Talk:Dyadics

In the definition of the double dot product I believe (ab):(xy) = (b.x)(a.y) source: Deen, William M. "Analysis of Transport Phenomena." Oxford University Press: New York, 1998. ISBN: 978-0-19-508494-8

I would change this but I don't have the software to render the expression neatly. 18.252.6.200 (talk) 00:37, 14 September 2009 (UTC)

You are correct. I will change this, as well as most of the notation on this page. As it stands, it's extremely ugly and non-standard.129.128.221.64 (talk) 18:08, 25 November 2009 (UTC)

Whew! That was enough work for me for now. I'm getting the two references mentioned here from my local library and I'll fix the notation on everything else, as well as check for correctness. This article -will- be the article people use to understand Dyads. 129.128.221.64 (talk) 18:50, 25 November 2009 (UTC)

Upon further research it seems that there are two different conventions in defining the double dot product. I'll put it in the article. 129.128.221.64 (talk) 23:19, 1 December 2009 (UTC)

Done with my edits. Hopefully this article looks a lot cleaner and makes a little more sense. I think we should probably merge the other two dyad articles with this one. 129.128.221.64 (talk) 23:30, 1 December 2009 (UTC)

Done (now in 2012?...). Maschen (talk) 23:45, 23 August 2012 (UTC)

This math is new to me. I was just wondering if there were any applications. It would be a helpful section to include. JKeck (talk) 17:46, 21 September 2011 (UTC)

Aren't the standard basis dyads at the bottom of the 3D Euclidean section transposed? — Preceding unsigned comment added by 173.25.54.191 (talk) 20:35, 9 September 2012 (UTC)

Well spotted. Fixed. — Quondum 06:51, 10 September 2012 (UTC)

The Identities section claims that the dyadic product is "compatible with inner product," but the identity given is the definition of the dot product given in the Dyadic algebra section. I'm removing the "identity" on the assumpion that it's actually a definition; if this is wrong, please let me know. Vectornaut (talk) 19:21, 18 November 2013 (UTC)

Should the article name be Dyadic per WP:SINGULAR / WP:PLURAL? --catslash (talk) 23:17, 25 February 2016 (UTC)

Agree, I think it should be either "Dyadic tensor" 145.94.184.187 (talk) 11:58, 30 April 2024 (UTC)

Agree, the page on Euclidean vectors has singular vector in the title, as well as vectors in physics and mathematics. Drlennartsson (talk) 20:06, 17 May 2025 (UTC)

Searching "double dot product" redirects here. But there are other "double dot" or "colon" products in tensor algebra which may or may not be the same as these ones here. In any case, even if they are equivalent, they are presented and defined in a totally different way, and really should have their own wikipedia article dedicated to them or at least mentioned in a page like Tensor contraction. But there seems to be no mention of them on wikipedia at all!!!

It looks like the double dot product of tensor algebra can be defined in two ways:

• As an operation that takes two rank-two tensors and gives a scalar, defined by: ${\displaystyle \mathbf {T}$ :\mathbf {U} =T_{ij}U_{ji}} . I am not 100% sure, but this may be equivalent to the first definition of the double dot product for dyadics.
• As an operation that takes two tensors in general and gives a tensor of rank two less, defined by: ${\displaystyle \mathbf {T}$ :\mathbf {U} =T_{ab\,\cdots \,hij}U_{ijk\,\cdots \,yz}} . So it's effectively contracting T and U twice. Once again, I'm not sure, but this may be equivalent to the second definition of the double dot product for dyadics when applied to two rank-two tensors.

http://physics.stackexchange.com/questions/167524/what-does-a-colon-mean-in-hydrodynamics-equations
http://math.stackexchange.com/questions/348739/double-dot-product-vs-double-inner-product
https://people.rit.edu/pnveme/EMEM851n/constitutive/tensors_rect.html
https://www.materials.uoc.gr/el/grad/courses/METY101/FLUID_DYNAMICS_CRETE.pdf
https://en.wikipedia.org/wiki/Colon_(punctuation)#Mathematics_and_logic

-- AndreRD (talk) 14:36, 29 September 2016 (UTC)

The third paragraph currently reads "...changing the order of the vectors results in a different dyadic." I think this is referring to the sequence in which the dyadic product is taken, not the order (e.g. 2nd order, 3rd order) of the dyadic tensor. Given the reference to tensor order in the preceding paragraph, I think the passage should read "...changing the sequence of the vectors results in a different dyadic." for clarity. — Preceding unsigned comment added by Kwaguirre (talk • contribs) 15:26, 23 May 2022 (UTC)

Merge? 68.134.243.51 (talk) 14:06, 21 October 2022 (UTC)

A (1-blade) bivector is the antisymmetric part of the corresponding dyad; IOW: no, they are not the same ~2025-40282-60 (talk) 19:07, 12 December 2025 (UTC)

"Dirac's bra–ket notation makes the use of dyads and dyadics intuitively clear, see Cahill (2013)."

