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!
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.