El teorema de la función implícita

Abstract

In this work we study several proofs of the classical implicit function theorem, which gives sufficient conditions for a vector equation to define a function. The first proof is the most elementary. It is obtained by induction on the number of dependent variables. The second proof is also based on calculus, by looking directly at the linear approximation of the mapping. Finally, Banach fixed point theorem is used in the more abstract functional analytic third proof. In the second part we show an application of the implicit function theorem. We give four different definitions of a smooth surface and prove all of them are equivalent using the range theorem, an important consequence of the implicit function theorem. We finish the work showing an alternative proof of the implicit function theorem using a classical existence theorem of solutions in ordinary differential equations.