Abstract
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra \({\mathbb{G}^3 \equiv C\ell_{3,0}}\) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford—for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan—for applications of geometric calculus.
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References
Ablamowicz, R., Fauser, B.: Mathematics of CLIFFORD—A Maple package for Clifford and Grassmann algebras. Adv. Appl. Clifford Algebras 15(2), 157– 181 (2005)
Abłamowicz, R, Fauser, B.: Clifford and Graßmann Hopf algebras via the BIGEBRA package for Maple. Comp. Phys. Commun. 170:115–130(2005)
Aragon-Camarasa, G., Aragon-Gonzalez, G., Aragon, J.L., Rodriguez-Andrade, MA.: Clifford algebra with mathematica. In: Rudas, I.J. ( ed.) Recent Advances in Applied Mathematics, volume 56 of Mathematics and Computers in Science and Engineering Series, pp. 64–73, Budapest, 12–14 Dec 2015. Proceedings of AMATH ’15, WSEAS Press.
Chappell, J.M., Drake, S.P., Seidel, C.L., Gunn, L.J., Iqbal, A., Allison, A., Abbott, D.: Geometric algebra for electrical and electronic engineers. 102(9), 1340–1363 (2014)
Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)
Dorst, L.: Geomteric Computing with Clifford Algebras, Chapter Honing Geometric Algebra for Its Use in the Computer Sciences. Springer, Berlin (2001)
Dorst, L., Fontijne, D, Mann, S.: Geometric Algebra for Computer Science. An Object-Oriented Approach to Geometry. Elsevier, Amsterdam (2007)
Fontijne, D.: Efficient implementation of geometric algebra. PhD Thesis, University of Amsterdam (2007)
Gsponer, A., Hurni, J.P.: The physical heritage of Sir W.R. Hamilton. “The Mathematical Heritage of Sir William Rowan Hamilton” Trinity College, Dublin, 17th–20th August, 1993 (1993)
Lasenby A., Doran C., Gull S.: A multivector derivative approach to lagrangian field theory. Found. Phys. 23(10), 1295–1327 (1993)
Maxima Project. http://maxima.sourceforge.net/docs/manual/maxima.html. Maxima 5.35.1 Manual (2014)
Porteus,I.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge, 2 edn (2000)
Toth,V.: Tensor manipulation in GPL Maxima. arXiv:cs/0503073 (2007)
van Vlaenderen, K.J., Waser, A.: Generalisation of classical electrodynamics to admit a scalar field and longitudinal waves. Hadron. J. 609–628 (2001)
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Prodanov, D., Toth, V.T. Sparse Representations of Clifford and Tensor Algebras in Maxima. Adv. Appl. Clifford Algebras 27, 661–683 (2017). https://doi.org/10.1007/s00006-016-0682-x
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DOI: https://doi.org/10.1007/s00006-016-0682-x