TITLE: Intuition behind functional dependence
QUESTION [16 upvotes]: What is the intuition behind functional independence ?
(This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally independent if the matrix whose columns are the gradients  $\nabla F_1,\ldots,\nabla F_k$ has full rank, i.e. rank $k$, on the whole domain of definition.
From what I gather from this answer, this is the same as saying that $F:=(F_1,\ldots,F_k):\mathbb{R}^n\rightarrow \mathbb{R}^k$ is submersion, but that doesn't help me much either, because I also don't have any intuition concerning submersions.)
So what does it really mean if the functions are functional indepedent - or conversely, dependent ? Is there, in the latter case, then also a relationship like $g(\nabla F_1,\ldots,\nabla F_k)=0$ -- or maybe like $g(F_1(x),\ldots F_k (x))=0$ for some $x$ -- for $g$ ranging in some specific set, similar to the case of linear independence ( in which $g$ would be from the set $\{g:\mathbb{R}^k\rightarrow \mathbb{R}:g(x_1,\ldots,x_k)=\sum \lambda_i x_i \text{ for some nonzero } \lambda_i \in \mathbb{R}\}$).

REPLY [2 votes]: Functional independence between two functions means that the level set of each of the functions intersects transversely the level set of the other function. In the plane, this means that a function $f$ functionally independent of another function $g$ cannot be written as $f=F(g)$ where $F$ is another function, because in this case, the level sets would be the same curves, for both $f$ and $g$. Similarly, in $k$ dimensions, functional independence means that the intersection between the $k$ $(k-1)$-dimensional level sets of the $k$ functions is a point, not a line or a plane, i. e., the functions $F_1$, $F_2$,...,$F_k$ contains $k$ independent informations of your ambient space, and can be taken locally as coordinates.