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.