Let $U$ be an open subset of $\mathbb{R}^d$, and let $F \colon U \to \mathbb{R}^{d +c}$ be a smooth map whose rank is everywhere equal to $d-1$:
$$\mathrm{rank} (dF_{p}) = d-1 \quad \text{for all $p \in U$.} \tag{1}\label{eq1}$$
Recall that the image of the differential $dF_{p}$ is spanned by the partial derivatives $\frac{\partial F(p)}{\partial u^{1}}, \dotsc, \frac{\partial F(p)}{\partial u^{d}}$.
I have some questions:
- Does condition \eqref{eq1} imply that one among the partial derivatives of $F$ vanishes identically?
- Could it happen that, while all partial derivatives are nowhere zero, one of them (say the first) is in the span of the others? Here I mean that
$$\dfrac{\partial F(p)}{\partial u^{1}} \in \mathrm{span} \left( \dfrac{\partial F(p)}{\partial u^{2}}, \dotsc, \dfrac{\partial F(p)}{\partial u^{d}}\right)\quad \text{for all $p \in U$.}$$
- If the answer to the second question is yes, can we then find an example?