Problem in notation: $(\vec{v}\cdot\vec{\nabla})\vec{v}$ versus $L_v(\vec{v})$ I've seen more than one time the use of the following notation: $(\vec{v}\cdot\vec{\nabla})\vec{v}$ to say $L_v(\vec{v})$ where the matrix $L_v$ is defined as follows: $(L_v)_{ij} := \partial_j v_i$.
For example this happens in the usual law of conservation of momentum of an eulerian fluid:
$$\rho\Big(\partial_t\vec{v} + (\vec{v}\cdot\vec{\nabla})\vec{v}\Big) = \rho\vec{b} + \vec{\nabla}p$$
My question is: how that notation could "help" us to understand "how things work"?
I think this is strange, but physicists are known to have strange notations that in some ways help to understand the nature of things. Take for example the divergence of a vector field $\vec{v}\ $: this is written in some texts as $\vec{\nabla}\cdot\vec{v}$ because if we think at $\vec{\nabla}$ as the vector $(\partial_1, \partial_2, \partial_3)$ then the scalar product $$\vec{\nabla}\cdot\vec{v} = (\partial_1, \partial_2, \partial_3) \cdot (v_1,v_2,v_3) = \sum_{i=1,2,3} \partial_i \ v_i = \text{div}(\vec{v})$$
Now, if I take for example a $2$ dimensional field $\vec{v} = (v_1,v_2)$ we have (correct me if I'm wrong)
$$L_v(\vec{v}) = 
\begin{pmatrix}
\partial_1 v_1 & \partial_2 v_1 \\ 
\partial_1 v_2 & \partial_2 v_2
\end{pmatrix}
\begin{pmatrix}
v_1 \\
v_2
\end{pmatrix}
=
\begin{pmatrix}
(\partial_1 v_1)v_1 + (\partial_2 v_1)v_2 \\ 
(\partial_1 v_2)v_1 + (\partial_2 v_2)v_2
\end{pmatrix}
$$
How could I derive "intuitively" the last vector using the notation $(\vec{v}\cdot\vec{\nabla})\vec{v}$?
 A: Okay, I think I got it.
I have to consider the scalar product $\vec{v}\cdot \vec{\nabla}$ as the "scalar" $v_1 \partial_1 + v_2\partial_2$ that multiplies the vector $(v_1,v_2)$:
$$(\vec{v}\cdot \vec{\nabla})\vec{v} = \big(v_1 \partial_1 + v_2\partial_2 \big)
\begin{pmatrix}
v_1 \\v_2
\end{pmatrix} = 
\begin{pmatrix}
v_1 \ \partial_1v_1 + v_2\ \partial_2 v_1 \\
v_1 \ \partial_1v_2 + v_2\ \partial_2 v_2
\end{pmatrix} = L_v(\vec v)
$$
This is really odd.
A: You already understand that in 2D we are to treat $\overrightarrow{\nabla}$ as $\left\langle \partial_{1},\partial_{2}\right\rangle$. The key for this and many other related notations is basically to treat something like $\partial_1$ as a scalar that's distributive and commutes with scalars (it's a linear operator, after all), and associative where it makes sense, but not commutative.
We have $\overrightarrow{v}\cdot\overrightarrow{\nabla}=\left\langle v_{1},v_{2}\right\rangle \cdot\left\langle \partial_{1},\partial_{2}\right\rangle =v_{1}\partial_{1}+v_{2}\partial_{2}$.
And then:
\begin{align}\left(\overrightarrow{v}\cdot\overrightarrow{\nabla}\right)\overrightarrow{v}&=\left(v_{1}\partial_{1}+v_{2}\partial_{2}\right)\left\langle v_{1},v_{2}\right\rangle \\&=\left(v_{1}\partial_{1}\right)\left\langle v_{1},v_{2}\right\rangle +v_{2}\partial_{2}\left\langle v_{1},v_{2}\right\rangle \\&=\left\langle \left(v_{1}\partial_{1}\right)v_{1},\left(v_{1}\partial_{1}\right)v_{2}\right\rangle +\left\langle \left(v_{2}\partial_{2}\right)v_{1},\left(v_{2}\partial_{2}\right)v_{2}\right\rangle \\&=\left\langle v_{1}\left(\partial_{1}v_{1}\right),v_{1}\left(\partial_{1}v_{2}\right)\right\rangle +\left\langle v_{2}\left(\partial_{2}v_{1}\right),v_{2}\left(\partial_{2}v_{2}\right)\right\rangle \\&=\left\langle v_{1}\left(\partial_{1}v_{1}\right)+v_{2}\left(\partial_{2}v_{1}\right),v_{1}\left(\partial_{1}v_{2}\right)+v_{2}\left(\partial_{2}v_{2}\right)\right\rangle \\&=\left\langle \left(\partial_{1}v_{1}\right)v_{1}+\left(\partial_{2}v_{1}\right)v_{2},\left(\partial_{1}v_{2}\right)v_{1}+\left(\partial_{2}v_{2}\right)v_{2}\right\rangle \\&=\begin{bmatrix}\left(\partial_{1}v_{1}\right)v_{1}+\left(\partial_{2}v_{1}\right)v_{2}\\
\left(\partial_{1}v_{2}\right)v_{1}+\left(\partial_{2}v_{2}\right)v_{2}
\end{bmatrix}\\&=\begin{bmatrix}\partial_{1}v_{1} & \partial_{2}v_{1}\\
\partial_{1}v_{2} & \partial_{2}v_{2}
\end{bmatrix}\begin{bmatrix}v_{1}\\
v_{2}
\end{bmatrix}\end{align}
