Dyad

Dyads extend Vectors to provide an alternative description to second Rank Tensors. A dyad $D({\bf A},{\bf B})$ of a pair of Vectors ${\bf A}$ and ${\bf B}$ is defined by $D({\bf A},{\bf B}) \equiv {\bf A}{\bf B}$. The Dot Product is defined by

$${\bf A}\cdot {\bf BC} \equiv ({\bf A}\cdot {\bf B}){\bf C}$$

$${\bf AB}\cdot {\bf C} \equiv {\bf A}({\bf B}\cdot {\bf C}),$$

and the Colon Product by

$${\bf AB}:{\bf CD} \equiv {\bf C}\cdot {\bf AB}\cdot {\bf D} = ({\bf A}\cdot{\bf C})({\bf B}\cdot {\bf D}).$$

References

Morse, P. M. and Feshbach, H. ``Dyadics and Other Vector Operators.'' §1.6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 54-92, 1953.