- #1
rsq_a
- 107
- 1
I can't figure out what an author means by this expression:
[tex]
\textbf{n} \cdot \textbf{e} \cdot (\textbf{I} - \textbf{nn})
[/tex]
and
[tex]
\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn})
[/tex]
Here, all I know is that [itex]\textbf{u}[/itex] and [itex]\textbf{U}[/itex] are vectors of length 3. [itex]\textbf{n}[/itex] is a unit normal, so also a vector of length 3. [itex]\textbf{I}[/itex] I'm assuming is a 3x3 identity matrix. The author has also written that [itex]\textbf{e} = 1/2 (\nabla \textbf{u} + (\nabla \textbf{u})^T)[/itex], so I guess that's a 3x3 matrix.
But that what does [itex]\textbf{nn}[/itex] even mean? [itex]\textbf{n}n^T[/itex] 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
[tex]
\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn}) = \left(\textbf{u} - \textbf{U}\right)(\textbf{I} - \textbf{nn}^T)
[/tex]
[tex]
\textbf{n} \cdot \textbf{e} \cdot (\textbf{I} - \textbf{nn})
[/tex]
and
[tex]
\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn})
[/tex]
Here, all I know is that [itex]\textbf{u}[/itex] and [itex]\textbf{U}[/itex] are vectors of length 3. [itex]\textbf{n}[/itex] is a unit normal, so also a vector of length 3. [itex]\textbf{I}[/itex] I'm assuming is a 3x3 identity matrix. The author has also written that [itex]\textbf{e} = 1/2 (\nabla \textbf{u} + (\nabla \textbf{u})^T)[/itex], so I guess that's a 3x3 matrix.
But that what does [itex]\textbf{nn}[/itex] even mean? [itex]\textbf{n}n^T[/itex] 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
[tex]
\left(\textbf{u} - \textbf{U}\right) \cdot (\textbf{I} - \textbf{nn}) = \left(\textbf{u} - \textbf{U}\right)(\textbf{I} - \textbf{nn}^T)
[/tex]