O Teorema da função implícita e suas aplicações

Abstract

In this paper we present the Implicit Function Theorem and some of its applications in many areas of mathematics. In chapter one we present the theorem in the most classical way. In the second chapter, we describe what a billard map is and show that given a (...) curve, our billard map is a (...) local diffeomorphism. We also calculade the derivative of the billard map. In chapter three we prove that the roots of a polynomial are dependent on the coefficients of the polynomial. So if we make a small perturbation in the coefficients of this polynomial we will also disturb the roots in a smooth way. In the fourth chapter, we present and prove the flow box theorem, using the theorem of the inverse function. In the fifth chapter, we show that the Poincaré map, near a periodic orbit of a (...) flow, is a (...)diffeomorphism. Concluding, in chapter six, we prove that given a differentiable function (...) where (...) is a real number is locally the trace of a regular parametrized surface. For this, we study some geometric properties