Question about the math notation when studying Euler equation I am trying to work out the Euler equation. However, I find some difficulties of the following step. Am I doing the right things and how could I get into the last step? Thanks a lot.
$$ \nabla (\vec{v} \otimes \vec{v}) $$
$$ = \nabla \left[ \begin{pmatrix} u \\ v \\ w \end{pmatrix} \begin{pmatrix} u & v & w \end{pmatrix} \right]$$
$$ =  \nabla \begin{pmatrix}
u^2 & uv & uw\\ uv & v^2 & vw \\ uw & vw & w^2 
\end{pmatrix}$$
$$ = \begin{pmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\end{pmatrix} \begin{pmatrix}
u^2 & uv & uw\\ uv & v^2 & vw \\ uw & vw & w^2 
\end{pmatrix}$$
The result I am looking for should be look like:
$$ = \frac{\partial}{\partial x} \begin{pmatrix} u^2 \\ uv \\uw \end{pmatrix} + 
 \frac{\partial}{\partial y} \begin{pmatrix} uv \\ v^2 \\vw \end{pmatrix} +
   \frac{\partial}{\partial z} \begin{pmatrix} uw \\ vw \\w^2 \end{pmatrix}$$
While when I keep working, I get the following row vector insteat.
$$  \begin{pmatrix} \frac{\partial}{\partial x} u^2 + \frac{\partial}{\partial y} (uv) + \frac{\partial}{\partial z} (uw) & \frac{\partial}{\partial x} (uv) + \frac{\partial}{\partial y} v^2 + \frac{\partial}{\partial z} (vw) & \frac{\partial}{\partial x} (uw) + \frac{\partial}{\partial y} (vw) + \frac{\partial}{\partial z} w^2 \end{pmatrix} $$
 A: One thing—it seems like the object you're attempting to calculate is $\nabla \cdot \left(\vec{v} \otimes \vec{v}\right)$, not $\nabla \left(\vec{v} \otimes \vec{v}\right)$.
Also, the object you're calculating is just the transpose of the other; you'll notice it more clearly if you bring the differentiation operator into the parentheses and add up. When it comes to the question of why you get a row vector vs. a column vector, it's because the "mnemonic" of defining $\nabla = \begin{pmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\end{pmatrix}$ with a row vector isn't technically correct and an abuse of notation, which leads to some dimensional inconsistencies if applied directly. A more accurate approach, which is coordinate-independent but maybe a bit more confusing, is this definition of $\nabla$:
$$\nabla = \lim_{V \rightarrow 0} \frac{1}{V}\int\int_S \vec{n}[...] dS$$
where the stuff inside the brackets can also be an operation (like $\cdot f$ for $\nabla \cdot f$).
Regardless, when the result is intrepreted as a vector (independent of row/column "orientation"), your result is correct.
A: I also had this problem. It is a notation problem.
It can be easily solved when using index notation, which I hate, but it is moore useful.
Acoording to index notation, you should be calculating:
$$ p_i= \sum_j \frac{\partial}{\partial x_j} v_i v_j  $$
And this would be more like
$$(\vec{v}\otimes \vec{v})\cdot \vec{\nabla}$$
But this notation is obviously not good.
In sum, use index notation instead.
