# Double dot product vs double inner product

Anything involving tensors has 47 different names and notations, and I am having trouble getting any consistency out of it.

This document (http://www.polymerprocessing.com/notes/root92a.pdf) clearly ascribes to the colon symbol (as "double dot product"):

$\mathbf{T}:\mathbf{U}=T_{ij} U_{ji}$

while this document (http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html) clearly ascribes to the colon symbol (as "double inner product"):

$\mathbf{T}:\mathbf{U}=T_{ij} U_{ij}$

Same symbol, two different definitions. To make matters worse, my textbook has:

$\mathbf{\epsilon}:\mathbf{T}$

where $\epsilon$ is the Levi-Civita symbol $\epsilon_{ijk}$ so who knows what that expression is supposed to represent.

Sorry for the rant/crankiness, but it's late, and I'm trying to study for a test which is apparently full of contradictions. Any help is greatly appreciated.

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What course is this for? I've never heard of these operations before. (Sorry, I know it's frustrating. There are a billion notations out there.) – Jesse Madnick Apr 2 '13 at 5:43
It's for a graduate transport processes course (for chemical engineering). – Nick Apr 2 '13 at 5:49

## 1 Answer

Sorry for such a late reply. I hope you did well on your test. Hopefully this response will help others.

The "double inner product" and "double dot product" are referring to the same thing- a double contraction over the last two indices of the first tensor and the first two indices of the second tensor. A double dot product between two tensors of orders m and n will result in a tensor of order (m+n-4).

So, in the case of the so called permutation tensor (signified with epsilon) double-dotted with some 2nd order tensor T, the result is a vector (because 3+2-4=1).

You are correct in that there is no universally-accepted notation for tensor-based expressions, unfortunately, so some people define their own inner (i.e. "dot") and outer (i.e. "tensor") products. But, this definition for the double dot product that I have described is the most widely accepted definition of that operation.

Hope this helps.

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