Generalized pseudotensor formulations of the Stokes' integral theorem

Oriented continua play an important role in micropolar elasticity modelling. All realizations of micropolar theories are conceptually possible only within the framework of the pseudotensor formalism and the orientable manifold notion. This particularly concerns the theory of micropolar hemitropic elastic media. In this paper, a pseudotensor description is used in contrast to Kartan's formalism. The pseudotensor formulation of Stokes' integral theorem is almost unknown in the current scientific literature. Here we consider various formulations of Stokes' integral theorem for an arbitrary asymmetric covariant pseudotensor field of a given weight and valency. This extends the theorem to the case of pseudotensors. This fact makes it possible to use the mentioned generalization for micropolar continua. The study mostly relies on the class of special coordinate systems often employed in classical physical field theories. A procedure for orientations consistency inside and on the boundary of a manifold is discussed for various formulations of Stokes' integral theorem.