volume | PIER Journals

Abstract

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.

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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.
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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.
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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.
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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.
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49. Caro, P. and T. Zhou, "On global uniqueness for an IBVP for the time-harmonic Maxwell equations," Mathematical Physics, 1210.7602, 2012.

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51. Deschamps, G., "Electromagnetics and differential forms," Proceedings of the IEEE, 676-696, June 1981.
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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.
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Dr. Karl Warnick

Professor Peter H. Russer