This statement seems subjective and also irrelevant since it's commenting on the ability of a textbook to make a concept clear. However, I think that it would still be relevant to mention Dirac notation in the context of dyads/dyadics. Would the article be better served if we change this sentence to say something about how bras, kets, and their outer products, etc., can be used for dyads/dyadics?

—QB2k (talk) 07:53, 20 December 2022 (UTC)

In the beginning of the section "Three-dimensional Euclidean space", vectors a and b are defined as linear combinations of basis vectors i, j and k. The components of a are a_1, a_2 and a_3, which can be collected in a 3x1 matrix (column vector), and similarly for b. This does not mean that a is equal to this matrix of its components, nor is b equal to its component matrix. Clearly the components of the base vector i is a 3x1 matrix with components (1,0,0) and so on for the others. However, a bit down it is stated that the basis vectors i, j and k, are each equal to a matrix with its components. This goes against the text of Gibbs and Wilson, see sections 13-17 in chapter 1. Therefore it is more clear to say each basis vector is equivalent or corresponds to such a matrix of components. This applies to the basis dyads too, ii is equivalent to the matrix [ 1 0 0 ; 0 0 0 ; 0 0 0] but on the current version of the page they are equal. And so on for the other basis dyads. Drlennartsson (talk) 20:04, 17 May 2025 (UTC)

(Sorry in advanced if this isn't a good use of the "Talk" page, I'm new to Wikipedia editing.)

It is my understanding that the singular "dyad" refers to the tensor product of two vectors (as stated by the article itself). Then, the "dyadic" is any sum of dyads; by that definition, it's certainly true that an individual "dyad" does constitute a monomial dyadic, but I think there might be cases in this article where it's more readable to say "dyad" when referring to a one-term dyadic. For instance, in a section on the dyadic product:

"...not a component of the vector as in a_i), then in algebraic form their dyadic product is:

${\displaystyle \mathbf {ab} =\sum _{j=1}^{N}\sum _{i=1}^{N}a_{i}b_{j}\mathbf {e} _{i}\mathbf {e} _{j}.}$

This is known as the nonion form of the dyadic."

If I'm reading this correctly, I believe it would be more suitable in this case to use "dyad"? Also, in a similar case:

"A dyad is a component of the dyadic (a monomial of the sum or equivalently an entry of the matrix)"

it seems a bit confusing to say that a dyad is an entry of the matrix (it can be, but under the specific conditions of being in R, not R3, which the article has been concerned with prior), at least as it is written presently. Yeeee haw (talk) 22:44, 12 August 2025 (UTC)

A dyad shouldn't correspond to an entry of the matrix, as the text says right now. If dyadics are represented as matrices, a dyad should be a zero or rank-one matrix (or, equivalently, a matrix that is the outer product of two vectors), not an entry of the matrix.

As for your first point, a dyad is a special kind of dyadic, so the page's statement isn't wrong, but the word 'dyad' is clearer. This issue probably arose because one can represent dyadics as a sum of dyads, so a dyad is a component of that sum. I've edited both of these.

(You're using the talk page perfectly. I feel obliged to mention Wikipedia's Be Bold policy---don't be afraid to just edit pages---but posting to the talk page to check info is fine.) The Boolean^_Talk 02:23, 30 December 2025 (UTC)

I am currently unaware of the other meanings of the word dyadic, though I found a use of it in the following quote

"Studies on doctor–patient communication focus predominantly on dyadic interactions between adults;" | as can be read in "Doctor–parent–child communication. A (re)view of the literature" by Kiek Tates and Ludwien Meeuwesen.

I am currently incapable of creating a page for this form of Dyadic on my own.. I just wanted to share so someone could do it if they want. ~2026-13495-81 (talk) 09:05, 2 March 2026 (UTC)