What are some applications of the  [Implicit Function Theorem](https://en.wikipedia.org/wiki/Implicit_function_theorem)  in calculus? The only applications I can think of are:


1. the result that the solution space of a non-degenerate system of equations naturally has the structure of a smooth manifold;
2. the Inverse Function Theorem.


I am looking for applications that would be interesting to an advanced US math undergraduate 
or 1-st year graduate student who is seeing the Implicit Function Theorem for the first or second time. The context is I will be explaining this result as part of 
a review of manifold theory.



What is the Implicit Function Theorem good for?
The infinite-dimensional implicit function theorem is used, among other things, to demonstrate the existence of solutions of nonlinear partial differential equations and parameterize the space of solutions. For equations of standard type (elliptic, parabolic, hyperbolic), the standard version on Banach spaces usually suffices, but you have to be clever about which Banach space to use. There is a generalization of the implicit function theorem, due to Nash who used it to demonstrate the existence of isometric embeddings of Riemannian manifolds in Euclidean space, that works for even more general types of PDE's. Moser stated and proved a simpler version of the theorem. There is a beautiful survey article by Richard Hamilton (who originally used the Nash-Moser implicit function theorem to prove the local-in-time existence of solutions to the Ricci flow) on the Nash-Moser implicit function theorem.


