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Given a bounded and open domain $\Omega\subset \mathbb{R}^n$ with $\Gamma:=\partial \Omega \in C^\infty$. Let $H$ be the mean curvature on $\Omega$ and $\nu$ the unit outer normal vector.

Furthermore we define the tangential Gradient for a smooth function $f:\overline{\Omega}\to\mathbb{R}$ and the tangential divergence for a smooth vector field $v:\overline{\Omega}\to\mathbb{R}^n$ as follows $$ \nabla_\Gamma f:=\nabla_x f-(\nu\cdot\nabla_xf))\nu $$ respectively $$ \operatorname{div}_\Gamma v := \operatorname{div}_x(v) - (\nu\cdot D_v)\nu, $$ where $(D_v)_j:=\sum_{i=1}\partial_jv_i$.

Is it now possible to show that the following theorem holds

$$ \int_\Gamma f~\operatorname{div}_\Gamma v~dS = \int_\Gamma (v\cdot \nabla_\Gamma f)~dS + (n-1)\int_\Gamma f~(v\cdot\nu)H~dS. $$

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    $\begingroup$ What’s your question? $\endgroup$ – Jack Lee Apr 2 at 15:10
  • $\begingroup$ How to prove (or at least where to find a proof) of the theorem $\int_\Gamma f~\operatorname{div}_\Gamma v~dS = \int_\Gamma (v\cdot \nabla_\Gamma f)~dS + (n-1)\int_\Gamma f~(v\cdot\nu)H~dS$; sorry for not making this clear. I tried to clarifie my question aove. $\endgroup$ – Bara Apr 3 at 7:22

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