O teorema da aplicação inversa e aplicações

Abstract

The Inverse Application Theorem (IAT) is an important analysis result that establishes, even if locally, the existence of a continuously differentiable inverse map. Given its importance, this work aims to show in detail its demonstration. Before, we will expose all the necessary prerequisites and, soon after, we will analyze some applications such as the Implicit Function Theorem and the Lagrange multipliers method. In order to give a broader idea and show that the TAI is valid in other environments, we will give a basic but sufficient view of Euclidean surfaces and then we will enunciate and give the due proof of the TAI between Euclidean surfaces of the same dimension. From there we will further extend our environment to differentiable manifolds.