Equivalence of the $H^1$ norm and the energy entropy norm

Question

Let $D \subset \mathbb R^2$ be a bounded domain with smooth boundary. Let $\mathbf v: D \to \mathbb R^2$ be a divergence free vector field tangent to the boundary, i.e., $\mbox{ div } \mathbf v = 0$ in $D$ and $\mathbf v \cdot \mathbf n = 0$ on the boundary of $D$, where $\mathbf n$ is the unit normal vector on the boundary. How does one show that the $H^1$ norm of $\mathbf v$, given by, $||\mathbf v||_2+ ||\nabla \mathbf v||_2 $ is equivalent to $||\mathbf v||_2+ ||\mbox{ curl }\mathbf v||_2$. Here $||\cdot||_2$ represents the $L^2$ norm. (The latter sum is sometimes called the energy and entropy norm.)