I seen in Evans that for $u\in H_0^1(\Omega)$ where $\partial\Omega$ is $C^1$ then $u|_{\partial\Omega}=Tu=0$, that is, $u$ vanishes at the boundary.
In the question "How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$", the spaces $H_0(div;\Omega)$ and $H_0(curl;\Omega)$ are introduced. Here, $H_0(div;\Omega)$ is the closure of $C_c^\infty(\Omega)$ in $H(div;\Omega):=\{u\in L_2(\Omega):div\,u\in L_2(\Omega)\}$ with respect to the norm $\|u\|=\|u\|_{L_2(\Omega)}+\|div\ u\|_{L_2(\Omega)}$. With $H_0(curl;\Omega)$ being defined analogously.
My problem: How does one see that $u\in H_0(div;\Omega)$ satisfies $u|_{\partial\Omega}\cdot n=0$ and that $v\in H_0(curl;\Omega)$ satisfies $v|_{\partial\Omega}\times n=0$?