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Dyadic

A dyadic, also known as a vector direct product, is a linear polynomial of dyads AB+CD+... consisting of nine components A_(ij) which transform as

(A_(ij))^'=sum_(m,n)(h_mh_n)/(h_i^'h_j^')(partialx_m)/(partialx_i^')(partialx_n)/(partialx_j^')A_(mn)
(1)
=sum_(m,n)(h_i^'h_j^')/(h_mh_n)(partialx_i^')/(partialx_m)(partialx_j^')/(partialx_n)A_(mn)
(2)
=sum_(m,n)(h_i^'h_n)/(h_mh_j^')(partialx_i^')/(partialx_m)(partialx_n)/(partialx_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

I·A=A.
(4)

In Cartesian coordinates,

I=x^^x^^+y^^y^^+z^^z^^,
(5)

and in spherical coordinates

I=del r.
(6)

See also

Dyad, Tensor, Tetradic, Vector Direct Product

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References

Arfken, G. "Dyadics." §3.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 137-140, 1985.Jeffreys, H. and Jeffreys, B. S. "Dyadic Notation." §3.04 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 89, 1988.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.

Referenced on Wolfram|Alpha

Dyadic

Cite this as:

Weisstein, Eric W. "Dyadic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dyadic.html

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