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Can anyone please help me with the understanding and proof of the statement found in
Theorem 2.10

Theorem: Assuming that $\Gamma$ is a hypersurface in $\mathbb{R}^{n+1}, n = 1,2$ with smooth boundary $\partial \Gamma$ and that $f \in C^1(\overline{\Gamma}).$ Then \begin{equation} \int_{\Gamma} \nabla_\Gamma f \, dA = \int_{\Gamma} f H \nu \, dA + \int_{\partial \Gamma} f \mu \, ds, \end{equation} where μ denotes the co-normal vector which is normal to ∂Γ and tangent to Γ, $\nu$ is the unit normal and $\nabla_\Gamma f= \nabla f - (\nabla f \cdot \nu )\nu$ and the curvature $H = \sum_{i=1}^{n+1}\partial_i \nu_i.$

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    $\begingroup$ What is the question exactly? $\endgroup$ – Daniel Robert-Nicoud Jun 20 '16 at 10:04
  • $\begingroup$ @DanielRobert-Nicoud , i want to proof that the right-hand-side holds using the definition of the tangential gradient. $\endgroup$ – uli.xu Jun 20 '16 at 10:43
  • $\begingroup$ your question is not clear. you have to reform it $\endgroup$ – A s Jun 20 '16 at 15:26
  • $\begingroup$ @As, is it ok now? $\endgroup$ – uli.xu Jun 21 '16 at 7:10

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