Dyadic

A linear Polynomial of Dyads ${\bf AB}+{\bf CD}+\ldots$ consisting of nine components $A_{ij}$ which transform as

$\displaystyle (A_{ij})'$	$\textstyle =$	$\displaystyle \sum_{m,n}{h_mh_n\over h_i'h_j'}{\partial x_m\over\partial x_i'}{\partial x_n\over\partial x_j'}A_{mn}$	(1)
	$\textstyle =$	$\displaystyle \sum_{m,n}{h_i'h_j'\over h_mh_n}{\partial x_i'\over\partial x_m}{\partial x_j'\over\partial x_n} A_{mn}$	(2)
	$\textstyle =$	$\displaystyle \sum_{m,n}{h_i'h_n\over h_mh_j'}{\partial x_i'\over\partial x_m}{\partial x_n\over\partial x_j'} A_{mn}.$	(3)

Dyadics are often represented by Gothic capital letters. The use of dyadics is nearly archaic since Tensors perform the same function but are notationally simpler.

A unit dyadic is also called the Idemfactor and is defined such that

$\begin{displaymath} {\bf I}\cdot{\bf A} \equiv {\bf A}. \end{displaymath}$

(4)

In Cartesian Coordinates,

$\begin{displaymath} {\bf I} = \hat {\bf x}\hat {\bf x}+\hat {\bf y}\hat {\bf y}+\hat {\bf z}\hat {\bf z}, \end{displaymath}$

(5)

and in Spherical Coordinates

$\begin{displaymath} {\bf I} = \nabla {\bf r}. \end{displaymath}$

(6)

See also Dyad, Tetradic

References

Arfken, G. ``Dyadics.'' §3.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 137-140, 1985.

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.