Implicit function Theorem Do you know some practical application of the Implicit & Inversion Theorem for functions?
I wonder if you've encountered a particular practical problem that is not too hard to understand (from physics( mechanichs), astronomy, chemistry etc) in which you made use of this theorem. 
Thanks!
 A: The implicit function theorem (IFT) encodes our intuitive feelings about "degrees of freedom" in a precise and definitive way, and as such becomes an ingrained part of our everyday mathematical (physical, economical, etc.) thinking. Its essential content is the following: When $n$ real variables $x_1$, $\ldots$, $x_n$ are bound among each other by $r$ so-called constitutional equations
$$F_i(x_1,x_2,\ldots, x_n)=0\qquad(1\leq i\leq r)\tag{1}$$
then  these equations define a certain set $S\subset{\mathbb R}^n$. Depending on the envisaged application this set may be interesting from a geometrical viewpoint, or it may be the set of admissible states of some system, etc.
According to the IFT this set $S$ is neither a set of finitely many points, nor a "sponge", nor a "full body", but is (under certain technical assumptions) a $d$-dimensional manifold, where $d=n-r$. This means that in order to "produce" this set $S$, instead of defining it implicitly by means of $(1)$, we need $d$ parameter variables $u_1$, $\ldots$, $u_d$. If need be we also can represent $S$ as a "graph" in the form
$$x_k=\phi_k(x_1,x_2,\ldots, x_d)\qquad(d+1\leq k\leq n)\ .$$
It is all important that for theoretical discussions about such things the IFT tells us the properties of the $\phi_k$ once and for all. We don't need explicit formulas for the $\phi_k$ appearing here – in fact, if in a concrete situation we have such formulas, we don't need the theorem to begin with.
