While studying Analysis on Manifolds in Elon Lages book, in the Chapter of Stokes Theorem there was this problem:
Let U $\subseteq$$\mathbb{R}^3$ be a open set and consider $F: U \rightarrow \mathbb{R}^3$ a smooth vector field, and let $M \subseteq U$ be a smooth manifold such that $F(p) = 0$ for every $p \in M$. Show that for every $p \in M$, $\nabla\times F(p) \in T_pM$
I couldn't solve this problem, and actually my only idea was trying to prove that $\nabla\times F(p)$ could be seen was a derivation for M, so I would have to prove it satisfies the Leibniz Rule. And perhaps the relation of this with Stokes Theorem is that you can see this vector field as a 1-form on $U$ and then it's curl it's just the derivative, so I could try using Stokes, but no further progress has been made.