Notes on Metric Independent Analysis of Classical Fields

Abstract

A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential .n 1/-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.

Material description of fluxes in terms of differential forms

“…(n−r )!r ! , where n = dimV [7,24]. With the help of the Tartaglia-Pascal triangle, the scheme below reports the dimension of r (V) for every order r and for every dimension n of the "mother" space V. dimV:…”

“…Because of this analogy, and of the fact that it is well established to study electromagnetism in terms of forms (see, e.g. [14,18,24]), it is interesting to report a treatment of Darcy's law too in this geometric formalism.…”

“…Therefore, the pull-back laws of pseudo-vectors and pseudo-scalars differ from those of vectors and scalars (the latter being invariant), respectively, by a multiplicative factor equal to the determinant of the deformation gradient. In fact, in the customary three-dimensional case, a pseudo-vector represents a secondorder differential form (or two-form) and a pseudo-scalar represents a (non-vanishing) third-order differential form (or non-vanishing three-form, or volume form) [7,24]. Differential forms are skew-symmetric covariant tensors and follow the regular transformation rules of tensors.…”

“…Exhaustive introductions to r -forms and spaces of r -forms can be found, e.g. in the works by Epstein [7] and Segev [24].…”

Continuum mechanics, stresses, currents and electrodynamics

“…For example, it may be shown [13,17] that, in the case r = n − 1, a differentiable traction stress density may be represented by a section of L(TS , TS ), a genuine tensor field.…”