What is the meaning of \textbf{nn} in matrix multiplication?

[rsq_a]

I can't figure out what an author means by this expression:

<br /> \textbf{n} \cdot \textbf{e} \cdot (\textbf{I} - \textbf{nn})<br />

and

<br /> \left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn})<br />

Here, all I know is that \textbf{u} and \textbf{U} are vectors of length 3. \textbf{n} is a unit normal, so also a vector of length 3. \textbf{I} I'm assuming is a 3x3 identity matrix. The author has also written that \textbf{e} = 1/2 (\nabla \textbf{u} + (\nabla \textbf{u})^T), so I guess that's a 3x3 matrix.

But that what does \textbf{nn} even mean? \textbf{n}n^T makes sense to me (giving a 3x3 matrix).

But then what does it mean to take the dot product of a 3x3 matrix with a 3x3 matrix? Is the author simply referring to matrix multiplication in
<br /> \left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn}) = \left(\textbf{u} - \textbf{U}\right)(\textbf{I} - \textbf{nn}^T)<br />

 

[arkajad]

Why are you so cryptic about "an author"? Telling us what is the exact context of all this may save our time.

 

[rsq_a]

Why are you so cryptic about "an author"? Telling us what is the exact context of all this may save our time.

I didn't see it as relevant. The equation(s) can be found on p.5 http://www.maths.nottingham.ac.uk/personal/pmzjb1/ejam_new.pdf.

 

[arkajad]

From other formulas in thise paper lot can be guessed. The dot means simply multiplication of one matrix by another matrix, for instance vector.matrix=vector. I am not sure whether there is a difference between row and columns vectors, but I guess there is one.

I could not decode

\textbf{e} = 1/2 (\nabla \textbf{u} + (\nabla \textbf{u})^T)

but that can be decoded looking somewhere else for "rate of strain tensor".

 

[rsq_a]

From other formulas in thise paper lot can be guessed.

Where? In (2.4) and (2.6), for example, dot is used consistently to mean the inner product (i.e. vector and vector).

\textbf{e} = 1/2 (\nabla \textbf{u} + (\nabla \textbf{u})^T)

This is easy. It's a 3x3 vector. The gradient of a vector is the transpose of the jacobian.

What about \textbf{nn}? I asked before how it makes sense to put two 3x1 (or 1x3) vectors together. Moreover, if you are correct, and

 

[arkajad]

I guess \mathbf{nn} is the 3x3 matrix n_in_j. So, for instance, \mathbf{u}\cdot\mathbf{nn} would be

\Sigma_i u_in_in_j