# American Institute of Mathematical Sciences

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

## The co-divergence of vector valued currents

 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

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.
Citation: Reuven Segev, Lior Falach. The co-divergence of vector valued currents. Discrete & 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, 266 (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. [4] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations. I. Cartesian Currents,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 37 (1998). [5] R. S. Palais, "Foundations of Global Non-Linear Analysis,'', W. A. Benjamin, (1968). [6] R. S. Palais, "The Geometrization of Physics,'', Lecture notes from a course at National Tsing Hua University, (1981). [7] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory,, Journal of Elasticity, 71 (2003), 183. 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. 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, 266 (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. [4] M. Giaquinta, G. Modica and J. Souček, "Cartesian Currents in the Calculus of Variations. I. Cartesian Currents,'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 37 (1998). [5] R. S. Palais, "Foundations of Global Non-Linear Analysis,'', W. A. Benjamin, (1968). [6] R. S. Palais, "The Geometrization of Physics,'', Lecture notes from a course at National Tsing Hua University, (1981). [7] G. Rodnay and R. Segev, Cauchy's flux theorem in light of geometric integration theory,, Journal of Elasticity, 71 (2003), 183. 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. doi: 10.1063/1.1475347. [9] H. Whitney, "Geometric Integration Theory,'', Princeton University Press, (1957).
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