I'm a beginner with vector calculus and analysis and I am looking at the following example problem. Please note that is is not a homework question, I am simply looking at some vector relationships. The problem is asking to show for any vector ${\overrightarrow{\mathbf{F}}}$
$$({\overrightarrow{\mathbf{F}}}\cdot{{\overrightarrow{\mathbf{\nabla}}}}){\overrightarrow{\mathbf{r}}}={\overrightarrow{\mathbf{F}}}$$
It is asking to show this in cartesian coordinates. I was wondering what the starting point is for such an example. I have looked quite intesively on the world wide web, as you'd expect, and can't seem to find a specific example which I could use to start this proof.
Denote \begin{equation*} \mathbf{r}=(x_{1},x_{2},x_{3}) \end{equation*} Now
\begin{eqnarray*} (\mathbf{F\cdot \nabla )r} &=&\mathbf{F} \\ (\sum_{l=1}^{3}F_{l}\frac{\partial }{\partial _{x_{l}}})x_{k} &=&F_{k} \end{eqnarray*} But \begin{equation*} \frac{\partial }{\partial _{x_{l}}}x_{k}=\delta _{kl} \end{equation*} so the left hand side equals the right hand side