What is meant by tangential operator? The context is as follows : Let $\Omega$ be a open set of $\mathbb{R}^n$, with smooth boundary $\Gamma$. Then there is the claim that for $\Psi \in C^1(\bar{\Omega})$, we have on $\Gamma$
$$ \frac{\partial \Psi}{\partial x_i} = \nu_i\frac{\partial \Psi}{\partial \nu} + \sigma_i\Psi$$ where $\sigma_i$ is a "first order tangential operator on $\Gamma$".  
What does this mean exactly?
 A: 
$\sigma_i$ is a "first order tangential operator on $\Gamma$".

For $\Psi \in C^1(\bar{\Omega})$, and $\partial \Omega = \Gamma$ is smooth, we can talk about the pointwise value of $\nabla \Psi$ on $\Gamma$, also do a decomposition with respect to the unit vector normal to $\Gamma$:
$$
\nabla \Psi\big\vert_{\Gamma} = (\text{Projection of }\nabla \Psi \text{ onto } \nu) + (\text{Tangential part}).
$$
This is 
$$
\nabla \Psi\big\vert_{\Gamma} = \nu\color{red}{(\nabla \Psi\cdot \nu)} + \color{blue}{\sigma( \Psi)},
$$
where the red term is the normal derivative $\partial \Psi/\partial \nu$, and the blue term is the "tangential part of $\nabla \Psi$":
$$
\sigma(\Psi) = \nu\wedge (\nabla \Psi\wedge \nu).\tag{1}
$$ 
The definition you gave is $\sigma$ in (1) component-wise.

Examples:


*

*When $n=2$, $ \nu\wedge (\nabla \Psi\wedge \nu)= \nu^{\perp} (\nabla \Psi \cdot \nu^{\perp})$, where $\nu^{\perp}$ is the unit tangential vector of the boundary curve rotating counterclockwisely.

*When $n=3$, $\nu\wedge (\nabla \Psi\wedge \nu)= \nu\times (\nabla \Psi\times \nu)$ in which the $\wedge$ is just the classical cross product.
If using the definition in (1), consider the spaces to be some Sobolev spaces, then $\sigma$ is a bounded linear operator under certain norm. For example, when $\Omega\subset \mathbb{R}^3$, $\Psi\in H^1(\Omega)$, and we have:
$$
\int_{\Omega} \nabla \times F\cdot \nabla \Psi = \int_{\Omega}  F\cdot \nabla \times (\nabla \Psi ) +  \int_{\partial \Omega}  (\nabla \Psi \times \nu)\cdot F\,dS
\\
=  \int_{\partial \Omega}  \nu\times (\nabla \Psi \times \nu)\cdot (\nu\times F)\,dS 
= \int_{\partial \Omega} \sigma(\Psi)\cdot (\nu\times F)\,dS,
$$
by some extension theorem about $F \in H^{1/2}(\Gamma)$ to $F:= E(F) \in H^1(\Omega)$, we have
$$
\|\sigma(\Psi)\|_{H^{-1/2}(\Gamma)} \leq C \|\Psi\|_{H^1(\Omega)}.
$$
