Coordinate-free operators based on one vector. I. Formal considerations

Abstract

In many systems, the tensors used to describe physical properties must acquire their structure from one vector. Knowledge of that fact alone leads to an interesting line of analysis for such systems. The analysis begins with a discussion of the types of dyadics that can be constructed from one vector. Attention is focused on certain exemplary dyadic operators, which, because of their geometrical properties, would appear particularly basic; the algebra of these dyadics is developed in detail. The algebra is then used in a derivation of the expansion of general functions of these dyadics. Several applications of the general expansion are presented: the inverse of a general dyadic operator (when it exists), a general rotation operator that includes strains and reflections, the determinant of a general dyadic operator, and operators, including some new ones, used in constructing Lorentz boosts. Finally, a coordinate-free representation of the Levi-Civita tensor is provided.