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

[rsq_a]

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

$$\textbf{n} \cdot \textbf{e} \cdot (\textbf{I} - \textbf{nn})$$

and

$$\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn})$$

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
$$\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn}) = \left(\textbf{u} - \textbf{U}\right)(\textbf{I} - \textbf{nn}^T)$$

 

[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

$$\Sigma_i u_in_in_j$$