A.8.1 Dyadic algebra

A dyadic is an operator whose properties are most easily understood by writing it as the juxtaposition of two vectors. Thus, given two vectors A and B, the dyadic D is an operator defined by

A is called the antecedent of the dyadic, and B is called the consequent.

If A and B are written in terms of the unit vectors i ₁, i ₂, i ₃ in a Cartesian coordinate system so that

and

then the dyadic D = AB is

The pairs of unit vectors i _r i _s are called dyads. The components of the dyadic are defined by

Two dyadics are equal if their corresponding components are equal. Thus, D = D ? means that D _rs = D ? _rs. AB is called the conjugate dyadic of BA and is often designated by a subscript c. Thus, if D = AB, then its conjugate is D _c = BA.

The sum of two dyadics is found by adding their components. Thus, F = D + E means F _rs = D _rs + E _rs. A multiplier on the left of a dyadic is called a prefactor, while a multiplier on the right is called a postfactor. Pre- and post-factors can be scalars, vectors, or dyadics,...