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I am supposed to show a proof on homework where and I have no idea what this expression is supposed to mean:

$(\rho \mathbf{uu}) $

where $\mathbf{u}$ is a velocity vector.

The only types of vector multiplication I have seen are dot product and cross product. I am confident that the professor did not mean dot product, and I do not think that it's supposed to be a cross product either. The professor also claims that there is not a typographical error.

Can somebody please explain how to interpret or simply that term?

For context, the problem is:

Show that: $\frac{\partial\rho \mathbf{u}}{\partial t} + \nabla \cdot (\rho \mathbf{uu}) = \rho\frac{D \mathbf{u}}{D t} $

The problem is written exactly how it is displayed above.

Also, the $ \rho \frac{D \mathbf{u}}{D t}$ is the material (total) derivative from fluid mechanics.

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Here, $\mathbf{uu}$ represents the dyadic product $\mathbf{uu} = \mathbf{uu}^T$ (a matrix). It is also often written as $\mathbf{u} \otimes \mathbf{u}$.

In coordinates, is $\mathbf{u} = [u_1, u_2, u_3]^T$, then $$\rho\mathbf{uu} = \left[\begin{matrix} \rho u_1u_1 & \rho u_1u_2 & \rho u_1 u_3\\ \rho u_2u_1 & \rho u_2u_2 & \rho u_2 u_3\\ \rho u_3 u_1 & \rho u_3 u_2 &\rho u_3 u_3\end{matrix}\right]$$ and $$\nabla \cdot (\rho\mathbf{uu}) = \left[\begin{matrix} \partial_1(\rho u_1u_1) + \partial_2( \rho u_1u_2) + \partial_3( \rho u_1 u_3)\\ \partial_1(\rho u_2u_1) + \partial_2( \rho u_2u_2) + \partial_3(\rho u_2 u_3)\\ \partial_1(\rho u_3 u_1) + \partial_2(\rho u_3 u_2) +\partial_3(\rho u_3 u_3)\end{matrix}\right]$$

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  • $\begingroup$ (+1) thank you, I should have been more careful in checking the notation specific to hydrodynamics $\endgroup$
    – Yuriy S
    Feb 8 '18 at 17:45
  • $\begingroup$ Thanks dude! you're a straight up G! $\endgroup$
    – dat boy
    Feb 8 '18 at 18:06

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