I've seen this vector identity from the book[1] in page 89, $$ (\nabla p)\times\nu =0,\ \text{on}\ \partial\Omega,$$ where $\nu $ is the outer normal vector of $\partial \Omega$, $ p \in H_0^1(\Omega).$ I try to calculate directly $(\nabla p)\times \nu =(\nu_3\partial_2 p-\nu_2\partial_3 p,\nu_1\partial_3 p-\nu_3\partial_1 p,\nu_2\partial_1 p-\nu_1\partial_2 p)^T,$ but I don't know how to show it equals $0$.
[1] Monk, Peter, Finite Element Methods for Maxwell's Equations (Oxford, 2003; online edn, Oxford Academic, 1 Sept. 2007)
Also, $(\nabla p)\times \nu=|\nabla p ||\nu|\sin(\theta)$ where $\theta$ is the angle between the two vectors. For the cross product to be zero the two vectors must be parallel. You must show that $\nabla p$ is parallel to $\nu$