ON THEORY OF ORIENTED TENSOR ELEMENTS OF AREA FOR A MICROPOLAR CONTINUUM IMMERSED IN AN EXTERNAL PLANE SPACE

The paper deals with the problems of the tensor element of volume (area) of an M -cell for a manifold immersed in a “plane” multidimensional space. The corresponding considerations imply the choice of the reper orientation, which determines the volume (area) element. The latter circumstance is of exceptional importance in micropolar theories of elasticity. The indicated theories can be correctly developed only within the framework of the pseudotensor formalism. This is especially true for the theory of hemitropic elastic continua. The governing information from the algebra of pseudotensors is presented. Manifolds given by Gaussian parametrization are considered. The concept of an M -cell on a manifold and its orientation is introduced. An algorithm for comparing the orientations of M -dimensional cells is described. The concept of the tensor element of volume (area) for the M - cell is determined. The formula for transformation a natural volume element to an invariant volume element is shown. Important applications for the mechanics of the micropolar continuum are noted and discussed.