Assume that $\Omega$ is a smooth domain with $C^2$ boundary $\Gamma$. # Local Extension Operator Let $\Gamma_{\epsilon}$ be a sufficiently small neighborhood of $\Gamma$. Define a unique projection operator $\mathcal{P}$ such that for $y\in\Gamma_{\epsilon}$ we have $\mathcal{P}(y)\in\Gamma$. Let $ \delta(y) := |y-\mathcal{P}(y)|. $ The line $y-P(y)$ is directed along the normal to $\Gamma$ at $y$. When $y$ is in the exterior of $\Omega$ it holds that $ \nabla \delta(y) = n(\mathcal{P}(y)). $ If $y$ is in the interior of $\Omega$ it holds that $ \nabla \delta(y) = -n(\mathcal{P}(y)). $ For any point $y\in\Gamma_{\epsilon}$ we can write $y = \mathcal{P}(y) + sn(\mathcal{P}(y))$ with $s=\pm\delta(y)$ (depending whether $y$ is in the interior or exterior of $\Omega$. For any function $u$ defined on $\Gamma$ can define a lifting operator by $ \tilde{u}(y) = u(\mathcal{P}(y)). $ We can define a gradient field $n(y)$ as $n(y):=\nabla s(y)$ for any $s\in\Gamma_{\epsilon}$. Now define the family $\Gamma_s$ of parallel surfaces as $ \Gamma_s = \{y; y = x+sn(x); x\in\Gamma\}. $ The field $n(y)$ is the field of normals to the surface $\Gamma_s$. # Surface Differential operators of scalar functions ## Surface Gradient The surface gradient $\nabla_{\Gamma}u$ is defined as $ \nabla_{\Gamma}u = \nabla\tilde{u}_{\Gamma}. $ ## Tangential curl The tangential curl is defined as $ \overrightarrow{\text{curl}}_{\Gamma} = \text{curl}\,(\tilde{u}n)_{\Gamma}. $ Since $n(y)$ is a gradient field it follows that $\text{curl}\, n = 0$ and therefore, using $\text{curl}\,(u\overrightarrow{v}) = \nabla u\times \overrightarrow{v} + u\,\text{curl}\,\overrightarrow{v}$, $ \overrightarrow{\text{curl}}_{\Gamma} = \nabla_{\Gamma}u\times n. $ # Lifting of tangential vector fields Let $\mathcal{R}_s(y) := \nabla n = D^2s$. This is called the curvature operator. One can show that $\mathcal{R}_s(y)n(y) = 0$. Hence, the normals are in the kernel of the curvature operator. With this operator one can for a tangential vector field $v$ define the lifted field $ \tilde{v}(y) = v(x) -s\mathcal{R}_s(y)v(x), $ which is tangential at each $\Gamma_s$. For the tangential derivative at $\Gamma_s$ we obtain $ \nabla_{\Gamma_s}\tilde{u} = (I -s\mathcal{R}_s)\nabla_{\Gamma}u. $ # Surface Divergence The surface divergence of the tangential vector field $v$ is defined as $ \text{div}_{\Gamma} = \text{div}\,\tilde{v}|_{\Gamma}, $ that is, we take the divergence of the extension and restrict it to $\Gamma$. # Surface curl We define the surface curl of a tangential vector field $v$ as $ \text{curl}_{\Gamma}\,v = (n\cdot \text{curl}\,\tilde{v})|_{\Gamma}. $ # Function Spaces The function spaces for boundary differential operators can be found in [[De Rham Sequence for Trace Spaces]]. # References Nédélec, _Acoustic and Electromagnetic Equations_, Springer, Chapter 2.5.6