On the Helmholtz decomposition for polyadics
Authors:
George Dassios and Ismo V. Lindell
Journal:
Quart. Appl. Math. 59 (2001), 787-796
MSC:
Primary 35J05; Secondary 15A69, 35J15, 35J25
DOI:
https://doi.org/10.1090/qam/1866557
MathSciNet review:
MR1866557
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Abstract: A polyadic field of rank $n$ is the tensor product of $n$ vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank $n$ is, again nonuniquely, decomposable into a term consisting of $n$ successive applications of the gradient to a scalar function, plus a term that consists of $\left ( n - 1 \right )$ successive applications of the gradient to the rotation of a vector function, plus a term that consists of $\left ( n - 2 \right )$ successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last $\left ( n + 1 \right )$th term, which consists of $n$ successive applications of the rotation operator to a polyadic function of rank $n$. Obviously, the $n = 1$ case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to ${n^{2}}$, are presented.
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L. Brand, Vector and Tensor Analysis, John Wiley and Sons, New York, London, 1947
P. Chadwick and E. A. Trowbridge, Elastic wave fields generated by scalar wave functions, Proc. Cambridge Philos. Soc. 63, 1177–1187 (1967)
P. Ciarlet, A decomposition of ${L^{2}}{\left ( \Omega \right )^{3}}$ and an application to magnetostatic equations, Math. Mod. Meth. Appl. Sci. 3, 289–301 (1993)
G. Dassios and R. Kleinman, Low Frequency Scattering, Oxford University Press, 2000
G. Dassios, K. Kiriaki, and D. Polyzos, Scattering theorems for complete dyadic fields, Internat. J. Engrg. Sci. 33, 269–277 (1995)
R. A. Eubanks and E. Sternberg, On the completeness of the Boussinesq-Papkovich stress functions, J. Rat. Mech. Anal. 5, 735–746 (1956)
B. Galerkin, Contribution à la solution générale du problème de la théorie de l’élasticité dans la cas de trois dimensions, Compt. Rend. 190, 1047–1048 (1930)
J. W. Gibbs, Vector Analysis, Yale University Press, New Haven, 1901
M. E. Gurtin, The linear theory of elasticity, in Encyclopaedia of Physics (C. Truesdell, editor), Vol. VIa/2, Springer-Verlag, Berlin, 1972
R. P. Kanwal, The existence and completeness of various potentials for the equations of Stokes flow, Internat. J. Engrg. Sci. 9, 375–386 (1971)
J. B. Keller, Simple proofs of the theorems of J. S. Lomont and H. E. Moses on the decomposition and representation of vector fields, Comm. Pure Appl. Math. 14, 77–80 (1960)
W. Kratz, On the representation of Stokes flows, SIAM J. Math. Anal. 22, 414–423 (1991)
I. V. Lindell, Methods for Electromagnetic Field Analysis, 2nd ed., Oxford University Press, 1995
J. S. Lomont and H. E. Moses, An angular momentum Helmholtz theorem, Comm. Pure Appl. Math. 14, 69–76 (1960)
J. Mathews, Gravitational multipole radiation, J. Soc. Indust. Appl. Math. 10, 768–780 (1962)
H. Neuber, Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie, Z. Angew. Math. Mech. 14, 203–212 (1934)
D. Palaniappan, S. D. Nigam, T. Amaranath, and R. Usha, Lamb’s solution of Stokes’s equations: A sphere theorem, Quart. J. Mech. Appl. Math. 45, 47–56 (1992)
P. F. Papkovich, The representation of the general integral of the fundamental equations of elasticity theory in terms of harmonic functions, Izv. Akad. Nauk SSSR, Ser. Mat. 10, 1425–1435 (1932) (in Russian)
P. F. Papkovich, Solution générale des équations différentielles fondamentales d’élasticité exprimée par trois fonctions harmoniques, Compt. Rend. 195, 513–515 (1932)
D. A. W. Pecknold, On the role of the Stokes-Helmholtz decomposition in the derivation of displacement potentials in classical elasticity, J. Elast. 1, 171–174 (1971)
E. Sternberg, On the integration of the equations of motion in the classical theory of elasticity, Arch. Rat. Mech. Anal. 6, 34–50 (1960)
T. Tran-Cong and J. R. Blake, General solutions of the Stokes’ flow equations, J. Mat. Anal. Appl. 90, 72–84 (1982)
X. Xinsheng and W. Minzhong, General complete solutions of the equations of spatial and axisymmetric Stokes flow, Quart. J. Mech. Appl. Math. 44, 537–548 (1991)
D. A. Woodside, Uniqueness theorems for classical four-vector fields in Euclidean and Minkowski spaces, J. Math. Phys. 40, 4911–4943 (1999)
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© Copyright 2001
American Mathematical Society