.
57. Elliott, Electromagnetics --- History, Theory, and Applications, IEEE Press, New York, 1991.
58. Collin, R. E., Field Theory of Guided Waves, IEEE Press, New York, 1991.
59. De Rham, G., Differentiable Manifolds, Springer, New York, 1984.
doi:10.1007/978-3-642-61752-2
Submit
