I think the following question is one simple but I need your help :)
So, how can I prove that :
$\left(\nu \nabla{u}\right)\nu=\frac{\partial u}{\partial \nu} \nu$ ?
and second question, why:
$\left(\nu \nabla{u}\right)\nu\frac{\partial u}{\partial \nu} =\left(\frac{\partial u}{\partial \nu}\right)^{2} x \nu $
These questions follow from Dirichlet's Problem:
\begin{cases} -\Delta{u}=f(u)& \text{in $\Omega$} \\ u=0& \text{on $\Gamma$} \\ \end{cases} where $\Omega \subset \mathbb{R}^{n}$ bounded domain with the border $\Gamma$ of $C^{1}$ class and $f : \mathbb{R} \to \mathbb{R}$ a continue function.
$\nu$ is exterior unit vector normal to the border $\Gamma$.
Thanks :)