Rank of a map restricted to the boundary of a manifold

Question

Let $M$ be a smooth manifold with boundary, $N$ a smooth manifold and $F : M \to N$ smooth. Define $$f = F|_{\partial M}.$$ If $df_x$ has rank $r$ at $x \in \partial M$, can we conclude that the rank of $F$ at $x$ is at least $r$?

Answer 1

Yes, because

$$ \left( dF|_x \right)|_{T_x \left( \partial M \right)} = df|_x $$

under the natural identification of the tangent space to $\partial M$ at $x$ as a subspace of the space to $M$ at $x$. The identity above implies that $$ \operatorname{Im} \left( df|_{x} \right) \subseteq \operatorname{Im} \left( dF|_{x} \right) $$ and so if the rank of $f$ at $x$ is $r$, the image of $df|_{x}$ contains $r$ linearly independent vectors and so does the image of $dF|_{x}$, hence $F$ has at least rank $r$ at $x$ (and, in fact, at most rank $r + 1$).