Functional dependence and independence

A function $\mathrm{\Phi }$ is functionally dependent on $h_1,\dots ,h_r$ if $\mathrm{\Phi }=G(h_1,\cdots ,h_r)$ for certain function $G$.

And $h_1,\dots ,h_r$ are called functionally independent if the differentials are linearly independent in a open set $U$, that is, the Jacobian of $H(x)=(h_1(x),\dots ,h_r(x))$ has maximal rank for $x\in U\subset {\mathbb{R}}^N$. It is the same as saying that $H$ is a submersion.
Or, in another way, every $x\in U$ is a regular point of $H$. So therefore, for every $y\in H(U)$, $H^{-1}(y)$ is an embedded manifold.

Idea: Consider an open subset $U\subseteq {\mathbb{R}}^3$ and three functions $f,g,h$ defined on it. At a point $p\in U$, the differential $d{f}_p$ is visualized as a set of parallel planes centered at $p$. The same is true for $d{g}_p$. If $d{f}_p,d{g}_p$ are linearly independent, these families fo planes are transversal, so $f=constant,g=constant$ defines a family of curves. If $h=G(f,g)$, then $h=constant$ does not allows us to define a point, i.e., the family of planes corresponding to $d{h}_p$ contain the intersection lines of $d{f}_p,d{g}_p$. Therefore $d{h}_p$ is a linear combination of $d{f}_p,d{g}_p$.
More simply, if $h$ depends on $f,g$ (or if $F(f,g,h)=0$), then $\{f=c_1,g=c_2,h=c_3\}$ is still a curve, and the normal vectors (assuming the standard metric to go from $df$ to $\mathrm{\nabla }f$) are dependent.

Particular case: Linear dependence.

A set of functions $f_1(x),f_2(x),...,f_n(x)$ is said to be linearly dependent on an interval I if there exist constants $c_1,c_2,...,c_n$, not all zero, such that

Otherwise, we say that the set $f_1(x),f_2(x),...,f_n(x)$ is linearly independent.

Related: Wronskian.