# integration by parts on hypersurfaces

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 $$\int_{\Gamma} \nabla_\Gamma f \, dA = \int_{\Gamma} f H \nu \, dA + \int_{\partial \Gamma} f \mu \, ds,$$ 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 $H = \sum_{i=1}^{n+1}\partial_i \nu_i.$