A subset $M \subset \mathbb{R}^n$ is called a smooth k-dimensional submanifold of $\mathbb{R}^n$, $k \leq n$, if any point $x \in M$ has a neighborhood $O_x$ in $\mathbb{R}^n$ in which $M$ is described as follows
- There exists a smooth vector-function $$F:O_x \to \mathbb{R}^{n-k}, \text{ } \text{ rank } \frac{dF}{dx}|_q=n-k $$ such that $O_x \cap M = F^{-1}(0)$
When we ask that $\text{ rank } \frac{dF}{dx}|_q=n-k$, what are we trying to say, what is the need for this in the definition?