The co-divergence of vector valued currents

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March 2012, 17(2): 687-698. doi: 10.3934/dcdsb.2012.17.687

The co-divergence of vector valued currents

Reuven Segev ^1, and Lior Falach ^1,

1.	Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva, 84105, Israel, Israel

Received June 2010 Revised July 2011 Published December 2011

Abstract
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In the context of stress theory of the mechanics of continuous media, a generalization of the boundary operator for de Rham currents---the co-divergence operator---is introduced. While the boundary operator of de Rham's theory applies to real valued currents, the co-divergence operator acts on vector valued currents, i.e., functionals dual to differential forms valued in a vector bundle. From the point of view of continuum mechanics, the framework presented here allows for the formulation of the principal notions of continuum mechanics on a manifold that does not have a Riemannian metric or a connection while at the same time allowing irregular bodies and velocity fields.

Keywords: balance equations, de Rham currents, Continuum mechanics, boundary operator, vector valued currents, differential operators..

Mathematics Subject Classification: 58A25, 74A1.

Citation: Reuven Segev, Lior Falach. The co-divergence of vector valued currents. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 687-698. doi: 10.3934/dcdsb.2012.17.687

References:

[1]	G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'' Translated from the French by F. R. Smith, With an introduction by S. S. Chern, Grundlehren der Mathematischen Wissenschaften, 266, Springer-Verlag, Berlin, 1984.
[2]	H. Federer, "Geometric Measure Theory,'' Springer-Verlag, 1969.
[3]	V. M. Gol'dshteĭn, V. I. Kuz'minow and I. A. Shvedov, Differential forms on Lipschitz manifold, Sibirskii Matematicheskii Zhurnal, 23 (1982), 16-30, 215.
[4]	M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations. I. Cartesian Currents,'' Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, A Series of Modern Surveys in Mathematics, 37, Springer-Verlag, Berlin, 1998.
[5]	R. S. Palais, "Foundations of Global Non-Linear Analysis,'' W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[6]	R. S. Palais, "The Geometrization of Physics,'' Lecture notes from a course at National Tsing Hua University, Hsinchu, Taiwan, 1981.
[7]	G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory, Journal of Elasticity, 71 (2003), 183-203. doi: 10.1023/B:ELAS.0000005545.46932.08.
[8]	R. Segev, Metric-independent analysis of the stress-energy tensor, Journal of Mathematical Physics, 43 (2002), 3220-3231. doi: 10.1063/1.1475347.
[9]	H. Whitney, "Geometric Integration Theory,'' Princeton University Press, 1957.

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References:

[1]	G. de Rham, "Differentiable Manifolds. Forms, Currents, Harmonic Forms,'' Translated from the French by F. R. Smith, With an introduction by S. S. Chern, Grundlehren der Mathematischen Wissenschaften, 266, Springer-Verlag, Berlin, 1984.
[2]	H. Federer, "Geometric Measure Theory,'' Springer-Verlag, 1969.
[3]	V. M. Gol'dshteĭn, V. I. Kuz'minow and I. A. Shvedov, Differential forms on Lipschitz manifold, Sibirskii Matematicheskii Zhurnal, 23 (1982), 16-30, 215.
[4]	M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations. I. Cartesian Currents,'' Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, A Series of Modern Surveys in Mathematics, 37, Springer-Verlag, Berlin, 1998.
[5]	R. S. Palais, "Foundations of Global Non-Linear Analysis,'' W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[6]	R. S. Palais, "The Geometrization of Physics,'' Lecture notes from a course at National Tsing Hua University, Hsinchu, Taiwan, 1981.
[7]	G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory, Journal of Elasticity, 71 (2003), 183-203. doi: 10.1023/B:ELAS.0000005545.46932.08.
[8]	R. Segev, Metric-independent analysis of the stress-energy tensor, Journal of Mathematical Physics, 43 (2002), 3220-3231. doi: 10.1063/1.1475347.
[9]	H. Whitney, "Geometric Integration Theory,'' Princeton University Press, 1957.

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