Material description of fluxes in terms of differential forms

Abstract

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.