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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 :)

1 Answers 1

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We have the definition $\frac{\partial u}{\partial \nu} := \nabla u \cdot \nu$ (think of directional derivatives).