I'm having a hard time trying to solve this question:
Let $F:U\rightarrow\mathbb{R}^3$ be of class $C^\infty$ defined on open $U\subseteq\mathbb{R}^3$ and let $M\subseteq U$ be a surface of class $C^\infty$ (here surface can mean curves, 2 dimensional surfaces or 3 dimensional surfaces, with or without boundaries) such that $F(p)=0$ for each $p\in M$. Show that $\nabla\times F(p)\in T_pM$
My reasoning goes as follows: let $\lambda:I\subset\mathbb{R}\rightarrow M$ be a curve of class $C^1$ such that $\lambda(0)=p$ (allowed because $M$ is of class $C^\infty$), then $\lambda'(0)\in T_pM$. Since $(F\circ\lambda)(t)=0$ we get that $F'(p)\lambda'(0)=0$ and, since $\lambda$ is arbitrary, we get that $T_pM\subseteq\ker F'(p)$. However, I'm stuck here, I cannot a relation between $\nabla\times F(p)$ and $T_pM\subseteq\ker F'(p)$. I tried to link $\nabla\times F(p)$ with the Jacobian of $F'(p)$ without success.
The exercise is on the generalized Stokes theorem section, but I didn't find any relation, specially because the surface is so generic. Maybe if given $p\in T_pM$ I take a parametrization $\phi: U_0\rightarrow U$ of a neighborhood $U$ of $p$ and take a compact $p\in K\subseteq U$ such that I can consider $K$ a surface with boundary?
Could anyone shed a light upon this question?