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WikiProject Mathematics (Rated Start-class, Low-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
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Field:  Geometry (historical)

## Double dot product

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)

## Applications?

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)

## Inner product

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)

## Plural in article name

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

## Other "double dot products" in mathematics

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