Let $F = u \nabla u $. Show that div $F = u \nabla^2 u + \nabla u \cdot \nabla u $ just a quick question, I'm just a bit off track as to how to derive this. This is what I have:
$$F = u \nabla u = u(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z})$$
$$ \implies \nabla F = \nabla (u \nabla u) = \nabla(u\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial z}) $$
$$ \implies \nabla F = (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z})\frac{\partial u}{\partial x}+(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z})\frac{\partial u}{\partial y}+(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z})\frac{\partial u}{\partial z}$$
Expanding this out, I get the $\nabla u \cdot \nabla u$ term, and I am left with, the terms such as $\frac{\partial u}{\partial y}\frac{\partial u}{\partial x}$, but can't seem to derive $u \nabla^2 u$ from these terms.
Any help or guidance would be greatly appreciated!
 A: Note that:


*

*$u$ is a scalar.

*$\nabla$ is a vector.

*$\nabla u$ is a vector.

*$u\nabla u$ is a vector.

*$\nabla u \cdot \nabla u$ is a scalar.

*$\nabla^2 u = \nabla \cdot \nabla u$ is a scalar.

*$\text{div}~ F = \nabla \cdot F$ is a scalar.


Thus, for the LHS, we have:
\begin{align*}
\text{div}~ F
&= \nabla \cdot (u\nabla u) \\
&= \begin{bmatrix}
\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z}
\end{bmatrix}
\cdot u
\begin{bmatrix}
\frac{\partial u}{\partial x} \\ \frac{\partial u}{\partial y} \\ \frac{\partial u}{\partial z}
\end{bmatrix} \\
&= \begin{bmatrix}
\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z}
\end{bmatrix}
\cdot 
\begin{bmatrix}
u\frac{\partial u}{\partial x} \\ u\frac{\partial u}{\partial y} \\ u\frac{\partial u}{\partial z}
\end{bmatrix} \\
&= \frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right) + \frac{\partial}{\partial y}\left(u\frac{\partial u}{\partial y}\right) + \frac{\partial }{\partial z}\left(u\frac{\partial u}{\partial z}\right) \\
&= \left(\frac{\partial u}{\partial x}\frac{\partial u}{\partial x} + u\frac{\partial^2 u}{\partial x^2}\right) + \left(\frac{\partial u}{\partial y}\frac{\partial u}{\partial y} + u\frac{\partial^2 u}{\partial y^2}\right) + \left(\frac{\partial u}{\partial z}\frac{\partial u}{\partial z} + u\frac{\partial^2 u}{\partial z^2}\right) \\
\end{align*}
I'm sure you can finish the rest.
