A function $\Phi$ is functionally dependent on $h_1, \ldots,h_{r}$ if $\Phi=G\left(h_{1}, \cdots, h_{r}\right)$ for certain function $G$.
And $h_1, \ldots,h_{r}$ are called functionally independent if the differentials are linearly independent in a open set $U$, that is, the Jacobian of $H(x)=(h_1(x), \ldots,h_{r}(x))$ 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.
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Author of the notes: Antonio J. Pan-Collantes
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