Let $u\in C^1(\Gamma)$ be a scalar function and $v\in (C^1(\Gamma))^2$ a tangential vector field defined on the surface $\Gamma\subset\mathbb{R}^3$. The following Stokes identities hold. $ \begin{aligned} \int_{\Gamma}\nabla_{\Gamma}u\cdot vd\,\Gamma &= -\int_{\Gamma}u\,\text{div}_{\Gamma}\,v\,d\Gamma\\ \int_{\Gamma}\left(\overrightarrow{\text{curl}}_{\Gamma}\,u\cdot v\right)d\Gamma &= \int_{\Gamma}u\,\text{curl}_{\Gamma}v\,d\Gamma\\ \text{div}_{\Gamma}\,\overrightarrow{\text{curl}}_{\Gamma}\,u &= 0\\ \text{curl}_{\Gamma}\nabla_{\Gamma}u &= 0\\ \text{div}_{\Gamma}\,(v\times n) &= \text{curl}_{\Gamma}\,v. \end{aligned} $ If $w\in C^2(\Gamma)$ then $ -\int_{\Gamma}\Delta_{\Gamma}w\,u\,d\Gamma = \int_{\Gamma}\nabla_{\Gamma}w\cdot\nabla_{\Gamma}u\,d\Gamma = \int_{\Gamma}\left(\overrightarrow{\text{curl}_{\Gamma}}\,w\cdot\overrightarrow{\text{curl}}_{\Gamma}u\right)d\Gamma. $ If $w$ is a tangential vector field with $w\in C^2(\Gamma)^2$ we have $ -\int_{\Gamma}\left(\Delta_{\Gamma}w\cdot v\right)d\Gamma = \int_{\Gamma}\text{div}_{\Gamma}\,w\,\text{div}_{\Gamma}\,v\,d\Gamma + \int_{\Gamma}\text{curl}_{\Gamma}\,w\,\text{curl}_{\Gamma}v\,d\Gamma. $ # Related Items [[Surface Differential Operators]] [[Stokes Theorem]] # References Nédélec, _Acoustic and Electromagnetic Equations_, Springer, Theorem 2.5.19