I've been working on this question:
Let $\mathbf F$ be a vector field on $\mathbb R^3$, and $\mathbf F_T$ be the projection of $\mathbf F$ on a tangent plane $T$ of a surface $S$. Show that the rotation of $\mathbf F_T$ equals $\nabla\times\mathbf F\cdot\mathbf n$, where $\mathbf n$ is a unit vector perpendicular with $T$.
I don't really get what this means nor how should I tackle it. I've done some tedious calculations, but I'm not sure if I'm making progress at all.
\begin{align*} \nabla\times\mathbf F_T\cdot\mathbf k&=\nabla\times(\mathbf F-(\mathbf n\cdot\mathbf F)\mathbf n)\cdot\mathbf k \\ &=\nabla\times\mathbf F\cdot\mathbf k-(\nabla(\mathbf n\cdot\mathbf F)\times\mathbf F)\cdot \mathbf k \\ &=\nabla\times\mathbf F\cdot\mathbf k-((\mathbf n\cdot\nabla)\times\mathbf F+(\mathbf n\times(\nabla\times\mathbf F))\times\mathbf F)\cdot\mathbf k \end{align*}
Edit $\mathbf k$ is the $z$-direction unit vector.